KILLED



    


Runtime Complexity (full) proof of /tmp/tmppvVQyV/LISTUTILITIES_nosorts_noand_C.xml









The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

0 CpxTRS


↳1 DecreasingLoopProof (⇔, 36.8 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), 7 ms)


    ↳10 typed CpxTrs


    ↳11 NoRewriteLemmaProof (LOWER BOUND(ID), 2 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), 9 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), 0 ms)


    ↳26 typed CpxTrs


    ↳27 NoRewriteLemmaProof (LOWER BOUND(ID), 14 ms)


    ↳28 typed CpxTrs


    ↳29 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)


    ↳30 typed CpxTrs


    ↳31 NoRewriteLemmaProof (LOWER BOUND(ID), 12 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), 0 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), 19 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), 0 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), 1 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









(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))

active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))

active(U21(tt, X)) → mark(U22(tt, X))

active(U22(tt, X)) → mark(X)

active(U31(tt, N)) → mark(U32(tt, N))

active(U32(tt, N)) → mark(N)

active(U41(tt, N, XS)) → mark(U42(tt, N, XS))

active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))

active(U51(tt, Y)) → mark(U52(tt, Y))

active(U52(tt, Y)) → mark(Y)

active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))

active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))

active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))

active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))

active(U71(tt, XS)) → mark(U72(tt, XS))

active(U72(tt, XS)) → mark(XS)

active(U81(tt, N, XS)) → mark(U82(tt, N, XS))

active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))

active(afterNth(N, XS)) → mark(U11(tt, N, XS))

active(fst(pair(X, Y))) → mark(U21(tt, X))

active(head(cons(N, XS))) → mark(U31(tt, N))

active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))

active(sel(N, XS)) → mark(U41(tt, N, XS))

active(snd(pair(X, Y))) → mark(U51(tt, Y))

active(splitAt(0, XS)) → mark(pair(nil, XS))

active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))

active(tail(cons(N, XS))) → mark(U71(tt, XS))

active(take(N, XS)) → mark(U81(tt, N, XS))

active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)

active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)

active(U22(X1, X2)) → U22(active(X1), X2)

active(U31(X1, X2)) → U31(active(X1), X2)

active(U32(X1, X2)) → U32(active(X1), X2)

active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)

active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)

active(head(X)) → head(active(X))

active(afterNth(X1, X2)) → afterNth(active(X1), X2)

active(afterNth(X1, X2)) → afterNth(X1, active(X2))

active(U51(X1, X2)) → U51(active(X1), X2)

active(U52(X1, X2)) → U52(active(X1), X2)

active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)

active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)

active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)

active(U64(X1, X2)) → U64(active(X1), X2)

active(pair(X1, X2)) → pair(active(X1), X2)

active(pair(X1, X2)) → pair(X1, active(X2))

active(cons(X1, X2)) → cons(active(X1), X2)

active(U71(X1, X2)) → U71(active(X1), X2)

active(U72(X1, X2)) → U72(active(X1), X2)

active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)

active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)

active(fst(X)) → fst(active(X))

active(natsFrom(X)) → natsFrom(active(X))

active(s(X)) → s(active(X))

active(sel(X1, X2)) → sel(active(X1), X2)

active(sel(X1, X2)) → sel(X1, active(X2))

active(tail(X)) → tail(active(X))

active(take(X1, X2)) → take(active(X1), X2)

active(take(X1, X2)) → take(X1, active(X2))

U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))

U12(mark(X1), X2, X3) → mark(U12(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))

U21(mark(X1), X2) → mark(U21(X1, X2))

U22(mark(X1), X2) → mark(U22(X1, X2))

U31(mark(X1), X2) → mark(U31(X1, X2))

U32(mark(X1), X2) → mark(U32(X1, X2))

U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))

U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))

head(mark(X)) → mark(head(X))

afterNth(mark(X1), X2) → mark(afterNth(X1, X2))

afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))

U51(mark(X1), X2) → mark(U51(X1, X2))

U52(mark(X1), X2) → mark(U52(X1, X2))

U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))

U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))

U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))

U64(mark(X1), X2) → mark(U64(X1, X2))

pair(mark(X1), X2) → mark(pair(X1, X2))

pair(X1, mark(X2)) → mark(pair(X1, X2))

cons(mark(X1), X2) → mark(cons(X1, X2))

U71(mark(X1), X2) → mark(U71(X1, X2))

U72(mark(X1), X2) → mark(U72(X1, X2))

U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))

U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))

fst(mark(X)) → mark(fst(X))

natsFrom(mark(X)) → mark(natsFrom(X))

s(mark(X)) → mark(s(X))

sel(mark(X1), X2) → mark(sel(X1, X2))

sel(X1, mark(X2)) → mark(sel(X1, X2))

tail(mark(X)) → mark(tail(X))

take(mark(X1), X2) → mark(take(X1, X2))

take(X1, mark(X2)) → mark(take(X1, X2))

proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))

proper(tt) → ok(tt)

proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))

proper(snd(X)) → snd(proper(X))

proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))

proper(U21(X1, X2)) → U21(proper(X1), proper(X2))

proper(U22(X1, X2)) → U22(proper(X1), proper(X2))

proper(U31(X1, X2)) → U31(proper(X1), proper(X2))

proper(U32(X1, X2)) → U32(proper(X1), proper(X2))

proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))

proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))

proper(head(X)) → head(proper(X))

proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))

proper(U51(X1, X2)) → U51(proper(X1), proper(X2))

proper(U52(X1, X2)) → U52(proper(X1), proper(X2))

proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))

proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))

proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))

proper(U64(X1, X2)) → U64(proper(X1), proper(X2))

proper(pair(X1, X2)) → pair(proper(X1), proper(X2))

proper(cons(X1, X2)) → cons(proper(X1), proper(X2))

proper(U71(X1, X2)) → U71(proper(X1), proper(X2))

proper(U72(X1, X2)) → U72(proper(X1), proper(X2))

proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))

proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))

proper(fst(X)) → fst(proper(X))

proper(natsFrom(X)) → natsFrom(proper(X))

proper(s(X)) → s(proper(X))

proper(sel(X1, X2)) → sel(proper(X1), proper(X2))

proper(0) → ok(0)

proper(nil) → ok(nil)

proper(tail(X)) → tail(proper(X))

proper(take(X1, X2)) → take(proper(X1), proper(X2))

U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))

U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))

snd(ok(X)) → ok(snd(X))

splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))

U21(ok(X1), ok(X2)) → ok(U21(X1, X2))

U22(ok(X1), ok(X2)) → ok(U22(X1, X2))

U31(ok(X1), ok(X2)) → ok(U31(X1, X2))

U32(ok(X1), ok(X2)) → ok(U32(X1, X2))

U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))

U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))

head(ok(X)) → ok(head(X))

afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))

U51(ok(X1), ok(X2)) → ok(U51(X1, X2))

U52(ok(X1), ok(X2)) → ok(U52(X1, X2))

U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))

U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))

U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))

U64(ok(X1), ok(X2)) → ok(U64(X1, X2))

pair(ok(X1), ok(X2)) → ok(pair(X1, X2))

cons(ok(X1), ok(X2)) → ok(cons(X1, X2))

U71(ok(X1), ok(X2)) → ok(U71(X1, X2))

U72(ok(X1), ok(X2)) → ok(U72(X1, X2))

U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))

U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))

fst(ok(X)) → ok(fst(X))

natsFrom(ok(X)) → ok(natsFrom(X))

s(ok(X)) → ok(s(X))

sel(ok(X1), ok(X2)) → ok(sel(X1, X2))

tail(ok(X)) → ok(tail(X))

take(ok(X1), ok(X2)) → ok(take(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
U11(mark(X1), X2, X3) →+ mark(U11(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(U11(tt, N, XS)) → mark(U12(tt, N, XS))

active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))

active(U21(tt, X)) → mark(U22(tt, X))

active(U22(tt, X)) → mark(X)

active(U31(tt, N)) → mark(U32(tt, N))

active(U32(tt, N)) → mark(N)

active(U41(tt, N, XS)) → mark(U42(tt, N, XS))

active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))

active(U51(tt, Y)) → mark(U52(tt, Y))

active(U52(tt, Y)) → mark(Y)

active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))

active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))

active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))

active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))

active(U71(tt, XS)) → mark(U72(tt, XS))

active(U72(tt, XS)) → mark(XS)

active(U81(tt, N, XS)) → mark(U82(tt, N, XS))

active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))

active(afterNth(N, XS)) → mark(U11(tt, N, XS))

active(fst(pair(X, Y))) → mark(U21(tt, X))

active(head(cons(N, XS))) → mark(U31(tt, N))

active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))

active(sel(N, XS)) → mark(U41(tt, N, XS))

active(snd(pair(X, Y))) → mark(U51(tt, Y))

active(splitAt(0', XS)) → mark(pair(nil, XS))

active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))

active(tail(cons(N, XS))) → mark(U71(tt, XS))

active(take(N, XS)) → mark(U81(tt, N, XS))

active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)

active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)

active(U22(X1, X2)) → U22(active(X1), X2)

active(U31(X1, X2)) → U31(active(X1), X2)

active(U32(X1, X2)) → U32(active(X1), X2)

active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)

active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)

active(head(X)) → head(active(X))

active(afterNth(X1, X2)) → afterNth(active(X1), X2)

active(afterNth(X1, X2)) → afterNth(X1, active(X2))

active(U51(X1, X2)) → U51(active(X1), X2)

active(U52(X1, X2)) → U52(active(X1), X2)

active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)

active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)

active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)

active(U64(X1, X2)) → U64(active(X1), X2)

active(pair(X1, X2)) → pair(active(X1), X2)

active(pair(X1, X2)) → pair(X1, active(X2))

active(cons(X1, X2)) → cons(active(X1), X2)

active(U71(X1, X2)) → U71(active(X1), X2)

active(U72(X1, X2)) → U72(active(X1), X2)

active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)

active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)

active(fst(X)) → fst(active(X))

active(natsFrom(X)) → natsFrom(active(X))

active(s(X)) → s(active(X))

active(sel(X1, X2)) → sel(active(X1), X2)

active(sel(X1, X2)) → sel(X1, active(X2))

active(tail(X)) → tail(active(X))

active(take(X1, X2)) → take(active(X1), X2)

active(take(X1, X2)) → take(X1, active(X2))

U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))

U12(mark(X1), X2, X3) → mark(U12(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))

U21(mark(X1), X2) → mark(U21(X1, X2))

U22(mark(X1), X2) → mark(U22(X1, X2))

U31(mark(X1), X2) → mark(U31(X1, X2))

U32(mark(X1), X2) → mark(U32(X1, X2))

U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))

U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))

head(mark(X)) → mark(head(X))

afterNth(mark(X1), X2) → mark(afterNth(X1, X2))

afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))

U51(mark(X1), X2) → mark(U51(X1, X2))

U52(mark(X1), X2) → mark(U52(X1, X2))

U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))

U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))

U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))

U64(mark(X1), X2) → mark(U64(X1, X2))

pair(mark(X1), X2) → mark(pair(X1, X2))

pair(X1, mark(X2)) → mark(pair(X1, X2))

cons(mark(X1), X2) → mark(cons(X1, X2))

U71(mark(X1), X2) → mark(U71(X1, X2))

U72(mark(X1), X2) → mark(U72(X1, X2))

U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))

U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))

fst(mark(X)) → mark(fst(X))

natsFrom(mark(X)) → mark(natsFrom(X))

s(mark(X)) → mark(s(X))

sel(mark(X1), X2) → mark(sel(X1, X2))

sel(X1, mark(X2)) → mark(sel(X1, X2))

tail(mark(X)) → mark(tail(X))

take(mark(X1), X2) → mark(take(X1, X2))

take(X1, mark(X2)) → mark(take(X1, X2))

proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))

proper(tt) → ok(tt)

proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))

proper(snd(X)) → snd(proper(X))

proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))

proper(U21(X1, X2)) → U21(proper(X1), proper(X2))

proper(U22(X1, X2)) → U22(proper(X1), proper(X2))

proper(U31(X1, X2)) → U31(proper(X1), proper(X2))

proper(U32(X1, X2)) → U32(proper(X1), proper(X2))

proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))

proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))

proper(head(X)) → head(proper(X))

proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))

proper(U51(X1, X2)) → U51(proper(X1), proper(X2))

proper(U52(X1, X2)) → U52(proper(X1), proper(X2))

proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))

proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))

proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))

proper(U64(X1, X2)) → U64(proper(X1), proper(X2))

proper(pair(X1, X2)) → pair(proper(X1), proper(X2))

proper(cons(X1, X2)) → cons(proper(X1), proper(X2))

proper(U71(X1, X2)) → U71(proper(X1), proper(X2))

proper(U72(X1, X2)) → U72(proper(X1), proper(X2))

proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))

proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))

proper(fst(X)) → fst(proper(X))

proper(natsFrom(X)) → natsFrom(proper(X))

proper(s(X)) → s(proper(X))

proper(sel(X1, X2)) → sel(proper(X1), proper(X2))

proper(0') → ok(0')

proper(nil) → ok(nil)

proper(tail(X)) → tail(proper(X))

proper(take(X1, X2)) → take(proper(X1), proper(X2))

U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))

U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))

snd(ok(X)) → ok(snd(X))

splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))

U21(ok(X1), ok(X2)) → ok(U21(X1, X2))

U22(ok(X1), ok(X2)) → ok(U22(X1, X2))

U31(ok(X1), ok(X2)) → ok(U31(X1, X2))

U32(ok(X1), ok(X2)) → ok(U32(X1, X2))

U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))

U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))

head(ok(X)) → ok(head(X))

afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))

U51(ok(X1), ok(X2)) → ok(U51(X1, X2))

U52(ok(X1), ok(X2)) → ok(U52(X1, X2))

U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))

U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))

U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))

U64(ok(X1), ok(X2)) → ok(U64(X1, X2))

pair(ok(X1), ok(X2)) → ok(pair(X1, X2))

cons(ok(X1), ok(X2)) → ok(cons(X1, X2))

U71(ok(X1), ok(X2)) → ok(U71(X1, X2))

U72(ok(X1), ok(X2)) → ok(U72(X1, X2))

U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))

U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))

fst(ok(X)) → ok(fst(X))

natsFrom(ok(X)) → ok(natsFrom(X))

s(ok(X)) → ok(s(X))

sel(ok(X1), ok(X2)) → ok(sel(X1, X2))

tail(ok(X)) → ok(tail(X))

take(ok(X1), ok(X2)) → ok(take(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(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok




(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
active, U12, snd, splitAt, U22, U32, U42, head, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
U12 < active
snd < active
splitAt < active
U22 < active
U32 < active
U42 < active
head < active
afterNth < active
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U12 < proper
snd < proper
splitAt < proper
U22 < proper
U32 < proper
U42 < proper
head < proper
afterNth < proper
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top




(8) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
U12, active, snd, splitAt, U22, U32, U42, head, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
U12 < active
snd < active
splitAt < active
U22 < active
U32 < active
U42 < active
head < active
afterNth < active
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U12 < proper
snd < proper
splitAt < proper
U22 < proper
U32 < proper
U42 < proper
head < proper
afterNth < proper
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top




(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U12.




(10) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
snd, active, splitAt, U22, U32, U42, head, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
snd < active
splitAt < active
U22 < active
U32 < active
U42 < active
head < active
afterNth < active
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
snd < proper
splitAt < proper
U22 < proper
U32 < proper
U42 < proper
head < proper
afterNth < proper
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top




(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol snd.




(12) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
splitAt, active, U22, U32, U42, head, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
splitAt < active
U22 < active
U32 < active
U42 < active
head < active
afterNth < active
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
splitAt < proper
U22 < proper
U32 < proper
U42 < proper
head < proper
afterNth < proper
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top




(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol splitAt.




(14) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
U22, active, U32, U42, head, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
U22 < active
U32 < active
U42 < active
head < active
afterNth < active
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U22 < proper
U32 < proper
U42 < proper
head < proper
afterNth < proper
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top




(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U22.




(16) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
U32, active, U42, head, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
U32 < active
U42 < active
head < active
afterNth < active
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U32 < proper
U42 < proper
head < proper
afterNth < proper
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top




(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U32.




(18) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
U42, active, head, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
U42 < active
head < active
afterNth < active
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U42 < proper
head < proper
afterNth < proper
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top




(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U42.




(20) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
head, active, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
head < active
afterNth < active
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
head < proper
afterNth < proper
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top




(21) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol head.




(22) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
afterNth, active, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
afterNth < active
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
afterNth < proper
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top




(23) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol afterNth.




(24) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
U52, active, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top




(25) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U52.




(26) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
U62, active, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top




(27) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U62.




(28) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
U63, active, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top




(29) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U63.




(30) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
U64, active, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top




(31) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U64.




(32) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
pair, active, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top




(33) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol pair.




(34) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
cons, active, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top




(35) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol cons.




(36) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
U72, active, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top




(37) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U72.




(38) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
U82, active, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top




(39) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U82.




(40) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
fst, active, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top




(41) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol fst.




(42) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
U11, active, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top




(43) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U11.




(44) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
U21, active, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top




(45) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U21.




(46) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
U31, active, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top




(47) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U31.




(48) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
natsFrom, active, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top




(49) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol natsFrom.




(50) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
s, active, U41, U51, U61, U71, U81, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top




(51) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol s.




(52) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
U41, active, U51, U61, U71, U81, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top




(53) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U41.




(54) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
U51, active, U61, U71, U81, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top




(55) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U51.




(56) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
U61, active, U71, U81, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top




(57) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U61.




(58) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
U71, active, U81, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top




(59) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U71.




(60) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
U81, active, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
U81 < active
sel < active
tail < active
take < active
active < top
U81 < proper
sel < proper
tail < proper
take < proper
proper < top




(61) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol U81.




(62) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
sel, active, tail, take, proper, top
They will be analysed ascendingly in the following order:
sel < active
tail < active
take < active
active < top
sel < proper
tail < proper
take < proper
proper < top




(63) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol sel.




(64) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
tail, active, take, proper, top
They will be analysed ascendingly in the following order:
tail < active
take < active
active < top
tail < proper
take < proper
proper < top




(65) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol tail.




(66) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
take, active, proper, top
They will be analysed ascendingly in the following order:
take < active
active < top
take < proper
proper < top




(67) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol take.




(68) Obligation:
TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(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(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(head(X)) → head(active(X))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(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))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
head(mark(X)) → mark(head(X))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
active, proper, top
They will be analysed ascendingly in the following order:
active < top
proper < top