KILLEDRuntime Complexity (full) proof of /tmp/tmpP9bE6n/LISTUTILITIES_complete_C.xml
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).0 CpxTRS↳1 DecreasingLoopProof (⇔, 121.3 s)↳2 BOUNDS(n^1, INF)↳3 RenamingProof (⇔, 0 ms)↳4 CpxRelTRS↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)↳6 typed CpxTrs↳7 OrderProof (LOWER BOUND(ID), 0 ms)↳8 typed CpxTrs↳9 NoRewriteLemmaProof (LOWER BOUND(ID), 43 ms)↳10 typed CpxTrs↳11 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳12 typed CpxTrs↳13 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳14 typed CpxTrs↳15 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳16 typed CpxTrs↳17 NoRewriteLemmaProof (LOWER BOUND(ID), 7 ms)↳18 typed CpxTrs↳19 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳20 typed CpxTrs↳21 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳22 typed CpxTrs↳23 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳24 typed CpxTrs↳25 NoRewriteLemmaProof (LOWER BOUND(ID), 5 ms)↳26 typed CpxTrs↳27 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳28 typed CpxTrs↳29 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳30 typed CpxTrs↳31 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳32 typed CpxTrs↳33 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳34 typed CpxTrs↳35 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳36 typed CpxTrs↳37 NoRewriteLemmaProof (LOWER BOUND(ID), 9 ms)↳38 typed CpxTrs↳39 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳40 typed CpxTrs↳41 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳42 typed CpxTrs↳43 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳44 typed CpxTrs↳45 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳46 typed CpxTrs↳47 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳48 typed CpxTrs↳49 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳50 typed CpxTrs↳51 NoRewriteLemmaProof (LOWER BOUND(ID), 10 ms)↳52 typed CpxTrs↳53 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳54 typed CpxTrs↳55 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳56 typed CpxTrs↳57 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳58 typed CpxTrs↳59 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳60 typed CpxTrs↳61 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳62 typed CpxTrs↳63 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳64 typed CpxTrs↳65 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳66 typed CpxTrs↳67 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳68 typed CpxTrs↳69 NoRewriteLemmaProof (LOWER BOUND(ID), 13 ms)↳70 typed CpxTrs↳71 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳72 typed CpxTrs↳73 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳74 typed CpxTrs↳75 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳76 typed CpxTrs↳77 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳78 typed CpxTrs↳79 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳80 typed CpxTrs↳81 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳82 typed CpxTrs↳83 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳84 typed CpxTrs↳85 NoRewriteLemmaProof (LOWER BOUND(ID), 6 ms)↳86 typed CpxTrs↳87 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳88 typed CpxTrs↳89 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳90 typed CpxTrs↳91 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳92 typed CpxTrs↳93 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳94 typed CpxTrs↳95 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳96 typed CpxTrs↳97 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳98 typed CpxTrs↳99 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳100 typed CpxTrs↳101 NoRewriteLemmaProof (LOWER BOUND(ID), 5 ms)↳102 typed CpxTrs↳103 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳104 typed CpxTrs↳105 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳106 typed CpxTrs↳107 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳108 typed CpxTrs↳109 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳110 typed CpxTrs↳111 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳112 typed CpxTrs↳113 NoRewriteLemmaProof (LOWER BOUND(ID), 7 ms)↳114 typed CpxTrs↳115 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳116 typed CpxTrs↳117 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳118 typed CpxTrs↳119 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳120 typed CpxTrs↳121 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳122 typed CpxTrs↳123 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳124 typed CpxTrs↳125 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳126 typed CpxTrs↳127 NoRewriteLemmaProof (LOWER BOUND(ID), 1 ms)↳128 typed CpxTrs(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0)) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0)) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0, XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0) → ok(0)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Rewrite Strategy: FULL(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
U101(mark(X1), X2, X3) →+ mark(U101(X1, X2, X3))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X1 / mark(X1)].
The result substitution is [ ].(2) BOUNDS(n^1, INF)
(3) RenamingProof (EQUIVALENT transformation)
Renamed function symbols to avoid clashes with predefined symbol.(4) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
S is empty.
Rewrite Strategy: FULL(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.(6) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':ok(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
active, U102, isNatural, U103, isLNat, snd, splitAt, U112, U122, U132, U133, U142, U143, U152, U153, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U102 < active
isNatural < active
U103 < active
isLNat < active
snd < active
splitAt < active
U112 < active
U122 < active
U132 < active
U133 < active
U142 < active
U143 < active
U152 < active
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U102 < proper
isNatural < proper
U103 < proper
isLNat < proper
snd < proper
splitAt < proper
U112 < proper
U122 < proper
U132 < proper
U133 < proper
U142 < proper
U143 < proper
U152 < proper
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(8) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U102, active, isNatural, U103, isLNat, snd, splitAt, U112, U122, U132, U133, U142, U143, U152, U153, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U102 < active
isNatural < active
U103 < active
isLNat < active
snd < active
splitAt < active
U112 < active
U122 < active
U132 < active
U133 < active
U142 < active
U143 < active
U152 < active
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U102 < proper
isNatural < proper
U103 < proper
isLNat < proper
snd < proper
splitAt < proper
U112 < proper
U122 < proper
U132 < proper
U133 < proper
U142 < proper
U143 < proper
U152 < proper
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U102.(10) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
isNatural, active, U103, isLNat, snd, splitAt, U112, U122, U132, U133, U142, U143, U152, U153, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
isNatural < active
U103 < active
isLNat < active
snd < active
splitAt < active
U112 < active
U122 < active
U132 < active
U133 < active
U142 < active
U143 < active
U152 < active
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
isNatural < proper
U103 < proper
isLNat < proper
snd < proper
splitAt < proper
U112 < proper
U122 < proper
U132 < proper
U133 < proper
U142 < proper
U143 < proper
U152 < proper
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol isNatural.(12) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U103, active, isLNat, snd, splitAt, U112, U122, U132, U133, U142, U143, U152, U153, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U103 < active
isLNat < active
snd < active
splitAt < active
U112 < active
U122 < active
U132 < active
U133 < active
U142 < active
U143 < active
U152 < active
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U103 < proper
isLNat < proper
snd < proper
splitAt < proper
U112 < proper
U122 < proper
U132 < proper
U133 < proper
U142 < proper
U143 < proper
U152 < proper
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U103.(14) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':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:
isLNat, active, snd, splitAt, U112, U122, U132, U133, U142, U143, U152, U153, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
isLNat < active
snd < active
splitAt < active
U112 < active
U122 < active
U132 < active
U133 < active
U142 < active
U143 < active
U152 < active
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
isLNat < proper
snd < proper
splitAt < proper
U112 < proper
U122 < proper
U132 < proper
U133 < proper
U142 < proper
U143 < proper
U152 < proper
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol isLNat.(16) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':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:
snd, active, splitAt, U112, U122, U132, U133, U142, U143, U152, U153, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
snd < active
splitAt < active
U112 < active
U122 < active
U132 < active
U133 < active
U142 < active
U143 < active
U152 < active
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
snd < proper
splitAt < proper
U112 < proper
U122 < proper
U132 < proper
U133 < proper
U142 < proper
U143 < proper
U152 < proper
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol snd.(18) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
splitAt, active, U112, U122, U132, U133, U142, U143, U152, U153, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
splitAt < active
U112 < active
U122 < active
U132 < active
U133 < active
U142 < active
U143 < active
U152 < active
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
splitAt < proper
U112 < proper
U122 < proper
U132 < proper
U133 < proper
U142 < proper
U143 < proper
U152 < proper
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol splitAt.(20) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U112, active, U122, U132, U133, U142, U143, U152, U153, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U112 < active
U122 < active
U132 < active
U133 < active
U142 < active
U143 < active
U152 < active
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U112 < proper
U122 < proper
U132 < proper
U133 < proper
U142 < proper
U143 < proper
U152 < proper
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(21) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U112.(22) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':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:
U122, active, U132, U133, U142, U143, U152, U153, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U122 < active
U132 < active
U133 < active
U142 < active
U143 < active
U152 < active
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U122 < proper
U132 < proper
U133 < proper
U142 < proper
U143 < proper
U152 < proper
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(23) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U122.(24) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U132, active, U133, U142, U143, U152, U153, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U132 < active
U133 < active
U142 < active
U143 < active
U152 < active
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U132 < proper
U133 < proper
U142 < proper
U143 < proper
U152 < proper
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(25) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U132.(26) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U133, active, U142, U143, U152, U153, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U133 < active
U142 < active
U143 < active
U152 < active
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U133 < proper
U142 < proper
U143 < proper
U152 < proper
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(27) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U133.(28) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':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:
U142, active, U143, U152, U153, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U142 < active
U143 < active
U152 < active
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U142 < proper
U143 < proper
U152 < proper
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(29) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U142.(30) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U143, active, U152, U153, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U143 < active
U152 < active
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U143 < proper
U152 < proper
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(31) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U143.(32) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U152, active, U153, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U152 < active
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U152 < proper
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(33) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U152.(34) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':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:
U153, active, cons, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U153 < active
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U153 < proper
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(35) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U153.(36) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
cons, active, natsFrom, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
cons < active
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
cons < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(37) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol cons.(38) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
natsFrom, active, s, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
natsFrom < active
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
natsFrom < proper
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(39) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol natsFrom.(40) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':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:
s, active, head, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
s < active
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
s < proper
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(41) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol s.(42) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
head, active, afterNth, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
head < active
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
head < proper
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(43) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol head.(44) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
afterNth, active, pair, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
afterNth < active
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
afterNth < proper
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(45) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol afterNth.(46) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':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:
pair, active, U202, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
pair < active
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
pair < proper
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(47) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol pair.(48) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U202, active, fst, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U202 < active
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U202 < proper
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(49) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U202.(50) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
fst, active, U42, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
fst < active
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
fst < proper
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(51) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol fst.(52) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':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:
U42, active, U43, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U42 < active
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U42 < proper
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(53) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U42.(54) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U43, active, U52, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U43 < active
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U43 < proper
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(55) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U43.(56) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U52, active, U53, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U52 < active
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U52 < proper
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(57) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U52.(58) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':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:
U53, active, U62, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U53 < active
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U53 < proper
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(59) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U53.(60) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U62, active, isPLNat, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U62 < active
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U62 < proper
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(61) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U62.(62) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
isPLNat, active, U72, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
isPLNat < active
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
isPLNat < proper
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(63) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol isPLNat.(64) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':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:
U72, active, U82, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U72 < active
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U72 < proper
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(65) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U72.(66) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U82, active, U92, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U82 < active
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U82 < proper
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(67) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U82.(68) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U92, active, U11, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U92 < active
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U92 < proper
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(69) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U92.(70) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':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:
U11, active, and, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U11 < active
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U11 < proper
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(71) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U11.(72) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
and, active, isNaturalKind, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
and < active
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
and < proper
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(73) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol and.(74) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
isNaturalKind, active, isLNatKind, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
isNaturalKind < active
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
isNaturalKind < proper
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(75) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol isNaturalKind.(76) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':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:
isLNatKind, active, U21, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
isLNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
isLNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(77) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol isLNatKind.(78) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U21, active, U31, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(79) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U21.(80) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U31, active, U41, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U31 < active
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U31 < proper
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(81) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U31.(82) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':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:
U41, active, U51, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U41 < active
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U41 < proper
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(83) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U41.(84) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U51, active, U61, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U51 < active
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U51 < proper
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(85) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U51.(86) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
U61, active, isPLNatKind, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
U61 < active
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
U61 < proper
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(87) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U61.(88) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
isPLNatKind, active, U71, U81, U91, U101, U111, U121, U131, U141, U151, U161, U171, U181, U191, U201, U211, U221, tail, take, sel, proper, topThey will be analysed ascendingly in the following order:
isPLNatKind < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
U141 < active
U151 < active
U161 < active
U171 < active
U181 < active
U191 < active
U201 < active
U211 < active
U221 < active
tail < active
take < active
sel < active
active < top
isPLNatKind < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
U141 < proper
U151 < proper
U161 < proper
U171 < proper
U181 < proper
U191 < proper
U201 < proper
U211 < proper
U221 < proper
tail < proper
take < proper
sel < proper
proper < top(89) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol isPLNatKind.(90) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':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:
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(91) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U71.(92) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
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(93) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U81.(94) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
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(95) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U91.(96) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':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:
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(97) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U101.(98) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
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(99) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U111.(100) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
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(101) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U121.(102) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':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:
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(103) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U131.(104) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':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:
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(105) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U141.(106) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':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:
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(107) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U151.(108) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
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(109) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U161.(110) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
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(111) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U171.(112) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':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:
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(113) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U181.(114) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
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(115) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U191.(116) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
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(117) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U201.(118) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':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:
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(119) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U211.(120) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
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(121) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U221.(122) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
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(123) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol tail.(124) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
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(125) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol take.(126) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':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:
sel, active, proper, topThey will be analysed ascendingly in the following order:
sel < active
active < top
sel < proper
proper < top(127) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol sel.(128) Obligation:
TRS:
Rules:
active(U101(tt, V1, V2)) → mark(U102(isNatural(V1), V2))
active(U102(tt, V2)) → mark(U103(isLNat(V2)))
active(U103(tt)) → mark(tt)
active(U11(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U111(tt, V1)) → mark(U112(isLNat(V1)))
active(U112(tt)) → mark(tt)
active(U121(tt, V1)) → mark(U122(isNatural(V1)))
active(U122(tt)) → mark(tt)
active(U131(tt, V1, V2)) → mark(U132(isNatural(V1), V2))
active(U132(tt, V2)) → mark(U133(isLNat(V2)))
active(U133(tt)) → mark(tt)
active(U141(tt, V1, V2)) → mark(U142(isLNat(V1), V2))
active(U142(tt, V2)) → mark(U143(isLNat(V2)))
active(U143(tt)) → mark(tt)
active(U151(tt, V1, V2)) → mark(U152(isNatural(V1), V2))
active(U152(tt, V2)) → mark(U153(isLNat(V2)))
active(U153(tt)) → mark(tt)
active(U161(tt, N)) → mark(cons(N, natsFrom(s(N))))
active(U171(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U181(tt, Y)) → mark(Y)
active(U191(tt, XS)) → mark(pair(nil, XS))
active(U201(tt, N, X, XS)) → mark(U202(splitAt(N, XS), X))
active(U202(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U21(tt, X)) → mark(X)
active(U211(tt, XS)) → mark(XS)
active(U221(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(U31(tt, N)) → mark(N)
active(U41(tt, V1, V2)) → mark(U42(isNatural(V1), V2))
active(U42(tt, V2)) → mark(U43(isLNat(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNatural(V1), V2))
active(U52(tt, V2)) → mark(U53(isLNat(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, V1)) → mark(U62(isPLNat(V1)))
active(U62(tt)) → mark(tt)
active(U71(tt, V1)) → mark(U72(isNatural(V1)))
active(U72(tt)) → mark(tt)
active(U81(tt, V1)) → mark(U82(isPLNat(V1)))
active(U82(tt)) → mark(tt)
active(U91(tt, V1)) → mark(U92(isLNat(V1)))
active(U92(tt)) → mark(tt)
active(afterNth(N, XS)) → mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X))
active(head(cons(N, XS))) → mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N))
active(isLNat(nil)) → mark(tt)
active(isLNat(afterNth(V1, V2))) → mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(cons(V1, V2))) → mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNat(fst(V1))) → mark(U61(isPLNatKind(V1), V1))
active(isLNat(natsFrom(V1))) → mark(U71(isNaturalKind(V1), V1))
active(isLNat(snd(V1))) → mark(U81(isPLNatKind(V1), V1))
active(isLNat(tail(V1))) → mark(U91(isLNatKind(V1), V1))
active(isLNat(take(V1, V2))) → mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isLNatKind(nil)) → mark(tt)
active(isLNatKind(afterNth(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(cons(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isLNatKind(fst(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(natsFrom(V1))) → mark(isNaturalKind(V1))
active(isLNatKind(snd(V1))) → mark(isPLNatKind(V1))
active(isLNatKind(tail(V1))) → mark(isLNatKind(V1))
active(isLNatKind(take(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isNatural(0')) → mark(tt)
active(isNatural(head(V1))) → mark(U111(isLNatKind(V1), V1))
active(isNatural(s(V1))) → mark(U121(isNaturalKind(V1), V1))
active(isNatural(sel(V1, V2))) → mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isNaturalKind(0')) → mark(tt)
active(isNaturalKind(head(V1))) → mark(isLNatKind(V1))
active(isNaturalKind(s(V1))) → mark(isNaturalKind(V1))
active(isNaturalKind(sel(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(isPLNat(pair(V1, V2))) → mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNat(splitAt(V1, V2))) → mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2))
active(isPLNatKind(pair(V1, V2))) → mark(and(isLNatKind(V1), isLNatKind(V2)))
active(isPLNatKind(splitAt(V1, V2))) → mark(and(isNaturalKind(V1), isLNatKind(V2)))
active(natsFrom(N)) → mark(U161(and(isNatural(N), isNaturalKind(N)), N))
active(sel(N, XS)) → mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(snd(pair(X, Y))) → mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y))
active(splitAt(0', XS)) → mark(U191(and(isLNat(XS), isLNatKind(XS)), XS))
active(splitAt(s(N), cons(X, XS))) → mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS))
active(tail(cons(N, XS))) → mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS))
active(take(N, XS)) → mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2)) → U102(active(X1), X2)
active(U103(X)) → U103(active(X))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U111(X1, X2)) → U111(active(X1), X2)
active(U112(X)) → U112(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3)) → U131(active(X1), X2, X3)
active(U132(X1, X2)) → U132(active(X1), X2)
active(U133(X)) → U133(active(X))
active(U141(X1, X2, X3)) → U141(active(X1), X2, X3)
active(U142(X1, X2)) → U142(active(X1), X2)
active(U143(X)) → U143(active(X))
active(U151(X1, X2, X3)) → U151(active(X1), X2, X3)
active(U152(X1, X2)) → U152(active(X1), X2)
active(U153(X)) → U153(active(X))
active(U161(X1, X2)) → U161(active(X1), X2)
active(cons(X1, X2)) → cons(active(X1), X2)
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(U171(X1, X2, X3)) → U171(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U181(X1, X2)) → U181(active(X1), X2)
active(U191(X1, X2)) → U191(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(U201(X1, X2, X3, X4)) → U201(active(X1), X2, X3, X4)
active(U202(X1, X2)) → U202(active(X1), X2)
active(U21(X1, X2)) → U21(active(X1), X2)
active(U211(X1, X2)) → U211(active(X1), X2)
active(U221(X1, X2, X3)) → U221(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X)) → U72(active(X))
active(U81(X1, X2)) → U81(active(X1), X2)
active(U82(X)) → U82(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2) → mark(U102(X1, X2))
U103(mark(X)) → mark(U103(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U111(mark(X1), X2) → mark(U111(X1, X2))
U112(mark(X)) → mark(U112(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3) → mark(U131(X1, X2, X3))
U132(mark(X1), X2) → mark(U132(X1, X2))
U133(mark(X)) → mark(U133(X))
U141(mark(X1), X2, X3) → mark(U141(X1, X2, X3))
U142(mark(X1), X2) → mark(U142(X1, X2))
U143(mark(X)) → mark(U143(X))
U151(mark(X1), X2, X3) → mark(U151(X1, X2, X3))
U152(mark(X1), X2) → mark(U152(X1, X2))
U153(mark(X)) → mark(U153(X))
U161(mark(X1), X2) → mark(U161(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
U171(mark(X1), X2, X3) → mark(U171(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U181(mark(X1), X2) → mark(U181(X1, X2))
U191(mark(X1), X2) → mark(U191(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4) → mark(U201(X1, X2, X3, X4))
U202(mark(X1), X2) → mark(U202(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U211(mark(X1), X2) → mark(U211(X1, X2))
U221(mark(X1), X2, X3) → mark(U221(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X)) → mark(U72(X))
U81(mark(X1), X2) → mark(U81(X1, X2))
U82(mark(X)) → mark(U82(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
and(mark(X1), X2) → mark(and(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2)) → U102(proper(X1), proper(X2))
proper(isNatural(X)) → isNatural(proper(X))
proper(U103(X)) → U103(proper(X))
proper(isLNat(X)) → isLNat(proper(X))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U111(X1, X2)) → U111(proper(X1), proper(X2))
proper(U112(X)) → U112(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(U131(X1, X2, X3)) → U131(proper(X1), proper(X2), proper(X3))
proper(U132(X1, X2)) → U132(proper(X1), proper(X2))
proper(U133(X)) → U133(proper(X))
proper(U141(X1, X2, X3)) → U141(proper(X1), proper(X2), proper(X3))
proper(U142(X1, X2)) → U142(proper(X1), proper(X2))
proper(U143(X)) → U143(proper(X))
proper(U151(X1, X2, X3)) → U151(proper(X1), proper(X2), proper(X3))
proper(U152(X1, X2)) → U152(proper(X1), proper(X2))
proper(U153(X)) → U153(proper(X))
proper(U161(X1, X2)) → U161(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(U171(X1, X2, X3)) → U171(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U181(X1, X2)) → U181(proper(X1), proper(X2))
proper(U191(X1, X2)) → U191(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U201(X1, X2, X3, X4)) → U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2)) → U202(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U211(X1, X2)) → U211(proper(X1), proper(X2))
proper(U221(X1, X2, X3)) → U221(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(isPLNat(X)) → isPLNat(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X)) → U72(proper(X))
proper(U81(X1, X2)) → U81(proper(X1), proper(X2))
proper(U82(X)) → U82(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNaturalKind(X)) → isNaturalKind(proper(X))
proper(isLNatKind(X)) → isLNatKind(proper(X))
proper(isPLNatKind(X)) → isPLNatKind(proper(X))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(0') → ok(0')
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2)) → ok(U102(X1, X2))
isNatural(ok(X)) → ok(isNatural(X))
U103(ok(X)) → ok(U103(X))
isLNat(ok(X)) → ok(isLNat(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U111(ok(X1), ok(X2)) → ok(U111(X1, X2))
U112(ok(X)) → ok(U112(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3)) → ok(U131(X1, X2, X3))
U132(ok(X1), ok(X2)) → ok(U132(X1, X2))
U133(ok(X)) → ok(U133(X))
U141(ok(X1), ok(X2), ok(X3)) → ok(U141(X1, X2, X3))
U142(ok(X1), ok(X2)) → ok(U142(X1, X2))
U143(ok(X)) → ok(U143(X))
U151(ok(X1), ok(X2), ok(X3)) → ok(U151(X1, X2, X3))
U152(ok(X1), ok(X2)) → ok(U152(X1, X2))
U153(ok(X)) → ok(U153(X))
U161(ok(X1), ok(X2)) → ok(U161(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
U171(ok(X1), ok(X2), ok(X3)) → ok(U171(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U181(ok(X1), ok(X2)) → ok(U181(X1, X2))
U191(ok(X1), ok(X2)) → ok(U191(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
U201(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2)) → ok(U202(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U211(ok(X1), ok(X2)) → ok(U211(X1, X2))
U221(ok(X1), ok(X2), ok(X3)) → ok(U221(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
isPLNat(ok(X)) → ok(isPLNat(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X)) → ok(U72(X))
U81(ok(X1), ok(X2)) → ok(U81(X1, X2))
U82(ok(X)) → ok(U82(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNaturalKind(ok(X)) → ok(isNaturalKind(X))
isLNatKind(ok(X)) → ok(isLNatKind(X))
isPLNatKind(ok(X)) → ok(isPLNatKind(X))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U101 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
tt :: tt:mark:nil:0':ok
mark :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U102 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNatural :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U103 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U11 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
snd :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
splitAt :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U111 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U112 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U121 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U122 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U131 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U132 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U133 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U141 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U142 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U143 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U151 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U152 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U153 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U161 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
cons :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
natsFrom :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
s :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U171 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
head :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
afterNth :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U181 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U191 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
pair :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
nil :: tt:mark:nil:0':ok
U201 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U202 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U21 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U211 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U221 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
fst :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U31 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U41 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U42 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U43 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U51 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U52 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U53 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U61 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U62 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNat :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U71 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U72 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U81 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U82 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
U91 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
U92 :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
and :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
isNaturalKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
isPLNatKind :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
tail :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
take :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
0' :: tt:mark:nil:0':ok
sel :: tt:mark:nil:0':ok → tt:mark:nil:0':ok → tt:mark:nil:0':ok
proper :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
ok :: tt:mark:nil:0':ok → tt:mark:nil:0':ok
top :: tt:mark:nil:0':ok → top
hole_tt:mark:nil:0':ok1_0 :: tt:mark:nil:0':ok
hole_top2_0 :: top
gen_tt:mark:nil:0':ok3_0 :: Nat → tt:mark:nil:0':okGenerator Equations:
gen_tt:mark:nil:0':ok3_0(0) ⇔ tt
gen_tt:mark:nil:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:nil:0':ok3_0(x))The following defined symbols remain to be analysed:
active, proper, topThey will be analysed ascendingly in the following order:
active < top
proper < top