KILLEDRuntime Complexity (full) proof of /tmp/tmpNcXnKG/LISTUTILITIES_complete_C.xml
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).0 CpxTRS↳1 RenamingProof (⇔, 0 ms)↳2 CpxRelTRS↳3 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)↳4 typed CpxTrs↳5 OrderProof (LOWER BOUND(ID), 0 ms)↳6 typed CpxTrs↳7 RewriteLemmaProof (LOWER BOUND(ID), 633 ms)↳8 BEST↳9 typed CpxTrs↳10 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳11 typed CpxTrs↳12 RewriteLemmaProof (LOWER BOUND(ID), 112 ms)↳13 BEST↳14 typed CpxTrs↳15 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳16 typed CpxTrs↳17 RewriteLemmaProof (LOWER BOUND(ID), 47 ms)↳18 BEST↳19 typed CpxTrs↳20 RewriteLemmaProof (LOWER BOUND(ID), 200 ms)↳21 BEST↳22 typed CpxTrs↳23 RewriteLemmaProof (LOWER BOUND(ID), 101 ms)↳24 BEST↳25 typed CpxTrs↳26 RewriteLemmaProof (LOWER BOUND(ID), 50 ms)↳27 BEST↳28 typed CpxTrs↳29 RewriteLemmaProof (LOWER BOUND(ID), 213 ms)↳30 BEST↳31 typed CpxTrs↳32 RewriteLemmaProof (LOWER BOUND(ID), 45 ms)↳33 BEST↳34 typed CpxTrs↳35 RewriteLemmaProof (LOWER BOUND(ID), 215 ms)↳36 BEST↳37 typed CpxTrs↳38 RewriteLemmaProof (LOWER BOUND(ID), 97 ms)↳39 BEST↳40 typed CpxTrs↳41 RewriteLemmaProof (LOWER BOUND(ID), 89 ms)↳42 BEST↳43 typed CpxTrs↳44 RewriteLemmaProof (LOWER BOUND(ID), 7 ms)↳45 BEST↳46 typed CpxTrs↳47 RewriteLemmaProof (LOWER BOUND(ID), 199 ms)↳48 BEST↳49 typed CpxTrs↳50 RewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳51 BEST↳52 typed CpxTrs↳53 RewriteLemmaProof (LOWER BOUND(ID), 10 ms)↳54 BEST↳55 typed CpxTrs↳56 RewriteLemmaProof (LOWER BOUND(ID), 10 ms)↳57 BEST↳58 typed CpxTrs↳59 RewriteLemmaProof (LOWER BOUND(ID), 122 ms)↳60 BEST↳61 typed CpxTrs↳62 RewriteLemmaProof (LOWER BOUND(ID), 110 ms)↳63 BEST↳64 typed CpxTrs↳65 RewriteLemmaProof (LOWER BOUND(ID), 138 ms)↳66 BEST↳67 typed CpxTrs↳68 RewriteLemmaProof (LOWER BOUND(ID), 77 ms)↳69 BEST↳70 typed CpxTrs↳71 RewriteLemmaProof (LOWER BOUND(ID), 73 ms)↳72 BEST↳73 typed CpxTrs↳74 RewriteLemmaProof (LOWER BOUND(ID), 18 ms)↳75 BEST↳76 typed CpxTrs↳77 RewriteLemmaProof (LOWER BOUND(ID), 125 ms)↳78 BEST↳79 typed CpxTrs↳80 RewriteLemmaProof (LOWER BOUND(ID), 69 ms)↳81 BEST↳82 typed CpxTrs↳83 RewriteLemmaProof (LOWER BOUND(ID), 70 ms)↳84 BEST↳85 typed CpxTrs↳86 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳87 typed CpxTrs↳88 RewriteLemmaProof (LOWER BOUND(ID), 70 ms)↳89 BEST↳90 typed CpxTrs↳91 RewriteLemmaProof (LOWER BOUND(ID), 80 ms)↳92 BEST↳93 typed CpxTrs↳94 RewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳95 BEST↳96 typed CpxTrs↳97 RewriteLemmaProof (LOWER BOUND(ID), 203 ms)↳98 BEST↳99 typed CpxTrs↳100 RewriteLemmaProof (LOWER BOUND(ID), 73 ms)↳101 BEST↳102 typed CpxTrs↳103 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳104 typed CpxTrs↳105 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳106 typed CpxTrs↳107 RewriteLemmaProof (LOWER BOUND(ID), 110 ms)↳108 BEST↳109 typed CpxTrs↳110 RewriteLemmaProof (LOWER BOUND(ID), 132 ms)↳111 BEST↳112 typed CpxTrs↳113 RewriteLemmaProof (LOWER BOUND(ID), 168 ms)↳114 BEST↳115 typed CpxTrs↳116 RewriteLemmaProof (LOWER BOUND(ID), 139 ms)↳117 BEST↳118 typed CpxTrs↳119 RewriteLemmaProof (LOWER BOUND(ID), 118 ms)↳120 BEST↳121 typed CpxTrs↳122 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳123 typed CpxTrs↳124 RewriteLemmaProof (LOWER BOUND(ID), 158 ms)↳125 BEST↳126 typed CpxTrs↳127 RewriteLemmaProof (LOWER BOUND(ID), 172 ms)↳128 BEST↳129 typed CpxTrs↳130 RewriteLemmaProof (LOWER BOUND(ID), 95 ms)↳131 BEST↳132 typed CpxTrs↳133 RewriteLemmaProof (LOWER BOUND(ID), 232 ms)↳134 BEST↳135 typed CpxTrs↳136 RewriteLemmaProof (LOWER BOUND(ID), 82 ms)↳137 BEST↳138 typed CpxTrs↳139 RewriteLemmaProof (LOWER BOUND(ID), 104 ms)↳140 BEST↳141 typed CpxTrs↳142 RewriteLemmaProof (LOWER BOUND(ID), 204 ms)↳143 BEST↳144 typed CpxTrs↳145 RewriteLemmaProof (LOWER BOUND(ID), 149 ms)↳146 BEST↳147 typed CpxTrs↳148 RewriteLemmaProof (LOWER BOUND(ID), 205 ms)↳149 BEST↳150 typed CpxTrs↳151 RewriteLemmaProof (LOWER BOUND(ID), 151 ms)↳152 BEST↳153 typed CpxTrs↳154 RewriteLemmaProof (LOWER BOUND(ID), 225 ms)↳155 BEST↳156 typed CpxTrs↳157 RewriteLemmaProof (LOWER BOUND(ID), 116 ms)↳158 BEST↳159 typed CpxTrs↳160 RewriteLemmaProof (LOWER BOUND(ID), 141 ms)↳161 BEST↳162 typed CpxTrs↳163 RewriteLemmaProof (LOWER BOUND(ID), 360 ms)↳164 BEST↳165 typed CpxTrs↳166 RewriteLemmaProof (LOWER BOUND(ID), 98 ms)↳167 BEST↳168 typed CpxTrs↳169 RewriteLemmaProof (LOWER BOUND(ID), 200 ms)↳170 BEST↳171 typed CpxTrs↳172 RewriteLemmaProof (LOWER BOUND(ID), 35 ms)↳173 BEST↳174 typed CpxTrs↳175 RewriteLemmaProof (LOWER BOUND(ID), 133 ms)↳176 BEST↳177 typed CpxTrs↳178 RewriteLemmaProof (LOWER BOUND(ID), 141 ms)↳179 BEST↳180 typed CpxTrs↳181 typed CpxTrs↳182 typed CpxTrs↳183 typed CpxTrs↳184 typed CpxTrs↳185 typed CpxTrs↳186 typed CpxTrs↳187 typed CpxTrs↳188 typed CpxTrs↳189 typed CpxTrs↳190 typed CpxTrs↳191 typed CpxTrs↳192 typed CpxTrs↳193 typed CpxTrs↳194 typed CpxTrs↳195 typed CpxTrs↳196 typed CpxTrs↳197 typed CpxTrs↳198 typed CpxTrs↳199 typed CpxTrs↳200 typed CpxTrs↳201 typed CpxTrs↳202 typed CpxTrs↳203 typed CpxTrs↳204 typed CpxTrs↳205 typed CpxTrs↳206 typed CpxTrs↳207 typed CpxTrs↳208 typed CpxTrs↳209 typed CpxTrs↳210 typed CpxTrs↳211 typed CpxTrs↳212 typed CpxTrs↳213 typed CpxTrs↳214 typed CpxTrs↳215 typed CpxTrs↳216 typed CpxTrs↳217 typed CpxTrs↳218 typed CpxTrs↳219 typed CpxTrs↳220 typed CpxTrs↳221 typed CpxTrs↳222 typed CpxTrs↳223 typed CpxTrs↳224 typed CpxTrs↳225 typed CpxTrs↳226 typed CpxTrs↳227 typed CpxTrs↳228 typed CpxTrs↳229 typed CpxTrs↳230 typed CpxTrs↳231 typed CpxTrs↳232 typed CpxTrs↳233 typed CpxTrs↳234 typed CpxTrs(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Rewrite Strategy: FULL(1) RenamingProof (EQUIVALENT transformation)
Renamed function symbols to avoid clashes with predefined symbol.(2) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
S is empty.
Rewrite Strategy: FULL(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.(4) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok(5) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
active, U102, isNatural, U103, isLNat, snd, splitAt, U112, U122, U132, U133, U142, U143, U152, U153, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U102 < active
isNatural < active
U103 < active
isLNat < active
snd < active
splitAt < active
U112 < active
U122 < active
U132 < active
U133 < active
U142 < active
U143 < active
U152 < active
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U102 < proper
isNatural < proper
U103 < proper
isLNat < proper
snd < proper
splitAt < proper
U112 < proper
U122 < proper
U132 < proper
U133 < proper
U142 < proper
U143 < proper
U152 < proper
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(6) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U102, active, isNatural, U103, isLNat, snd, splitAt, U112, U122, U132, U133, U142, U143, U152, U153, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U102 < active
isNatural < active
U103 < active
isLNat < active
snd < active
splitAt < active
U112 < active
U122 < active
U132 < active
U133 < active
U142 < active
U143 < active
U152 < active
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U102 < proper
isNatural < proper
U103 < proper
isLNat < proper
snd < proper
splitAt < proper
U112 < proper
U122 < proper
U132 < proper
U133 < proper
U142 < proper
U143 < proper
U152 < proper
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)Induction Base:
U102(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U102(gen_tt:mark:nil:0':ok3_0(+(1, +(n5_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(8) Complex Obligation (BEST)
(9) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
isNatural, active, U103, isLNat, snd, splitAt, U112, U122, U132, U133, U142, U143, U152, U153, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
isNatural < active
U103 < active
isLNat < active
snd < active
splitAt < active
U112 < active
U122 < active
U132 < active
U133 < active
U142 < active
U143 < active
U152 < active
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
isNatural < proper
U103 < proper
isLNat < proper
snd < proper
splitAt < proper
U112 < proper
U122 < proper
U132 < proper
U133 < proper
U142 < proper
U143 < proper
U152 < proper
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol isNatural.(11) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U103, active, isLNat, snd, splitAt, U112, U122, U132, U133, U142, U143, U152, U153, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U103 < active
isLNat < active
snd < active
splitAt < active
U112 < active
U122 < active
U132 < active
U133 < active
U142 < active
U143 < active
U152 < active
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U103 < proper
isLNat < proper
snd < proper
splitAt < proper
U112 < proper
U122 < proper
U132 < proper
U133 < proper
U142 < proper
U143 < proper
U152 < proper
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)Induction Base:
U103(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
U103(gen_tt:mark:nil:0':ok3_0(+(1, +(n4427_0, 1)))) →RΩ(1)
mark(U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(13) Complex Obligation (BEST)
(14) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
isLNat, active, snd, splitAt, U112, U122, U132, U133, U142, U143, U152, U153, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
isLNat < active
snd < active
splitAt < active
U112 < active
U122 < active
U132 < active
U133 < active
U142 < active
U143 < active
U152 < active
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
isLNat < proper
snd < proper
splitAt < proper
U112 < proper
U122 < proper
U132 < proper
U133 < proper
U142 < proper
U143 < proper
U152 < proper
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol isLNat.(16) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
snd, active, splitAt, U112, U122, U132, U133, U142, U143, U152, U153, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
snd < active
splitAt < active
U112 < active
U122 < active
U132 < active
U133 < active
U142 < active
U143 < active
U152 < active
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
snd < proper
splitAt < proper
U112 < proper
U122 < proper
U132 < proper
U133 < proper
U142 < proper
U143 < proper
U152 < proper
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(17) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)Induction Base:
snd(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
snd(gen_tt:mark:nil:0':ok3_0(+(1, +(n5825_0, 1)))) →RΩ(1)
mark(snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(18) Complex Obligation (BEST)
(19) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
splitAt, active, U112, U122, U132, U133, U142, U143, U152, U153, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
splitAt < active
U112 < active
U122 < active
U132 < active
U133 < active
U142 < active
U143 < active
U152 < active
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
splitAt < proper
U112 < proper
U122 < proper
U132 < proper
U133 < proper
U142 < proper
U143 < proper
U152 < proper
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(20) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)Induction Base:
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, +(n7312_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(21) Complex Obligation (BEST)
(22) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U112, active, U122, U132, U133, U142, U143, U152, U153, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U112 < active
U122 < active
U132 < active
U133 < active
U142 < active
U143 < active
U152 < active
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U112 < proper
U122 < proper
U132 < proper
U133 < proper
U142 < proper
U143 < proper
U152 < proper
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(23) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)Induction Base:
U112(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
U112(gen_tt:mark:nil:0':ok3_0(+(1, +(n12640_0, 1)))) →RΩ(1)
mark(U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(24) Complex Obligation (BEST)
(25) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U122, active, U132, U133, U142, U143, U152, U153, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U122 < active
U132 < active
U133 < active
U142 < active
U143 < active
U152 < active
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U122 < proper
U132 < proper
U133 < proper
U142 < proper
U143 < proper
U152 < proper
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(26) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)Induction Base:
U122(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
U122(gen_tt:mark:nil:0':ok3_0(+(1, +(n14378_0, 1)))) →RΩ(1)
mark(U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(27) Complex Obligation (BEST)
(28) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U132, active, U133, U142, U143, U152, U153, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U132 < active
U133 < active
U142 < active
U143 < active
U152 < active
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U132 < proper
U133 < proper
U142 < proper
U143 < proper
U152 < proper
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(29) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)Induction Base:
U132(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U132(gen_tt:mark:nil:0':ok3_0(+(1, +(n16217_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(30) Complex Obligation (BEST)
(31) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U133, active, U142, U143, U152, U153, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U133 < active
U142 < active
U143 < active
U152 < active
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U133 < proper
U142 < proper
U143 < proper
U152 < proper
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(32) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)Induction Base:
U133(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
U133(gen_tt:mark:nil:0':ok3_0(+(1, +(n22062_0, 1)))) →RΩ(1)
mark(U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(33) Complex Obligation (BEST)
(34) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U142, active, U143, U152, U153, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U142 < active
U143 < active
U152 < active
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U142 < proper
U143 < proper
U152 < proper
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(35) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)Induction Base:
U142(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U142(gen_tt:mark:nil:0':ok3_0(+(1, +(n24152_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(36) Complex Obligation (BEST)
(37) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U143, active, U152, U153, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U143 < active
U152 < active
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U143 < proper
U152 < proper
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(38) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)Induction Base:
U143(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
U143(gen_tt:mark:nil:0':ok3_0(+(1, +(n30507_0, 1)))) →RΩ(1)
mark(U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(39) Complex Obligation (BEST)
(40) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U152, active, U153, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U152 < active
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U152 < proper
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(41) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)Induction Base:
U152(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U152(gen_tt:mark:nil:0':ok3_0(+(1, +(n32848_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(42) Complex Obligation (BEST)
(43) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U153, active, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(44) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)Induction Base:
U153(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
U153(gen_tt:mark:nil:0':ok3_0(+(1, +(n39713_0, 1)))) →RΩ(1)
mark(U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(45) Complex Obligation (BEST)
(46) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
cons, active, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(47) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)Induction Base:
cons(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
cons(gen_tt:mark:nil:0':ok3_0(+(1, +(n42305_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(48) Complex Obligation (BEST)
(49) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
natsFrom, active, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(50) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)Induction Base:
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, +(n49680_0, 1)))) →RΩ(1)
mark(natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(51) Complex Obligation (BEST)
(52) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
s, active, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(53) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)Induction Base:
s(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
s(gen_tt:mark:nil:0':ok3_0(+(1, +(n52523_0, 1)))) →RΩ(1)
mark(s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(54) Complex Obligation (BEST)
(55) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
head, active, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(56) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)Induction Base:
head(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
head(gen_tt:mark:nil:0':ok3_0(+(1, +(n55467_0, 1)))) →RΩ(1)
mark(head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(57) Complex Obligation (BEST)
(58) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
afterNth, active, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(59) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)Induction Base:
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, +(n58512_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(60) Complex Obligation (BEST)
(61) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
pair, active, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(62) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)Induction Base:
pair(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
pair(gen_tt:mark:nil:0':ok3_0(+(1, +(n67008_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(63) Complex Obligation (BEST)
(64) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U202, active, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(65) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)Induction Base:
U202(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U202(gen_tt:mark:nil:0':ok3_0(+(1, +(n75808_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(66) Complex Obligation (BEST)
(67) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
fst, active, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(68) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)Induction Base:
fst(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
fst(gen_tt:mark:nil:0':ok3_0(+(1, +(n84713_0, 1)))) →RΩ(1)
mark(fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(69) Complex Obligation (BEST)
(70) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U42, active, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(71) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)Induction Base:
U42(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U42(gen_tt:mark:nil:0':ok3_0(+(1, +(n88309_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(72) Complex Obligation (BEST)
(73) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U43, active, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(74) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)Induction Base:
U43(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
U43(gen_tt:mark:nil:0':ok3_0(+(1, +(n97724_0, 1)))) →RΩ(1)
mark(U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(75) Complex Obligation (BEST)
(76) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U52, active, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(77) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)Induction Base:
U52(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U52(gen_tt:mark:nil:0':ok3_0(+(1, +(n101571_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(78) Complex Obligation (BEST)
(79) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U53, active, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(80) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)Induction Base:
U53(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
U53(gen_tt:mark:nil:0':ok3_0(+(1, +(n111496_0, 1)))) →RΩ(1)
mark(U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(81) Complex Obligation (BEST)
(82) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U62, active, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(83) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)Induction Base:
U62(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
U62(gen_tt:mark:nil:0':ok3_0(+(1, +(n115594_0, 1)))) →RΩ(1)
mark(U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(84) Complex Obligation (BEST)
(85) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
isPLNat, active, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(86) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol isPLNat.(87) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U72, active, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(88) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)Induction Base:
U72(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
U72(gen_tt:mark:nil:0':ok3_0(+(1, +(n119894_0, 1)))) →RΩ(1)
mark(U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(89) Complex Obligation (BEST)
(90) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U82, active, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(91) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)Induction Base:
U82(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
U82(gen_tt:mark:nil:0':ok3_0(+(1, +(n124194_0, 1)))) →RΩ(1)
mark(U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(92) Complex Obligation (BEST)
(93) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U92, active, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(94) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)Induction Base:
U92(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
U92(gen_tt:mark:nil:0':ok3_0(+(1, +(n128595_0, 1)))) →RΩ(1)
mark(U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(95) Complex Obligation (BEST)
(96) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U11, active, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(97) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)Induction Base:
U11(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))Induction Step:
U11(gen_tt:mark:nil:0':ok3_0(+(1, +(n133097_0, 1))), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) →RΩ(1)
mark(U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(98) Complex Obligation (BEST)
(99) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
and, active, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(100) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)Induction Base:
and(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
and(gen_tt:mark:nil:0':ok3_0(+(1, +(n151116_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(101) Complex Obligation (BEST)
(102) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
isNaturalKind, active, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(103) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol isNaturalKind.(104) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
isLNatKind, active, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(105) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol isLNatKind.(106) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U21, active, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(107) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)Induction Base:
U21(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U21(gen_tt:mark:nil:0':ok3_0(+(1, +(n163021_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(108) Complex Obligation (BEST)
(109) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U31, active, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(110) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)Induction Base:
U31(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U31(gen_tt:mark:nil:0':ok3_0(+(1, +(n174986_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(111) Complex Obligation (BEST)
(112) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U41, active, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(113) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)Induction Base:
U41(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))Induction Step:
U41(gen_tt:mark:nil:0':ok3_0(+(1, +(n187255_0, 1))), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) →RΩ(1)
mark(U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(114) Complex Obligation (BEST)
(115) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U51, active, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(116) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)Induction Base:
U51(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))Induction Step:
U51(gen_tt:mark:nil:0':ok3_0(+(1, +(n207253_0, 1))), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) →RΩ(1)
mark(U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(117) Complex Obligation (BEST)
(118) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U61, active, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(119) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)Induction Base:
U61(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U61(gen_tt:mark:nil:0':ok3_0(+(1, +(n227856_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(120) Complex Obligation (BEST)
(121) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
isPLNatKind, active, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(122) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol isPLNatKind.(123) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U71, active, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(124) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)Induction Base:
U71(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U71(gen_tt:mark:nil:0':ok3_0(+(1, +(n241384_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(125) Complex Obligation (BEST)
(126) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U81, active, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(127) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)Induction Base:
U81(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U81(gen_tt:mark:nil:0':ok3_0(+(1, +(n255065_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(128) Complex Obligation (BEST)
(129) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U91, active, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(130) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)Induction Base:
U91(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U91(gen_tt:mark:nil:0':ok3_0(+(1, +(n269050_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(131) Complex Obligation (BEST)
(132) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U101, active, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(133) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)Induction Base:
U101(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))Induction Step:
U101(gen_tt:mark:nil:0':ok3_0(+(1, +(n283339_0, 1))), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) →RΩ(1)
mark(U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(134) Complex Obligation (BEST)
(135) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U111, active, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(136) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)Induction Base:
U111(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U111(gen_tt:mark:nil:0':ok3_0(+(1, +(n306379_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(137) Complex Obligation (BEST)
(138) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U121, active, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(139) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3213740)Induction Base:
U121(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U121(gen_tt:mark:nil:0':ok3_0(+(1, +(n321374_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(140) Complex Obligation (BEST)
(141) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3213740)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U131, active, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(142) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U131(gen_tt:mark:nil:0':ok3_0(+(1, n336673_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3366730)Induction Base:
U131(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))Induction Step:
U131(gen_tt:mark:nil:0':ok3_0(+(1, +(n336673_0, 1))), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) →RΩ(1)
mark(U131(gen_tt:mark:nil:0':ok3_0(+(1, n336673_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(143) Complex Obligation (BEST)
(144) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3213740)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n336673_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3366730)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U141, active, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(145) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U141(gen_tt:mark:nil:0':ok3_0(+(1, n361234_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3612340)Induction Base:
U141(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))Induction Step:
U141(gen_tt:mark:nil:0':ok3_0(+(1, +(n361234_0, 1))), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) →RΩ(1)
mark(U141(gen_tt:mark:nil:0':ok3_0(+(1, n361234_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(146) Complex Obligation (BEST)
(147) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3213740)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n336673_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3366730)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n361234_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3612340)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U151, active, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(148) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U151(gen_tt:mark:nil:0':ok3_0(+(1, n386400_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3864000)Induction Base:
U151(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))Induction Step:
U151(gen_tt:mark:nil:0':ok3_0(+(1, +(n386400_0, 1))), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) →RΩ(1)
mark(U151(gen_tt:mark:nil:0':ok3_0(+(1, n386400_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(149) Complex Obligation (BEST)
(150) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3213740)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n336673_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3366730)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n361234_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3612340)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n386400_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3864000)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U161, active, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(151) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U161(gen_tt:mark:nil:0':ok3_0(+(1, n412171_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4121710)Induction Base:
U161(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U161(gen_tt:mark:nil:0':ok3_0(+(1, +(n412171_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U161(gen_tt:mark:nil:0':ok3_0(+(1, n412171_0)), gen_tt:mark:nil:0':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(152) Complex Obligation (BEST)
(153) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3213740)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n336673_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3366730)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n361234_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3612340)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n386400_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3864000)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n412171_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4121710)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U171, active, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(154) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U171(gen_tt:mark:nil:0':ok3_0(+(1, n428980_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n4289800)Induction Base:
U171(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))Induction Step:
U171(gen_tt:mark:nil:0':ok3_0(+(1, +(n428980_0, 1))), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) →RΩ(1)
mark(U171(gen_tt:mark:nil:0':ok3_0(+(1, n428980_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(155) Complex Obligation (BEST)
(156) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3213740)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n336673_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3366730)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n361234_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3612340)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n386400_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3864000)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n412171_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4121710)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n428980_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n4289800)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U181, active, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(157) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U181(gen_tt:mark:nil:0':ok3_0(+(1, n455814_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4558140)Induction Base:
U181(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U181(gen_tt:mark:nil:0':ok3_0(+(1, +(n455814_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U181(gen_tt:mark:nil:0':ok3_0(+(1, n455814_0)), gen_tt:mark:nil:0':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(158) Complex Obligation (BEST)
(159) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3213740)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n336673_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3366730)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n361234_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3612340)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n386400_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3864000)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n412171_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4121710)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n428980_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n4289800)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n455814_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4558140)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U191, active, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(160) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U191(gen_tt:mark:nil:0':ok3_0(+(1, n473329_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4733290)Induction Base:
U191(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U191(gen_tt:mark:nil:0':ok3_0(+(1, +(n473329_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U191(gen_tt:mark:nil:0':ok3_0(+(1, n473329_0)), gen_tt:mark:nil:0':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(161) Complex Obligation (BEST)
(162) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3213740)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n336673_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3366730)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n361234_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3612340)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n386400_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3864000)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n412171_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4121710)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n428980_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n4289800)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n455814_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4558140)
U191(gen_tt:mark:nil:0':ok3_0(+(1, n473329_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4733290)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U201, active, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(163) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U201(gen_tt:mark:nil:0':ok3_0(+(1, n491148_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) → *4_0, rt ∈ Ω(n4911480)Induction Base:
U201(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d))Induction Step:
U201(gen_tt:mark:nil:0':ok3_0(+(1, +(n491148_0, 1))), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) →RΩ(1)
mark(U201(gen_tt:mark:nil:0':ok3_0(+(1, n491148_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(164) Complex Obligation (BEST)
(165) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3213740)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n336673_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3366730)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n361234_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3612340)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n386400_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3864000)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n412171_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4121710)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n428980_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n4289800)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n455814_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4558140)
U191(gen_tt:mark:nil:0':ok3_0(+(1, n473329_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4733290)
U201(gen_tt:mark:nil:0':ok3_0(+(1, n491148_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) → *4_0, rt ∈ Ω(n4911480)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U211, active, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(166) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U211(gen_tt:mark:nil:0':ok3_0(+(1, n529943_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5299430)Induction Base:
U211(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U211(gen_tt:mark:nil:0':ok3_0(+(1, +(n529943_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U211(gen_tt:mark:nil:0':ok3_0(+(1, n529943_0)), gen_tt:mark:nil:0':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(167) Complex Obligation (BEST)
(168) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3213740)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n336673_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3366730)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n361234_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3612340)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n386400_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3864000)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n412171_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4121710)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n428980_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n4289800)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n455814_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4558140)
U191(gen_tt:mark:nil:0':ok3_0(+(1, n473329_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4733290)
U201(gen_tt:mark:nil:0':ok3_0(+(1, n491148_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) → *4_0, rt ∈ Ω(n4911480)
U211(gen_tt:mark:nil:0':ok3_0(+(1, n529943_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5299430)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U221, active, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U221 < active
tail < active
take < active
sel < active
active < top
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(169) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U221(gen_tt:mark:nil:0':ok3_0(+(1, n548566_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n5485660)Induction Base:
U221(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))Induction Step:
U221(gen_tt:mark:nil:0':ok3_0(+(1, +(n548566_0, 1))), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) →RΩ(1)
mark(U221(gen_tt:mark:nil:0':ok3_0(+(1, n548566_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(170) Complex Obligation (BEST)
(171) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3213740)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n336673_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3366730)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n361234_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3612340)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n386400_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3864000)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n412171_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4121710)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n428980_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n4289800)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n455814_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4558140)
U191(gen_tt:mark:nil:0':ok3_0(+(1, n473329_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4733290)
U201(gen_tt:mark:nil:0':ok3_0(+(1, n491148_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) → *4_0, rt ∈ Ω(n4911480)
U211(gen_tt:mark:nil:0':ok3_0(+(1, n529943_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5299430)
U221(gen_tt:mark:nil:0':ok3_0(+(1, n548566_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n5485660)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
tail, active, take, sel, proper, topThey will be analysed ascendingly in the following order:
tail < active
take < active
sel < active
active < top
tail < proper
take < proper
sel < proper
proper < top(172) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
tail(gen_tt:mark:nil:0':ok3_0(+(1, n578131_0))) → *4_0, rt ∈ Ω(n5781310)Induction Base:
tail(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
tail(gen_tt:mark:nil:0':ok3_0(+(1, +(n578131_0, 1)))) →RΩ(1)
mark(tail(gen_tt:mark:nil:0':ok3_0(+(1, n578131_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(173) Complex Obligation (BEST)
(174) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3213740)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n336673_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3366730)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n361234_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3612340)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n386400_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3864000)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n412171_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4121710)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n428980_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n4289800)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n455814_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4558140)
U191(gen_tt:mark:nil:0':ok3_0(+(1, n473329_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4733290)
U201(gen_tt:mark:nil:0':ok3_0(+(1, n491148_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) → *4_0, rt ∈ Ω(n4911480)
U211(gen_tt:mark:nil:0':ok3_0(+(1, n529943_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5299430)
U221(gen_tt:mark:nil:0':ok3_0(+(1, n548566_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n5485660)
tail(gen_tt:mark:nil:0':ok3_0(+(1, n578131_0))) → *4_0, rt ∈ Ω(n5781310)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
take, active, sel, proper, topThey will be analysed ascendingly in the following order:
take < active
sel < active
active < top
take < proper
sel < proper
proper < top(175) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
take(gen_tt:mark:nil:0':ok3_0(+(1, n586723_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5867230)Induction Base:
take(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
take(gen_tt:mark:nil:0':ok3_0(+(1, +(n586723_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(take(gen_tt:mark:nil:0':ok3_0(+(1, n586723_0)), gen_tt:mark:nil:0':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(176) Complex Obligation (BEST)
(177) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3213740)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n336673_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3366730)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n361234_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3612340)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n386400_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3864000)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n412171_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4121710)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n428980_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n4289800)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n455814_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4558140)
U191(gen_tt:mark:nil:0':ok3_0(+(1, n473329_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4733290)
U201(gen_tt:mark:nil:0':ok3_0(+(1, n491148_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) → *4_0, rt ∈ Ω(n4911480)
U211(gen_tt:mark:nil:0':ok3_0(+(1, n529943_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5299430)
U221(gen_tt:mark:nil:0':ok3_0(+(1, n548566_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n5485660)
tail(gen_tt:mark:nil:0':ok3_0(+(1, n578131_0))) → *4_0, rt ∈ Ω(n5781310)
take(gen_tt:mark:nil:0':ok3_0(+(1, n586723_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5867230)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
sel, active, proper, topThey will be analysed ascendingly in the following order:
sel < active
active < top
sel < proper
proper < top(178) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
sel(gen_tt:mark:nil:0':ok3_0(+(1, n606457_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n6064570)Induction Base:
sel(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
sel(gen_tt:mark:nil:0':ok3_0(+(1, +(n606457_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(sel(gen_tt:mark:nil:0':ok3_0(+(1, n606457_0)), gen_tt:mark:nil:0':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(179) Complex Obligation (BEST)
(180) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3213740)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n336673_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3366730)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n361234_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3612340)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n386400_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3864000)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n412171_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4121710)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n428980_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n4289800)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n455814_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4558140)
U191(gen_tt:mark:nil:0':ok3_0(+(1, n473329_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4733290)
U201(gen_tt:mark:nil:0':ok3_0(+(1, n491148_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) → *4_0, rt ∈ Ω(n4911480)
U211(gen_tt:mark:nil:0':ok3_0(+(1, n529943_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5299430)
U221(gen_tt:mark:nil:0':ok3_0(+(1, n548566_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n5485660)
tail(gen_tt:mark:nil:0':ok3_0(+(1, n578131_0))) → *4_0, rt ∈ Ω(n5781310)
take(gen_tt:mark:nil:0':ok3_0(+(1, n586723_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5867230)
sel(gen_tt:mark:nil:0':ok3_0(+(1, n606457_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n6064570)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
active, proper, topThey will be analysed ascendingly in the following order:
active < top
proper < top(181) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3213740)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n336673_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3366730)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n361234_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3612340)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n386400_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3864000)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n412171_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4121710)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n428980_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n4289800)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n455814_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4558140)
U191(gen_tt:mark:nil:0':ok3_0(+(1, n473329_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4733290)
U201(gen_tt:mark:nil:0':ok3_0(+(1, n491148_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) → *4_0, rt ∈ Ω(n4911480)
U211(gen_tt:mark:nil:0':ok3_0(+(1, n529943_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5299430)
U221(gen_tt:mark:nil:0':ok3_0(+(1, n548566_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n5485660)
tail(gen_tt:mark:nil:0':ok3_0(+(1, n578131_0))) → *4_0, rt ∈ Ω(n5781310)
take(gen_tt:mark:nil:0':ok3_0(+(1, n586723_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5867230)
sel(gen_tt:mark:nil:0':ok3_0(+(1, n606457_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n6064570)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(182) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3213740)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n336673_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3366730)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n361234_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3612340)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n386400_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3864000)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n412171_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4121710)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n428980_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n4289800)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n455814_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4558140)
U191(gen_tt:mark:nil:0':ok3_0(+(1, n473329_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4733290)
U201(gen_tt:mark:nil:0':ok3_0(+(1, n491148_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) → *4_0, rt ∈ Ω(n4911480)
U211(gen_tt:mark:nil:0':ok3_0(+(1, n529943_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5299430)
U221(gen_tt:mark:nil:0':ok3_0(+(1, n548566_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n5485660)
tail(gen_tt:mark:nil:0':ok3_0(+(1, n578131_0))) → *4_0, rt ∈ Ω(n5781310)
take(gen_tt:mark:nil:0':ok3_0(+(1, n586723_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5867230)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(183) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3213740)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n336673_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3366730)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n361234_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3612340)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n386400_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3864000)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n412171_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4121710)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n428980_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n4289800)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n455814_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4558140)
U191(gen_tt:mark:nil:0':ok3_0(+(1, n473329_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4733290)
U201(gen_tt:mark:nil:0':ok3_0(+(1, n491148_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) → *4_0, rt ∈ Ω(n4911480)
U211(gen_tt:mark:nil:0':ok3_0(+(1, n529943_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5299430)
U221(gen_tt:mark:nil:0':ok3_0(+(1, n548566_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n5485660)
tail(gen_tt:mark:nil:0':ok3_0(+(1, n578131_0))) → *4_0, rt ∈ Ω(n5781310)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(184) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3213740)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n336673_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3366730)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n361234_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3612340)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n386400_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3864000)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n412171_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4121710)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n428980_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n4289800)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n455814_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4558140)
U191(gen_tt:mark:nil:0':ok3_0(+(1, n473329_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4733290)
U201(gen_tt:mark:nil:0':ok3_0(+(1, n491148_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) → *4_0, rt ∈ Ω(n4911480)
U211(gen_tt:mark:nil:0':ok3_0(+(1, n529943_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5299430)
U221(gen_tt:mark:nil:0':ok3_0(+(1, n548566_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n5485660)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(185) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3213740)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n336673_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3366730)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n361234_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3612340)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n386400_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3864000)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n412171_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4121710)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n428980_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n4289800)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n455814_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4558140)
U191(gen_tt:mark:nil:0':ok3_0(+(1, n473329_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4733290)
U201(gen_tt:mark:nil:0':ok3_0(+(1, n491148_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) → *4_0, rt ∈ Ω(n4911480)
U211(gen_tt:mark:nil:0':ok3_0(+(1, n529943_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5299430)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(186) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3213740)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n336673_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3366730)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n361234_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3612340)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n386400_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3864000)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n412171_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4121710)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n428980_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n4289800)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n455814_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4558140)
U191(gen_tt:mark:nil:0':ok3_0(+(1, n473329_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4733290)
U201(gen_tt:mark:nil:0':ok3_0(+(1, n491148_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c), gen_tt:mark:nil:0':ok3_0(d)) → *4_0, rt ∈ Ω(n4911480)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(187) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3213740)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n336673_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3366730)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n361234_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3612340)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n386400_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3864000)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n412171_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4121710)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n428980_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n4289800)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n455814_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4558140)
U191(gen_tt:mark:nil:0':ok3_0(+(1, n473329_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4733290)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(188) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3213740)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n336673_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3366730)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n361234_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3612340)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n386400_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3864000)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n412171_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4121710)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n428980_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n4289800)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n455814_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4558140)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(189) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3213740)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n336673_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3366730)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n361234_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3612340)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n386400_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3864000)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n412171_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4121710)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n428980_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n4289800)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(190) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3213740)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n336673_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3366730)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n361234_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3612340)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n386400_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3864000)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n412171_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4121710)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(191) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3213740)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n336673_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3366730)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n361234_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3612340)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n386400_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3864000)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(192) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3213740)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n336673_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3366730)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n361234_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3612340)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(193) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3213740)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n336673_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3366730)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(194) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n321374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3213740)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(195) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n306379_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3063790)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(196) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n283339_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2833390)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(197) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n269050_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2690500)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(198) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n255065_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2550650)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(199) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n241384_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2413840)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(200) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n227856_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2278560)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(201) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n207253_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2072530)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(202) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n187255_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1872550)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(203) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n174986_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1749860)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(204) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n163021_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1630210)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(205) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)
and(gen_tt:mark:nil:0':ok3_0(+(1, n151116_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1511160)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(206) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n133097_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1330970)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(207) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)
U92(gen_tt:mark:nil:0':ok3_0(+(1, n128595_0))) → *4_0, rt ∈ Ω(n1285950)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(208) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)
U82(gen_tt:mark:nil:0':ok3_0(+(1, n124194_0))) → *4_0, rt ∈ Ω(n1241940)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(209) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)
U72(gen_tt:mark:nil:0':ok3_0(+(1, n119894_0))) → *4_0, rt ∈ Ω(n1198940)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(210) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)
U62(gen_tt:mark:nil:0':ok3_0(+(1, n115594_0))) → *4_0, rt ∈ Ω(n1155940)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(211) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)
U53(gen_tt:mark:nil:0':ok3_0(+(1, n111496_0))) → *4_0, rt ∈ Ω(n1114960)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(212) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n101571_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1015710)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(213) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)
U43(gen_tt:mark:nil:0':ok3_0(+(1, n97724_0))) → *4_0, rt ∈ Ω(n977240)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(214) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n88309_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n883090)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(215) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n84713_0))) → *4_0, rt ∈ Ω(n847130)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(216) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n75808_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n758080)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(217) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n67008_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n670080)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(218) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n58512_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n585120)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(219) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)
head(gen_tt:mark:nil:0':ok3_0(+(1, n55467_0))) → *4_0, rt ∈ Ω(n554670)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(220) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)
s(gen_tt:mark:nil:0':ok3_0(+(1, n52523_0))) → *4_0, rt ∈ Ω(n525230)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(221) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n49680_0))) → *4_0, rt ∈ Ω(n496800)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(222) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n42305_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n423050)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(223) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)
U153(gen_tt:mark:nil:0':ok3_0(+(1, n39713_0))) → *4_0, rt ∈ Ω(n397130)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(224) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n32848_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n328480)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(225) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)
U143(gen_tt:mark:nil:0':ok3_0(+(1, n30507_0))) → *4_0, rt ∈ Ω(n305070)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(226) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n24152_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n241520)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(227) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)
U133(gen_tt:mark:nil:0':ok3_0(+(1, n22062_0))) → *4_0, rt ∈ Ω(n220620)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(228) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n16217_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n162170)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(229) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)
U122(gen_tt:mark:nil:0':ok3_0(+(1, n14378_0))) → *4_0, rt ∈ Ω(n143780)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(230) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)
U112(gen_tt:mark:nil:0':ok3_0(+(1, n12640_0))) → *4_0, rt ∈ Ω(n126400)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(231) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n7312_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n73120)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(232) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n5825_0))) → *4_0, rt ∈ Ω(n58250)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(233) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
U103(gen_tt:mark:nil:0':ok3_0(+(1, n4427_0))) → *4_0, rt ∈ Ω(n44270)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.
(234) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okLemmas:
U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)Generator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))No more defined symbols left to analyse.