KILLEDRuntime Complexity (full) proof of /tmp/tmpoQjpRr/LISTUTILITIES_nokinds_noand_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), 519 ms)↳8 BEST↳9 typed CpxTrs↳10 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳11 typed CpxTrs↳12 RewriteLemmaProof (LOWER BOUND(ID), 486 ms)↳13 BEST↳14 typed CpxTrs↳15 RewriteLemmaProof (LOWER BOUND(ID), 66 ms)↳16 BEST↳17 typed CpxTrs↳18 RewriteLemmaProof (LOWER BOUND(ID), 222 ms)↳19 BEST↳20 typed CpxTrs↳21 RewriteLemmaProof (LOWER BOUND(ID), 83 ms)↳22 BEST↳23 typed CpxTrs↳24 RewriteLemmaProof (LOWER BOUND(ID), 54 ms)↳25 BEST↳26 typed CpxTrs↳27 RewriteLemmaProof (LOWER BOUND(ID), 124 ms)↳28 BEST↳29 typed CpxTrs↳30 RewriteLemmaProof (LOWER BOUND(ID), 120 ms)↳31 BEST↳32 typed CpxTrs↳33 RewriteLemmaProof (LOWER BOUND(ID), 12 ms)↳34 BEST↳35 typed CpxTrs↳36 RewriteLemmaProof (LOWER BOUND(ID), 1 ms)↳37 BEST↳38 typed CpxTrs↳39 RewriteLemmaProof (LOWER BOUND(ID), 199 ms)↳40 BEST↳41 typed CpxTrs↳42 RewriteLemmaProof (LOWER BOUND(ID), 35 ms)↳43 BEST↳44 typed CpxTrs↳45 RewriteLemmaProof (LOWER BOUND(ID), 233 ms)↳46 BEST↳47 typed CpxTrs↳48 RewriteLemmaProof (LOWER BOUND(ID), 162 ms)↳49 BEST↳50 typed CpxTrs↳51 RewriteLemmaProof (LOWER BOUND(ID), 109 ms)↳52 BEST↳53 typed CpxTrs↳54 RewriteLemmaProof (LOWER BOUND(ID), 233 ms)↳55 BEST↳56 typed CpxTrs↳57 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳58 typed CpxTrs↳59 RewriteLemmaProof (LOWER BOUND(ID), 245 ms)↳60 BEST↳61 typed CpxTrs↳62 RewriteLemmaProof (LOWER BOUND(ID), 78 ms)↳63 BEST↳64 typed CpxTrs↳65 RewriteLemmaProof (LOWER BOUND(ID), 132 ms)↳66 BEST↳67 typed CpxTrs↳68 RewriteLemmaProof (LOWER BOUND(ID), 93 ms)↳69 BEST↳70 typed CpxTrs↳71 RewriteLemmaProof (LOWER BOUND(ID), 201 ms)↳72 BEST↳73 typed CpxTrs↳74 RewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳75 BEST↳76 typed CpxTrs↳77 RewriteLemmaProof (LOWER BOUND(ID), 108 ms)↳78 BEST↳79 typed CpxTrs↳80 RewriteLemmaProof (LOWER BOUND(ID), 72 ms)↳81 BEST↳82 typed CpxTrs↳83 RewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳84 BEST↳85 typed CpxTrs↳86 RewriteLemmaProof (LOWER BOUND(ID), 112 ms)↳87 BEST↳88 typed CpxTrs↳89 RewriteLemmaProof (LOWER BOUND(ID), 202 ms)↳90 BEST↳91 typed CpxTrs↳92 RewriteLemmaProof (LOWER BOUND(ID), 185 ms)↳93 BEST↳94 typed CpxTrs↳95 RewriteLemmaProof (LOWER BOUND(ID), 69 ms)↳96 BEST↳97 typed CpxTrs↳98 RewriteLemmaProof (LOWER BOUND(ID), 118 ms)↳99 BEST↳100 typed CpxTrs↳101 RewriteLemmaProof (LOWER BOUND(ID), 67 ms)↳102 BEST↳103 typed CpxTrs↳104 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳105 typed CpxTrs↳106 RewriteLemmaProof (LOWER BOUND(ID), 80 ms)↳107 BEST↳108 typed CpxTrs↳109 RewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳110 BEST↳111 typed CpxTrs↳112 RewriteLemmaProof (LOWER BOUND(ID), 6 ms)↳113 BEST↳114 typed CpxTrs↳115 RewriteLemmaProof (LOWER BOUND(ID), 87 ms)↳116 BEST↳117 typed CpxTrs↳118 RewriteLemmaProof (LOWER BOUND(ID), 2 ms)↳119 BEST↳120 typed CpxTrs↳121 RewriteLemmaProof (LOWER BOUND(ID), 14 ms)↳122 BEST↳123 typed CpxTrs↳124 RewriteLemmaProof (LOWER BOUND(ID), 114 ms)↳125 BEST↳126 typed CpxTrs↳127 RewriteLemmaProof (LOWER BOUND(ID), 138 ms)↳128 BEST↳129 typed CpxTrs↳130 RewriteLemmaProof (LOWER BOUND(ID), 76 ms)↳131 BEST↳132 typed CpxTrs↳133 RewriteLemmaProof (LOWER BOUND(ID), 112 ms)↳134 BEST↳135 typed CpxTrs↳136 RewriteLemmaProof (LOWER BOUND(ID), 220 ms)↳137 BEST↳138 typed CpxTrs↳139 RewriteLemmaProof (LOWER BOUND(ID), 162 ms)↳140 BEST↳141 typed CpxTrs↳142 RewriteLemmaProof (LOWER BOUND(ID), 91 ms)↳143 BEST↳144 typed CpxTrs↳145 RewriteLemmaProof (LOWER BOUND(ID), 308 ms)↳146 BEST↳147 typed CpxTrs↳148 RewriteLemmaProof (LOWER BOUND(ID), 150 ms)↳149 BEST↳150 typed CpxTrs↳151 RewriteLemmaProof (LOWER BOUND(ID), 199 ms)↳152 BEST↳153 typed CpxTrs↳154 RewriteLemmaProof (LOWER BOUND(ID), 32 ms)↳155 BEST↳156 typed CpxTrs↳157 RewriteLemmaProof (LOWER BOUND(ID), 133 ms)↳158 BEST↳159 typed CpxTrs↳160 RewriteLemmaProof (LOWER BOUND(ID), 144 ms)↳161 BEST↳162 typed CpxTrs↳163 typed CpxTrs↳164 typed CpxTrs↳165 typed CpxTrs↳166 typed CpxTrs↳167 typed CpxTrs↳168 typed CpxTrs↳169 typed CpxTrs↳170 typed CpxTrs↳171 typed CpxTrs↳172 typed CpxTrs↳173 typed CpxTrs↳174 typed CpxTrs↳175 typed CpxTrs↳176 typed CpxTrs↳177 typed CpxTrs↳178 typed CpxTrs↳179 typed CpxTrs↳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(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0, XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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, isLNat, U12, snd, splitAt, U132, U142, U152, cons, natsFrom, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, 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
isLNat < active
U12 < active
snd < active
splitAt < active
U132 < active
U142 < active
U152 < active
cons < active
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < 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
isLNat < proper
U12 < proper
snd < proper
splitAt < proper
U132 < proper
U142 < proper
U152 < proper
cons < proper
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < 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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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, isLNat, U12, snd, splitAt, U132, U142, U152, cons, natsFrom, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, 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
isLNat < active
U12 < active
snd < active
splitAt < active
U132 < active
U142 < active
U152 < active
cons < active
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < 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
isLNat < proper
U12 < proper
snd < proper
splitAt < proper
U132 < proper
U142 < proper
U152 < proper
cons < proper
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < 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))) → *4_0, rt ∈ Ω(n50)Induction Base:
U102(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
U102(gen_tt:mark:nil:0':ok3_0(+(1, +(n5_0, 1)))) →RΩ(1)
mark(U102(gen_tt:mark:nil:0':ok3_0(+(1, n5_0)))) →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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *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:
isLNat, active, U12, snd, splitAt, U132, U142, U152, cons, natsFrom, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, 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
U12 < active
snd < active
splitAt < active
U132 < active
U142 < active
U152 < active
cons < active
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < 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
U12 < proper
snd < proper
splitAt < proper
U132 < proper
U142 < proper
U152 < proper
cons < proper
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < 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 isLNat.(11) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *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:
U12, active, snd, splitAt, U132, U142, U152, cons, natsFrom, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, 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:
U12 < active
snd < active
splitAt < active
U132 < active
U142 < active
U152 < active
cons < active
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < 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
U12 < proper
snd < proper
splitAt < proper
U132 < proper
U142 < proper
U152 < proper
cons < proper
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < 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:
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)Induction Base:
U12(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:
U12(gen_tt:mark:nil:0':ok3_0(+(1, +(n1128_0, 1))), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) →RΩ(1)
mark(U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_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).
(13) Complex Obligation (BEST)
(14) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)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, U132, U142, U152, cons, natsFrom, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, 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
U132 < active
U142 < active
U152 < active
cons < active
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < 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
U132 < proper
U142 < proper
U152 < proper
cons < proper
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < 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) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)Induction Base:
snd(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
snd(gen_tt:mark:nil:0':ok3_0(+(1, +(n8293_0, 1)))) →RΩ(1)
mark(snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(16) Complex Obligation (BEST)
(17) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)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, U132, U142, U152, cons, natsFrom, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, 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
U132 < active
U142 < active
U152 < active
cons < active
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < 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
U132 < proper
U142 < proper
U152 < proper
cons < proper
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < 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(18) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)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, +(n9709_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_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).
(19) Complex Obligation (BEST)
(20) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)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, U142, U152, cons, natsFrom, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, 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
U142 < active
U152 < active
cons < active
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < 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
U142 < proper
U152 < proper
cons < proper
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < 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(21) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)Induction Base:
U132(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
U132(gen_tt:mark:nil:0':ok3_0(+(1, +(n14655_0, 1)))) →RΩ(1)
mark(U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(22) Complex Obligation (BEST)
(23) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)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, U152, cons, natsFrom, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, 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
U152 < active
cons < active
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < 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
U152 < proper
cons < proper
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < 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(24) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)Induction Base:
U142(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
U142(gen_tt:mark:nil:0':ok3_0(+(1, +(n16322_0, 1)))) →RΩ(1)
mark(U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(25) Complex Obligation (BEST)
(26) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)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, cons, natsFrom, s, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, 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
cons < active
natsFrom < active
s < active
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < 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
cons < proper
natsFrom < proper
s < proper
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < 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(27) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)Induction Base:
U152(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
U152(gen_tt:mark:nil:0':ok3_0(+(1, +(n18090_0, 1)))) →RΩ(1)
mark(U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(28) Complex Obligation (BEST)
(29) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)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, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, 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
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < 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
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < 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(30) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)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, +(n19959_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_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).
(31) Complex Obligation (BEST)
(32) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)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, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, 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
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < 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
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < 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(33) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)Induction Base:
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, +(n25628_0, 1)))) →RΩ(1)
mark(natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(34) Complex Obligation (BEST)
(35) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)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, U172, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, 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
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < 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
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < 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(36) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)Induction Base:
s(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
s(gen_tt:mark:nil:0':ok3_0(+(1, +(n27748_0, 1)))) →RΩ(1)
mark(s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(37) Complex Obligation (BEST)
(38) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)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:
U172, active, head, afterNth, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, 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:
U172 < active
head < active
afterNth < active
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < 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
U172 < proper
head < proper
afterNth < proper
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < 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(39) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)Induction Base:
U172(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:
U172(gen_tt:mark:nil:0':ok3_0(+(1, +(n29969_0, 1))), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) →RΩ(1)
mark(U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_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).
(40) Complex Obligation (BEST)
(41) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)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, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, 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
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < 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
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < 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(42) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)Induction Base:
head(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
head(gen_tt:mark:nil:0':ok3_0(+(1, +(n40521_0, 1)))) →RΩ(1)
mark(head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(43) Complex Obligation (BEST)
(44) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)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, U182, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, 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
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < 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
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < 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(45) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)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, +(n43042_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_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).
(46) Complex Obligation (BEST)
(47) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)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:
U182, active, pair, U202, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, 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:
U182 < active
pair < active
U202 < active
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < 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
U182 < proper
pair < proper
U202 < proper
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < 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(48) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)Induction Base:
U182(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U182(gen_tt:mark:nil:0':ok3_0(+(1, +(n50234_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_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).
(49) Complex Obligation (BEST)
(50) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)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, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, 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
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < 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
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < 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(51) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)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, +(n57531_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_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).
(52) Complex Obligation (BEST)
(53) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)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, isNatural, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, 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
isNatural < active
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < 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
isNatural < proper
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < 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(54) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)Induction Base:
U202(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:
U202(gen_tt:mark:nil:0':ok3_0(+(1, +(n65331_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(U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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).
(55) Complex Obligation (BEST)
(56) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)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, U203, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, 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
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < 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
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < 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(57) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol isNatural.(58) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)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:
U203, active, U204, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, 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:
U203 < active
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < 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
U203 < proper
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < 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:
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)Induction Base:
U203(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:
U203(gen_tt:mark:nil:0':ok3_0(+(1, +(n83394_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(U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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).
(60) Complex Obligation (BEST)
(61) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)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:
U204, active, U22, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, 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:
U204 < active
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < 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
U204 < proper
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < 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:
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)Induction Base:
U204(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U204(gen_tt:mark:nil:0':ok3_0(+(1, +(n102385_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)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:
U22, active, U212, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, 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:
U22 < active
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < 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
U22 < proper
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < 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:
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)Induction Base:
U22(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U22(gen_tt:mark:nil:0':ok3_0(+(1, +(n111290_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)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:
U212, active, U222, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, 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:
U212 < active
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < 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
U212 < proper
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < 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:
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)Induction Base:
U212(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U212(gen_tt:mark:nil:0':ok3_0(+(1, +(n120499_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_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).
(69) Complex Obligation (BEST)
(70) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)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:
U222, active, fst, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, 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:
U222 < active
fst < active
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < 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
U222 < proper
fst < proper
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < 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:
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)Induction Base:
U222(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:
U222(gen_tt:mark:nil:0':ok3_0(+(1, +(n130012_0, 1))), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) →RΩ(1)
mark(U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_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).
(72) Complex Obligation (BEST)
(73) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)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, U32, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, 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
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < 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
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < 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:
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)Induction Base:
fst(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
fst(gen_tt:mark:nil:0':ok3_0(+(1, +(n145732_0, 1)))) →RΩ(1)
mark(fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)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:
U32, active, U42, U52, U11, U21, U31, U41, U51, U61, isPLNat, 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:
U32 < active
U42 < active
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < 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
U32 < proper
U42 < proper
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < 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:
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)Induction Base:
U32(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U32(gen_tt:mark:nil:0':ok3_0(+(1, +(n149949_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)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, U52, U11, U21, U31, U41, U51, U61, isPLNat, 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
U52 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < 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
U52 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < 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:
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)Induction Base:
U42(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
U42(gen_tt:mark:nil:0':ok3_0(+(1, +(n160374_0, 1)))) →RΩ(1)
mark(U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)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, U11, U21, U31, U41, U51, U61, isPLNat, 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
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < 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
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < 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:
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)Induction Base:
U52(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
U52(gen_tt:mark:nil:0':ok3_0(+(1, +(n164842_0, 1)))) →RΩ(1)
mark(U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)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, U21, U31, U41, U51, U61, isPLNat, 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
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNat < 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
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNat < 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) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)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, +(n169411_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, n169411_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).
(87) Complex Obligation (BEST)
(88) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)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, isPLNat, 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
isPLNat < 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
isPLNat < 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(89) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)Induction Base:
U21(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:
U21(gen_tt:mark:nil:0':ok3_0(+(1, +(n187127_0, 1))), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) →RΩ(1)
mark(U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_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).
(90) Complex Obligation (BEST)
(91) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)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, isPLNat, 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
isPLNat < 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
isPLNat < 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(92) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)Induction Base:
U31(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:
U31(gen_tt:mark:nil:0':ok3_0(+(1, +(n205448_0, 1))), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) →RΩ(1)
mark(U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_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).
(93) Complex Obligation (BEST)
(94) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)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, isPLNat, 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
isPLNat < 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
isPLNat < 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(95) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)Induction Base:
U41(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U41(gen_tt:mark:nil:0':ok3_0(+(1, +(n224374_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_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).
(96) Complex Obligation (BEST)
(97) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)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, isPLNat, 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
isPLNat < 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
isPLNat < 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(98) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)Induction Base:
U51(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U51(gen_tt:mark:nil:0':ok3_0(+(1, +(n236721_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_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).
(99) Complex Obligation (BEST)
(100) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)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, isPLNat, 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
isPLNat < 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
isPLNat < 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(101) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)Induction Base:
U61(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
U61(gen_tt:mark:nil:0':ok3_0(+(1, +(n249372_0, 1)))) →RΩ(1)
mark(U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(102) Complex Obligation (BEST)
(103) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)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, 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
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
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(104) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol isPLNat.(105) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)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(106) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)Induction Base:
U71(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
U71(gen_tt:mark:nil:0':ok3_0(+(1, +(n255094_0, 1)))) →RΩ(1)
mark(U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(107) Complex Obligation (BEST)
(108) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)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(109) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)Induction Base:
U81(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
U81(gen_tt:mark:nil:0':ok3_0(+(1, +(n260762_0, 1)))) →RΩ(1)
mark(U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(110) Complex Obligation (BEST)
(111) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)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(112) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)Induction Base:
U91(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
U91(gen_tt:mark:nil:0':ok3_0(+(1, +(n266531_0, 1)))) →RΩ(1)
mark(U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(113) Complex Obligation (BEST)
(114) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)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(115) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)Induction Base:
U101(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U101(gen_tt:mark:nil:0':ok3_0(+(1, +(n272401_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_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).
(116) Complex Obligation (BEST)
(117) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)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(118) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)Induction Base:
U111(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
U111(gen_tt:mark:nil:0':ok3_0(+(1, +(n286180_0, 1)))) →RΩ(1)
mark(U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(119) Complex Obligation (BEST)
(120) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)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(121) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0))) → *4_0, rt ∈ Ω(n2923010)Induction Base:
U121(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
U121(gen_tt:mark:nil:0':ok3_0(+(1, +(n292301_0, 1)))) →RΩ(1)
mark(U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(122) Complex Obligation (BEST)
(123) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0))) → *4_0, rt ∈ Ω(n2923010)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(124) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U131(gen_tt:mark:nil:0':ok3_0(+(1, n298523_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2985230)Induction Base:
U131(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U131(gen_tt:mark:nil:0':ok3_0(+(1, +(n298523_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U131(gen_tt:mark:nil:0':ok3_0(+(1, n298523_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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0))) → *4_0, rt ∈ Ω(n2923010)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n298523_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2985230)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(127) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U141(gen_tt:mark:nil:0':ok3_0(+(1, n313018_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3130180)Induction Base:
U141(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U141(gen_tt:mark:nil:0':ok3_0(+(1, +(n313018_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U141(gen_tt:mark:nil:0':ok3_0(+(1, n313018_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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0))) → *4_0, rt ∈ Ω(n2923010)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n298523_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2985230)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n313018_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3130180)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(130) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U151(gen_tt:mark:nil:0':ok3_0(+(1, n327817_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3278170)Induction Base:
U151(gen_tt:mark:nil:0':ok3_0(+(1, 0)), gen_tt:mark:nil:0':ok3_0(b))Induction Step:
U151(gen_tt:mark:nil:0':ok3_0(+(1, +(n327817_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U151(gen_tt:mark:nil:0':ok3_0(+(1, n327817_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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0))) → *4_0, rt ∈ Ω(n2923010)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n298523_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2985230)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n313018_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3130180)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n327817_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3278170)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(133) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U161(gen_tt:mark:nil:0':ok3_0(+(1, n342920_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3429200)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, +(n342920_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U161(gen_tt:mark:nil:0':ok3_0(+(1, n342920_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).
(134) Complex Obligation (BEST)
(135) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0))) → *4_0, rt ∈ Ω(n2923010)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n298523_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2985230)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n313018_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3130180)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n327817_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3278170)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n342920_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3429200)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(136) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U171(gen_tt:mark:nil:0':ok3_0(+(1, n358327_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3583270)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, +(n358327_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, n358327_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).
(137) Complex Obligation (BEST)
(138) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0))) → *4_0, rt ∈ Ω(n2923010)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n298523_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2985230)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n313018_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3130180)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n327817_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3278170)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n342920_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3429200)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n358327_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3583270)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(139) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U181(gen_tt:mark:nil:0':ok3_0(+(1, n382930_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3829300)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, +(n382930_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U181(gen_tt:mark:nil:0':ok3_0(+(1, n382930_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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0))) → *4_0, rt ∈ Ω(n2923010)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n298523_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2985230)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n313018_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3130180)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n327817_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3278170)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n342920_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3429200)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n358327_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3583270)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n382930_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3829300)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(142) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U191(gen_tt:mark:nil:0':ok3_0(+(1, n399043_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3990430)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, +(n399043_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U191(gen_tt:mark:nil:0':ok3_0(+(1, n399043_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).
(143) Complex Obligation (BEST)
(144) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0))) → *4_0, rt ∈ Ω(n2923010)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n298523_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2985230)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n313018_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3130180)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n327817_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3278170)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n342920_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3429200)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n358327_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3583270)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n382930_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3829300)
U191(gen_tt:mark:nil:0':ok3_0(+(1, n399043_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3990430)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(145) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U201(gen_tt:mark:nil:0':ok3_0(+(1, n415460_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(n4154600)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, +(n415460_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, n415460_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).
(146) Complex Obligation (BEST)
(147) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0))) → *4_0, rt ∈ Ω(n2923010)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n298523_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2985230)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n313018_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3130180)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n327817_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3278170)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n342920_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3429200)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n358327_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3583270)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n382930_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3829300)
U191(gen_tt:mark:nil:0':ok3_0(+(1, n399043_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3990430)
U201(gen_tt:mark:nil:0':ok3_0(+(1, n415460_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(n4154600)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(148) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U211(gen_tt:mark:nil:0':ok3_0(+(1, n451195_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4511950)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, +(n451195_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(U211(gen_tt:mark:nil:0':ok3_0(+(1, n451195_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).
(149) Complex Obligation (BEST)
(150) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0))) → *4_0, rt ∈ Ω(n2923010)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n298523_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2985230)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n313018_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3130180)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n327817_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3278170)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n342920_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3429200)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n358327_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3583270)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n382930_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3829300)
U191(gen_tt:mark:nil:0':ok3_0(+(1, n399043_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3990430)
U201(gen_tt:mark:nil:0':ok3_0(+(1, n415460_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(n4154600)
U211(gen_tt:mark:nil:0':ok3_0(+(1, n451195_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4511950)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(151) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U221(gen_tt:mark:nil:0':ok3_0(+(1, n468416_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n4684160)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, +(n468416_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, n468416_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).
(152) Complex Obligation (BEST)
(153) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0))) → *4_0, rt ∈ Ω(n2923010)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n298523_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2985230)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n313018_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3130180)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n327817_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3278170)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n342920_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3429200)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n358327_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3583270)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n382930_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3829300)
U191(gen_tt:mark:nil:0':ok3_0(+(1, n399043_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3990430)
U201(gen_tt:mark:nil:0':ok3_0(+(1, n415460_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(n4154600)
U211(gen_tt:mark:nil:0':ok3_0(+(1, n451195_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4511950)
U221(gen_tt:mark:nil:0':ok3_0(+(1, n468416_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n4684160)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(154) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
tail(gen_tt:mark:nil:0':ok3_0(+(1, n495750_0))) → *4_0, rt ∈ Ω(n4957500)Induction Base:
tail(gen_tt:mark:nil:0':ok3_0(+(1, 0)))Induction Step:
tail(gen_tt:mark:nil:0':ok3_0(+(1, +(n495750_0, 1)))) →RΩ(1)
mark(tail(gen_tt:mark:nil:0':ok3_0(+(1, n495750_0)))) →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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0))) → *4_0, rt ∈ Ω(n2923010)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n298523_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2985230)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n313018_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3130180)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n327817_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3278170)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n342920_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3429200)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n358327_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3583270)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n382930_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3829300)
U191(gen_tt:mark:nil:0':ok3_0(+(1, n399043_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3990430)
U201(gen_tt:mark:nil:0':ok3_0(+(1, n415460_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(n4154600)
U211(gen_tt:mark:nil:0':ok3_0(+(1, n451195_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4511950)
U221(gen_tt:mark:nil:0':ok3_0(+(1, n468416_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n4684160)
tail(gen_tt:mark:nil:0':ok3_0(+(1, n495750_0))) → *4_0, rt ∈ Ω(n4957500)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(157) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
take(gen_tt:mark:nil:0':ok3_0(+(1, n503769_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5037690)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, +(n503769_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(take(gen_tt:mark:nil:0':ok3_0(+(1, n503769_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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0))) → *4_0, rt ∈ Ω(n2923010)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n298523_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2985230)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n313018_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3130180)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n327817_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3278170)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n342920_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3429200)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n358327_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3583270)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n382930_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3829300)
U191(gen_tt:mark:nil:0':ok3_0(+(1, n399043_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3990430)
U201(gen_tt:mark:nil:0':ok3_0(+(1, n415460_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(n4154600)
U211(gen_tt:mark:nil:0':ok3_0(+(1, n451195_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4511950)
U221(gen_tt:mark:nil:0':ok3_0(+(1, n468416_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n4684160)
tail(gen_tt:mark:nil:0':ok3_0(+(1, n495750_0))) → *4_0, rt ∈ Ω(n4957500)
take(gen_tt:mark:nil:0':ok3_0(+(1, n503769_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5037690)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(160) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
sel(gen_tt:mark:nil:0':ok3_0(+(1, n522101_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5221010)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, +(n522101_0, 1))), gen_tt:mark:nil:0':ok3_0(b)) →RΩ(1)
mark(sel(gen_tt:mark:nil:0':ok3_0(+(1, n522101_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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0))) → *4_0, rt ∈ Ω(n2923010)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n298523_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2985230)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n313018_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3130180)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n327817_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3278170)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n342920_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3429200)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n358327_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3583270)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n382930_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3829300)
U191(gen_tt:mark:nil:0':ok3_0(+(1, n399043_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3990430)
U201(gen_tt:mark:nil:0':ok3_0(+(1, n415460_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(n4154600)
U211(gen_tt:mark:nil:0':ok3_0(+(1, n451195_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4511950)
U221(gen_tt:mark:nil:0':ok3_0(+(1, n468416_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n4684160)
tail(gen_tt:mark:nil:0':ok3_0(+(1, n495750_0))) → *4_0, rt ∈ Ω(n4957500)
take(gen_tt:mark:nil:0':ok3_0(+(1, n503769_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5037690)
sel(gen_tt:mark:nil:0':ok3_0(+(1, n522101_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5221010)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(163) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0))) → *4_0, rt ∈ Ω(n2923010)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n298523_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2985230)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n313018_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3130180)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n327817_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3278170)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n342920_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3429200)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n358327_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3583270)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n382930_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3829300)
U191(gen_tt:mark:nil:0':ok3_0(+(1, n399043_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3990430)
U201(gen_tt:mark:nil:0':ok3_0(+(1, n415460_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(n4154600)
U211(gen_tt:mark:nil:0':ok3_0(+(1, n451195_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4511950)
U221(gen_tt:mark:nil:0':ok3_0(+(1, n468416_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n4684160)
tail(gen_tt:mark:nil:0':ok3_0(+(1, n495750_0))) → *4_0, rt ∈ Ω(n4957500)
take(gen_tt:mark:nil:0':ok3_0(+(1, n503769_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5037690)
sel(gen_tt:mark:nil:0':ok3_0(+(1, n522101_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5221010)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.
(164) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0))) → *4_0, rt ∈ Ω(n2923010)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n298523_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2985230)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n313018_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3130180)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n327817_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3278170)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n342920_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3429200)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n358327_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3583270)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n382930_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3829300)
U191(gen_tt:mark:nil:0':ok3_0(+(1, n399043_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3990430)
U201(gen_tt:mark:nil:0':ok3_0(+(1, n415460_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(n4154600)
U211(gen_tt:mark:nil:0':ok3_0(+(1, n451195_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4511950)
U221(gen_tt:mark:nil:0':ok3_0(+(1, n468416_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n4684160)
tail(gen_tt:mark:nil:0':ok3_0(+(1, n495750_0))) → *4_0, rt ∈ Ω(n4957500)
take(gen_tt:mark:nil:0':ok3_0(+(1, n503769_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5037690)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.
(165) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0))) → *4_0, rt ∈ Ω(n2923010)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n298523_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2985230)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n313018_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3130180)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n327817_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3278170)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n342920_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3429200)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n358327_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3583270)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n382930_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3829300)
U191(gen_tt:mark:nil:0':ok3_0(+(1, n399043_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3990430)
U201(gen_tt:mark:nil:0':ok3_0(+(1, n415460_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(n4154600)
U211(gen_tt:mark:nil:0':ok3_0(+(1, n451195_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4511950)
U221(gen_tt:mark:nil:0':ok3_0(+(1, n468416_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n4684160)
tail(gen_tt:mark:nil:0':ok3_0(+(1, n495750_0))) → *4_0, rt ∈ Ω(n4957500)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.
(166) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0))) → *4_0, rt ∈ Ω(n2923010)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n298523_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2985230)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n313018_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3130180)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n327817_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3278170)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n342920_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3429200)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n358327_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3583270)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n382930_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3829300)
U191(gen_tt:mark:nil:0':ok3_0(+(1, n399043_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3990430)
U201(gen_tt:mark:nil:0':ok3_0(+(1, n415460_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(n4154600)
U211(gen_tt:mark:nil:0':ok3_0(+(1, n451195_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4511950)
U221(gen_tt:mark:nil:0':ok3_0(+(1, n468416_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n4684160)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.
(167) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0))) → *4_0, rt ∈ Ω(n2923010)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n298523_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2985230)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n313018_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3130180)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n327817_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3278170)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n342920_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3429200)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n358327_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3583270)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n382930_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3829300)
U191(gen_tt:mark:nil:0':ok3_0(+(1, n399043_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3990430)
U201(gen_tt:mark:nil:0':ok3_0(+(1, n415460_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(n4154600)
U211(gen_tt:mark:nil:0':ok3_0(+(1, n451195_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n4511950)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.
(168) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0))) → *4_0, rt ∈ Ω(n2923010)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n298523_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2985230)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n313018_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3130180)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n327817_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3278170)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n342920_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3429200)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n358327_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3583270)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n382930_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3829300)
U191(gen_tt:mark:nil:0':ok3_0(+(1, n399043_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3990430)
U201(gen_tt:mark:nil:0':ok3_0(+(1, n415460_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(n4154600)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.
(169) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0))) → *4_0, rt ∈ Ω(n2923010)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n298523_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2985230)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n313018_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3130180)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n327817_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3278170)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n342920_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3429200)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n358327_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3583270)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n382930_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3829300)
U191(gen_tt:mark:nil:0':ok3_0(+(1, n399043_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3990430)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.
(170) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0))) → *4_0, rt ∈ Ω(n2923010)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n298523_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2985230)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n313018_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3130180)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n327817_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3278170)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n342920_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3429200)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n358327_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3583270)
U181(gen_tt:mark:nil:0':ok3_0(+(1, n382930_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3829300)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.
(171) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0))) → *4_0, rt ∈ Ω(n2923010)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n298523_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2985230)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n313018_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3130180)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n327817_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3278170)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n342920_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3429200)
U171(gen_tt:mark:nil:0':ok3_0(+(1, n358327_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n3583270)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.
(172) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0))) → *4_0, rt ∈ Ω(n2923010)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n298523_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2985230)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n313018_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3130180)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n327817_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3278170)
U161(gen_tt:mark:nil:0':ok3_0(+(1, n342920_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3429200)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.
(173) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0))) → *4_0, rt ∈ Ω(n2923010)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n298523_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2985230)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n313018_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3130180)
U151(gen_tt:mark:nil:0':ok3_0(+(1, n327817_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3278170)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.
(174) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0))) → *4_0, rt ∈ Ω(n2923010)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n298523_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2985230)
U141(gen_tt:mark:nil:0':ok3_0(+(1, n313018_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3130180)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.
(175) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0))) → *4_0, rt ∈ Ω(n2923010)
U131(gen_tt:mark:nil:0':ok3_0(+(1, n298523_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2985230)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.
(176) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)
U121(gen_tt:mark:nil:0':ok3_0(+(1, n292301_0))) → *4_0, rt ∈ Ω(n2923010)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.
(177) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)
U111(gen_tt:mark:nil:0':ok3_0(+(1, n286180_0))) → *4_0, rt ∈ Ω(n2861800)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.
(178) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)
U101(gen_tt:mark:nil:0':ok3_0(+(1, n272401_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2724010)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.
(179) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)
U91(gen_tt:mark:nil:0':ok3_0(+(1, n266531_0))) → *4_0, rt ∈ Ω(n2665310)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.
(180) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)
U81(gen_tt:mark:nil:0':ok3_0(+(1, n260762_0))) → *4_0, rt ∈ Ω(n2607620)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.
(181) Obligation:
TRS:
Rules:
active(U101(tt, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)
U71(gen_tt:mark:nil:0':ok3_0(+(1, n255094_0))) → *4_0, rt ∈ Ω(n2550940)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)
U61(gen_tt:mark:nil:0':ok3_0(+(1, n249372_0))) → *4_0, rt ∈ Ω(n2493720)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)
U51(gen_tt:mark:nil:0':ok3_0(+(1, n236721_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2367210)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)
U41(gen_tt:mark:nil:0':ok3_0(+(1, n224374_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n2243740)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)
U31(gen_tt:mark:nil:0':ok3_0(+(1, n205448_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n2054480)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)
U21(gen_tt:mark:nil:0':ok3_0(+(1, n187127_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1871270)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)
U11(gen_tt:mark:nil:0':ok3_0(+(1, n169411_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1694110)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)
U52(gen_tt:mark:nil:0':ok3_0(+(1, n164842_0))) → *4_0, rt ∈ Ω(n1648420)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)
U42(gen_tt:mark:nil:0':ok3_0(+(1, n160374_0))) → *4_0, rt ∈ Ω(n1603740)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)
U32(gen_tt:mark:nil:0':ok3_0(+(1, n149949_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1499490)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)
fst(gen_tt:mark:nil:0':ok3_0(+(1, n145732_0))) → *4_0, rt ∈ Ω(n1457320)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)
U222(gen_tt:mark:nil:0':ok3_0(+(1, n130012_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n1300120)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)
U212(gen_tt:mark:nil:0':ok3_0(+(1, n120499_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1204990)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)
U22(gen_tt:mark:nil:0':ok3_0(+(1, n111290_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1112900)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)
U204(gen_tt:mark:nil:0':ok3_0(+(1, n102385_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n1023850)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)
U203(gen_tt:mark:nil:0':ok3_0(+(1, n83394_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(n833940)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)
U202(gen_tt:mark:nil:0':ok3_0(+(1, n65331_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(n653310)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)
pair(gen_tt:mark:nil:0':ok3_0(+(1, n57531_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n575310)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)
U182(gen_tt:mark:nil:0':ok3_0(+(1, n50234_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n502340)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)
afterNth(gen_tt:mark:nil:0':ok3_0(+(1, n43042_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n430420)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)
head(gen_tt:mark:nil:0':ok3_0(+(1, n40521_0))) → *4_0, rt ∈ Ω(n405210)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)
U172(gen_tt:mark:nil:0':ok3_0(+(1, n29969_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299690)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)
s(gen_tt:mark:nil:0':ok3_0(+(1, n27748_0))) → *4_0, rt ∈ Ω(n277480)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)
natsFrom(gen_tt:mark:nil:0':ok3_0(+(1, n25628_0))) → *4_0, rt ∈ Ω(n256280)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)
cons(gen_tt:mark:nil:0':ok3_0(+(1, n19959_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n199590)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)
U152(gen_tt:mark:nil:0':ok3_0(+(1, n18090_0))) → *4_0, rt ∈ Ω(n180900)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)
U142(gen_tt:mark:nil:0':ok3_0(+(1, n16322_0))) → *4_0, rt ∈ Ω(n163220)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)
U132(gen_tt:mark:nil:0':ok3_0(+(1, n14655_0))) → *4_0, rt ∈ Ω(n146550)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)
splitAt(gen_tt:mark:nil:0':ok3_0(+(1, n9709_0)), gen_tt:mark:nil:0':ok3_0(b)) → *4_0, rt ∈ Ω(n97090)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)
snd(gen_tt:mark:nil:0':ok3_0(+(1, n8293_0))) → *4_0, rt ∈ Ω(n82930)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *4_0, rt ∈ Ω(n50)
U12(gen_tt:mark:nil:0':ok3_0(+(1, n1128_0)), gen_tt:mark:nil:0':ok3_0(b), gen_tt:mark:nil:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11280)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, V2)) → mark(U102(isLNat(V2)))
active(U102(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(U12(isLNat(XS), N, XS))
active(U111(tt)) → mark(tt)
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U121(tt)) → mark(tt)
active(U131(tt, V2)) → mark(U132(isLNat(V2)))
active(U132(tt)) → mark(tt)
active(U141(tt, V2)) → mark(U142(isLNat(V2)))
active(U142(tt)) → mark(tt)
active(U151(tt, V2)) → mark(U152(isLNat(V2)))
active(U152(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(U182(isLNat(Y), Y))
active(U182(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(isNatural(X), N, X, XS))
active(U202(tt, N, X, XS)) → mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS)) → mark(U204(splitAt(N, XS), X))
active(U204(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y)) → mark(U22(isLNat(Y), X))
active(U211(tt, XS)) → mark(U212(isLNat(XS), XS))
active(U212(tt, XS)) → mark(XS)
active(U22(tt, X)) → mark(X)
active(U221(tt, N, XS)) → mark(U222(isLNat(XS), N, XS))
active(U222(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS)) → mark(U32(isLNat(XS), N))
active(U32(tt, N)) → mark(N)
active(U41(tt, V2)) → mark(U42(isLNat(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isLNat(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(isNatural(N), N, XS))
active(fst(pair(X, Y))) → mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS))) → mark(U31(isNatural(N), N, XS))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(isNatural(V1), V2))
active(isLNat(cons(V1, V2))) → mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1))) → mark(U61(isPLNat(V1)))
active(isLNat(natsFrom(V1))) → mark(U71(isNatural(V1)))
active(isLNat(snd(V1))) → mark(U81(isPLNat(V1)))
active(isLNat(tail(V1))) → mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2))) → mark(U101(isNatural(V1), V2))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNat(V1)))
active(isNatural(s(V1))) → mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2))) → mark(U131(isNatural(V1), V2))
active(isPLNat(pair(V1, V2))) → mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(isNatural(V1), V2))
active(natsFrom(N)) → mark(U161(isNatural(N), N))
active(sel(N, XS)) → mark(U171(isNatural(N), N, XS))
active(snd(pair(X, Y))) → mark(U181(isLNat(X), Y))
active(splitAt(0', XS)) → mark(U191(isLNat(XS), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(isNatural(N), XS))
active(take(N, XS)) → mark(U221(isNatural(N), N, XS))
active(U101(X1, X2)) → U101(active(X1), X2)
active(U102(X)) → U102(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U111(X)) → U111(active(X))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U121(X)) → U121(active(X))
active(U131(X1, X2)) → U131(active(X1), X2)
active(U132(X)) → U132(active(X))
active(U141(X1, X2)) → U141(active(X1), X2)
active(U142(X)) → U142(active(X))
active(U151(X1, X2)) → U151(active(X1), X2)
active(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(active(X1), X2, X3, X4)
active(U203(X1, X2, X3, X4)) → U203(active(X1), X2, X3, X4)
active(U204(X1, X2)) → U204(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U212(X1, X2)) → U212(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(U222(X1, X2, X3)) → U222(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X)) → U91(active(X))
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) → mark(U101(X1, X2))
U102(mark(X)) → mark(U102(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U111(mark(X)) → mark(U111(X))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U121(mark(X)) → mark(U121(X))
U131(mark(X1), X2) → mark(U131(X1, X2))
U132(mark(X)) → mark(U132(X))
U141(mark(X1), X2) → mark(U141(X1, X2))
U142(mark(X)) → mark(U142(X))
U151(mark(X1), X2) → mark(U151(X1, X2))
U152(mark(X)) → mark(U152(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))
U172(mark(X1), X2, X3) → mark(U172(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))
U182(mark(X1), X2) → mark(U182(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, X3, X4) → mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4) → mark(U203(X1, X2, X3, X4))
U204(mark(X1), X2) → mark(U204(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U212(mark(X1), X2) → mark(U212(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
U222(mark(X1), X2, X3) → mark(U222(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X)) → mark(U91(X))
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)) → U101(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U102(X)) → U102(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U111(X)) → U111(proper(X))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U121(X)) → U121(proper(X))
proper(U131(X1, X2)) → U131(proper(X1), proper(X2))
proper(U132(X)) → U132(proper(X))
proper(U141(X1, X2)) → U141(proper(X1), proper(X2))
proper(U142(X)) → U142(proper(X))
proper(U151(X1, X2)) → U151(proper(X1), proper(X2))
proper(U152(X)) → U152(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(U172(X1, X2, X3)) → U172(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(U182(X1, X2)) → U182(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, X3, X4)) → U202(proper(X1), proper(X2), proper(X3), proper(X4))
proper(isNatural(X)) → isNatural(proper(X))
proper(U203(X1, X2, X3, X4)) → U203(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U204(X1, X2)) → U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U212(X1, X2)) → U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(U222(X1, X2, X3)) → U222(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X)) → U91(proper(X))
proper(isPLNat(X)) → isPLNat(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(U101(X1, X2))
U102(ok(X)) → ok(U102(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U111(ok(X)) → ok(U111(X))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U121(ok(X)) → ok(U121(X))
U131(ok(X1), ok(X2)) → ok(U131(X1, X2))
U132(ok(X)) → ok(U132(X))
U141(ok(X1), ok(X2)) → ok(U141(X1, X2))
U142(ok(X)) → ok(U142(X))
U151(ok(X1), ok(X2)) → ok(U151(X1, X2))
U152(ok(X)) → ok(U152(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))
U172(ok(X1), ok(X2), ok(X3)) → ok(U172(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))
U182(ok(X1), ok(X2)) → ok(U182(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(X3), ok(X4)) → ok(U202(X1, X2, X3, X4))
isNatural(ok(X)) → ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U203(X1, X2, X3, X4))
U204(ok(X1), ok(X2)) → ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U212(ok(X1), ok(X2)) → ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
U222(ok(X1), ok(X2), ok(X3)) → ok(U222(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X)) → ok(U91(X))
isPLNat(ok(X)) → ok(isPLNat(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 :: 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
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
U12 :: tt:mark:nil:0':ok → 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
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
U121 :: 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
U132 :: 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
U142 :: 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
U152 :: 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
U172 :: 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
U182 :: 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 → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U203 :: 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
U204 :: 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 → tt:mark:nil:0':ok
U22 :: 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
U212 :: 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
U222 :: 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 → tt:mark:nil:0':ok
U32 :: 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
U42 :: 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
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: 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))) → *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.