Termination w.r.t. Q proof of /tmp/tmph6NNgD/OvConsOS_complete_noand

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS

↳1 QTRSToCSRProof (⇔)

↳2 CSR

↳3 PoloCSRProof (⇔)

↳4 CSR

↳5 PoloCSRProof (⇔)

↳6 CSR

↳7 PoloCSRProof (⇔)

↳8 CSR

↳9 CSDependencyPairsProof (⇔)

↳10 QCSDP

↳11 QCSDependencyGraphProof (⇔)

↳12 AND

  ↳13 QCSDP

    ↳14 QCSUsableRulesProof (⇔)

    ↳15 QCSDP

    ↳16 QCSDPMuMonotonicPoloProof (⇔)

    ↳17 QCSDP

    ↳18 QCSDependencyGraphProof (⇔)

    ↳19 TRUE

  ↳20 QCSDP

    ↳21 QCSUsableRulesProof (⇔)

    ↳22 QCSDP

    ↳23 QCSDPMuMonotonicPoloProof (⇔)

    ↳24 QCSDP

    ↳25 QCSDependencyGraphProof (⇔)

    ↳26 TRUE

  ↳27 QCSDP

    ↳28 QCSDPSubtermProof (⇔)

    ↳29 QCSDP

    ↳30 QCSDependencyGraphProof (⇔)

    ↳31 TRUE

  ↳32 QCSDP

    ↳33 QCSDPReductionPairProof (⇔)

    ↳34 QCSDP

    ↳35 QCSDependencyGraphProof (⇔)

    ↳36 TRUE

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0, IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
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(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

(1) QTRSToCSRProof (EQUIVALENT transformation)

The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U101(tt, V1, V2)) → mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) → mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) → mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) → mark(U105(isNat(V1), V2))
active(U105(tt, V2)) → mark(U106(isNatIList(V2)))
active(U106(tt)) → mark(tt)
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) → mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) → mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) → mark(U114(isNatKind(N), L))
active(U114(tt, L)) → mark(s(length(L)))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U121(tt, IL)) → mark(U122(isNatIListKind(IL)))
active(U122(tt)) → mark(nil)
active(U13(tt)) → mark(tt)
active(U131(tt, IL, M, N)) → mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) → mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) → mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) → mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) → mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIListKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt)) → mark(tt)
active(U91(tt, V1, V2)) → mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) → mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) → mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) → mark(U95(isNat(V1), V2))
active(U95(tt, V2)) → mark(U96(isNatList(V2)))
active(U96(tt)) → mark(tt)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U81(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) → mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U111(isNatList(L), L, N))
active(take(0, IL)) → mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) → mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(U105(X1, X2)) → U105(active(X1), X2)
active(U106(X)) → U106(active(X))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U111(X1, X2, X3)) → U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) → U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) → U113(active(X1), X2, X3)
active(U114(X1, X2)) → U114(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U13(X)) → U13(active(X))
active(U121(X1, X2)) → U121(active(X1), X2)
active(U122(X)) → U122(active(X))
active(U131(X1, X2, X3, X4)) → U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) → U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) → U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) → U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) → U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) → U136(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X)) → U71(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) → U94(active(X1), X2, X3)
active(U95(X1, X2)) → U95(active(X1), X2)
active(U96(X)) → U96(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U105(mark(X1), X2) → mark(U105(X1, X2))
U106(mark(X)) → mark(U106(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U111(mark(X1), X2, X3) → mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) → mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) → mark(U113(X1, X2, X3))
U114(mark(X1), X2) → mark(U114(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U13(mark(X)) → mark(U13(X))
U121(mark(X1), X2) → mark(U121(X1, X2))
U122(mark(X)) → mark(U122(X))
U131(mark(X1), X2, X3, X4) → mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) → mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) → mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) → mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) → mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) → mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) → mark(U94(X1, X2, X3))
U95(mark(X1), X2) → mark(U95(X1, X2))
U96(mark(X)) → mark(U96(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) → U105(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U106(X)) → U106(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) → U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) → U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) → U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) → U114(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U121(X1, X2)) → U121(proper(X1), proper(X2))
proper(U122(X)) → U122(proper(X))
proper(nil) → ok(nil)
proper(U131(X1, X2, X3, X4)) → U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) → U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) → U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) → U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) → U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) → U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
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(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X)) → U81(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) → U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) → U95(proper(X1), proper(X2))
proper(U96(X)) → U96(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) → ok(U105(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U106(ok(X)) → ok(U106(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) → ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) → ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) → ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) → ok(U114(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U121(ok(X1), ok(X2)) → ok(U121(X1, X2))
U122(ok(X)) → ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) → ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) → ok(U95(X1, X2))
U96(ok(X)) → ok(U96(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
Special symbols used for the transformation (see [GM04]):
top: top, active: active, mark: mark, ok: ok, proper: proper
The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U101: {1}
tt: empty set
U102: {1}
isNatKind: empty set
U103: {1}
isNatIListKind: empty set
U104: {1}
U105: {1}
isNat: empty set
U106: {1}
isNatIList: empty set
U11: {1}
U12: {1}
U111: {1}
U112: {1}
U113: {1}
U114: {1}
s: {1}
length: {1}
U13: {1}
isNatList: empty set
U121: {1}
U122: {1}
nil: empty set
U131: {1}
U132: {1}
U133: {1}
U134: {1}
U135: {1}
U136: {1}
take: {1, 2}
U21: {1}
U22: {1}
U23: {1}
U31: {1}
U32: {1}
U33: {1}
U41: {1}
U42: {1}
U43: {1}
U44: {1}
U45: {1}
U46: {1}
U51: {1}
U52: {1}
U61: {1}
U62: {1}
U71: {1}
U81: {1}
U91: {1}
U92: {1}
U93: {1}
U94: {1}
U95: {1}
U96: {1}
The QTRS contained all rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is complete (and sound).

(2) Obligation:

Context-sensitive rewrite system:
The TRS R consists of the following rules:

zeros → cons(0, zeros)
U101(tt, V1, V2) → U102(isNatKind(V1), V1, V2)
U102(tt, V1, V2) → U103(isNatIListKind(V2), V1, V2)
U103(tt, V1, V2) → U104(isNatIListKind(V2), V1, V2)
U104(tt, V1, V2) → U105(isNat(V1), V2)
U105(tt, V2) → U106(isNatIList(V2))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(V1), V1)
U111(tt, L, N) → U112(isNatIListKind(L), L, N)
U112(tt, L, N) → U113(isNat(N), L, N)
U113(tt, L, N) → U114(isNatKind(N), L)
U114(tt, L) → s(length(L))
U12(tt, V1) → U13(isNatList(V1))
U121(tt, IL) → U122(isNatIListKind(IL))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(IL), IL, M, N)
U132(tt, IL, M, N) → U133(isNat(M), IL, M, N)
U133(tt, IL, M, N) → U134(isNatKind(M), IL, M, N)
U134(tt, IL, M, N) → U135(isNat(N), IL, M, N)
U135(tt, IL, M, N) → U136(isNatKind(N), IL, M, N)
U136(tt, IL, M, N) → cons(N, take(M, IL))
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(V), V)
U32(tt, V) → U33(isNatList(V))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(V1), V1, V2)
U42(tt, V1, V2) → U43(isNatIListKind(V2), V1, V2)
U43(tt, V1, V2) → U44(isNatIListKind(V2), V1, V2)
U44(tt, V1, V2) → U45(isNat(V1), V2)
U45(tt, V2) → U46(isNatIList(V2))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(V2))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(V2))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(V1), V1, V2)
U92(tt, V1, V2) → U93(isNatIListKind(V2), V1, V2)
U93(tt, V1, V2) → U94(isNatIListKind(V2), V1, V2)
U94(tt, V1, V2) → U95(isNat(V1), V2)
U95(tt, V2) → U96(isNatList(V2))
U96(tt) → tt
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(V) → U31(isNatIListKind(V), V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatIListKind(take(V1, V2)) → U61(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U71(isNatIListKind(V1))
isNatKind(s(V1)) → U81(isNatKind(V1))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U91(isNatKind(V1), V1, V2)
isNatList(take(V1, V2)) → U101(isNatKind(V1), V1, V2)
length(nil) → 0
length(cons(N, L)) → U111(isNatList(L), L, N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(IL), IL, M, N)

The replacement map contains the following entries:

(3) PoloCSRProof (EQUIVALENT transformation)

The following rules can be removed because they are oriented strictly by a µ-monotonic polynomial ordering:

length(nil) → 0

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(U101(x₁, x₂, x₃)) = x₁
POL(U102(x₁, x₂, x₃)) = 2·x₁
POL(U103(x₁, x₂, x₃)) = x₁
POL(U104(x₁, x₂, x₃)) = x₁
POL(U105(x₁, x₂)) = 2·x₁
POL(U106(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁
POL(U111(x₁, x₂, x₃)) = 2 + 2·x₁ + x₂ + 2·x₃
POL(U112(x₁, x₂, x₃)) = 2 + 2·x₁ + x₂ + 2·x₃
POL(U113(x₁, x₂, x₃)) = 2 + 2·x₁ + x₂ + 2·x₃
POL(U114(x₁, x₂)) = 2 + x₁ + x₂
POL(U12(x₁, x₂)) = 2·x₁
POL(U121(x₁, x₂)) = 2·x₁ + x₂
POL(U122(x₁)) = 2·x₁
POL(U13(x₁)) = x₁
POL(U131(x₁, x₂, x₃, x₄)) = 2·x₁ + 2·x₂ + 2·x₃ + 2·x₄
POL(U132(x₁, x₂, x₃, x₄)) = 2·x₁ + 2·x₂ + 2·x₃ + 2·x₄
POL(U133(x₁, x₂, x₃, x₄)) = x₁ + 2·x₂ + 2·x₃ + 2·x₄
POL(U134(x₁, x₂, x₃, x₄)) = 2·x₁ + 2·x₂ + 2·x₃ + 2·x₄
POL(U135(x₁, x₂, x₃, x₄)) = 2·x₁ + 2·x₂ + 2·x₃ + 2·x₄
POL(U136(x₁, x₂, x₃, x₄)) = 2·x₁ + 2·x₂ + 2·x₃ + 2·x₄
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁, x₂)) = 2·x₁
POL(U23(x₁)) = 2·x₁
POL(U31(x₁, x₂)) = 2·x₁
POL(U32(x₁, x₂)) = x₁
POL(U33(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = 2·x₁
POL(U42(x₁, x₂, x₃)) = 2·x₁
POL(U43(x₁, x₂, x₃)) = x₁
POL(U44(x₁, x₂, x₃)) = 2·x₁
POL(U45(x₁, x₂)) = x₁
POL(U46(x₁)) = 2·x₁
POL(U51(x₁, x₂)) = 2·x₁
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂)) = x₁
POL(U62(x₁)) = 2·x₁
POL(U71(x₁)) = x₁
POL(U81(x₁)) = 2·x₁
POL(U91(x₁, x₂, x₃)) = 2·x₁
POL(U92(x₁, x₂, x₃)) = 2·x₁
POL(U93(x₁, x₂, x₃)) = x₁
POL(U94(x₁, x₂, x₃)) = x₁
POL(U95(x₁, x₂)) = 2·x₁
POL(U96(x₁)) = x₁
POL(cons(x₁, x₂)) = 2·x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 2 + x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 2·x₁ + 2·x₂
POL(tt) = 0
POL(zeros) = 0

(4) Obligation:

Context-sensitive rewrite system:
The TRS R consists of the following rules:

zeros → cons(0, zeros)
U101(tt, V1, V2) → U102(isNatKind(V1), V1, V2)
U102(tt, V1, V2) → U103(isNatIListKind(V2), V1, V2)
U103(tt, V1, V2) → U104(isNatIListKind(V2), V1, V2)
U104(tt, V1, V2) → U105(isNat(V1), V2)
U105(tt, V2) → U106(isNatIList(V2))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(V1), V1)
U111(tt, L, N) → U112(isNatIListKind(L), L, N)
U112(tt, L, N) → U113(isNat(N), L, N)
U113(tt, L, N) → U114(isNatKind(N), L)
U114(tt, L) → s(length(L))
U12(tt, V1) → U13(isNatList(V1))
U121(tt, IL) → U122(isNatIListKind(IL))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(IL), IL, M, N)
U132(tt, IL, M, N) → U133(isNat(M), IL, M, N)
U133(tt, IL, M, N) → U134(isNatKind(M), IL, M, N)
U134(tt, IL, M, N) → U135(isNat(N), IL, M, N)
U135(tt, IL, M, N) → U136(isNatKind(N), IL, M, N)
U136(tt, IL, M, N) → cons(N, take(M, IL))
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(V), V)
U32(tt, V) → U33(isNatList(V))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(V1), V1, V2)
U42(tt, V1, V2) → U43(isNatIListKind(V2), V1, V2)
U43(tt, V1, V2) → U44(isNatIListKind(V2), V1, V2)
U44(tt, V1, V2) → U45(isNat(V1), V2)
U45(tt, V2) → U46(isNatIList(V2))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(V2))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(V2))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(V1), V1, V2)
U92(tt, V1, V2) → U93(isNatIListKind(V2), V1, V2)
U93(tt, V1, V2) → U94(isNatIListKind(V2), V1, V2)
U94(tt, V1, V2) → U95(isNat(V1), V2)
U95(tt, V2) → U96(isNatList(V2))
U96(tt) → tt
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(V) → U31(isNatIListKind(V), V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatIListKind(take(V1, V2)) → U61(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U71(isNatIListKind(V1))
isNatKind(s(V1)) → U81(isNatKind(V1))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U91(isNatKind(V1), V1, V2)
isNatList(take(V1, V2)) → U101(isNatKind(V1), V1, V2)
length(cons(N, L)) → U111(isNatList(L), L, N)
take(0, IL) → U121(isNatIList(IL), IL)
take(s(M), cons(N, IL)) → U131(isNatIList(IL), IL, M, N)

The replacement map contains the following entries:

(5) PoloCSRProof (EQUIVALENT transformation)

The following rules can be removed because they are oriented strictly by a µ-monotonic polynomial ordering:

take(0, IL) → U121(isNatIList(IL), IL)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(U101(x₁, x₂, x₃)) = 2·x₁
POL(U102(x₁, x₂, x₃)) = 2·x₁
POL(U103(x₁, x₂, x₃)) = x₁
POL(U104(x₁, x₂, x₃)) = x₁
POL(U105(x₁, x₂)) = x₁
POL(U106(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁
POL(U111(x₁, x₂, x₃)) = 2·x₁ + x₂
POL(U112(x₁, x₂, x₃)) = 2·x₁ + x₂
POL(U113(x₁, x₂, x₃)) = 2·x₁ + x₂
POL(U114(x₁, x₂)) = 2·x₁ + x₂
POL(U12(x₁, x₂)) = 2·x₁
POL(U121(x₁, x₂)) = 2·x₁
POL(U122(x₁)) = 2·x₁
POL(U13(x₁)) = 2·x₁
POL(U131(x₁, x₂, x₃, x₄)) = 1 + 2·x₁ + 2·x₂ + 2·x₃ + 2·x₄
POL(U132(x₁, x₂, x₃, x₄)) = 1 + 2·x₁ + 2·x₂ + 2·x₃ + 2·x₄
POL(U133(x₁, x₂, x₃, x₄)) = 1 + 2·x₁ + 2·x₂ + 2·x₃ + 2·x₄
POL(U134(x₁, x₂, x₃, x₄)) = 1 + 2·x₁ + 2·x₂ + 2·x₃ + 2·x₄
POL(U135(x₁, x₂, x₃, x₄)) = 1 + 2·x₁ + 2·x₂ + 2·x₃ + 2·x₄
POL(U136(x₁, x₂, x₃, x₄)) = 1 + 2·x₁ + 2·x₂ + 2·x₃ + 2·x₄
POL(U21(x₁, x₂)) = 2·x₁
POL(U22(x₁, x₂)) = x₁
POL(U23(x₁)) = 2·x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁, x₂)) = 2·x₁
POL(U33(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = 2·x₁
POL(U42(x₁, x₂, x₃)) = 2·x₁
POL(U43(x₁, x₂, x₃)) = x₁
POL(U44(x₁, x₂, x₃)) = 2·x₁
POL(U45(x₁, x₂)) = 2·x₁
POL(U46(x₁)) = x₁
POL(U51(x₁, x₂)) = 2·x₁
POL(U52(x₁)) = 2·x₁
POL(U61(x₁, x₂)) = 2·x₁
POL(U62(x₁)) = x₁
POL(U71(x₁)) = 2·x₁
POL(U81(x₁)) = 2·x₁
POL(U91(x₁, x₂, x₃)) = 2·x₁
POL(U92(x₁, x₂, x₃)) = x₁
POL(U93(x₁, x₂, x₃)) = 2·x₁
POL(U94(x₁, x₂, x₃)) = x₁
POL(U95(x₁, x₂)) = 2·x₁
POL(U96(x₁)) = 2·x₁
POL(cons(x₁, x₂)) = 2·x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(tt) = 0
POL(zeros) = 0

(6) Obligation:

Context-sensitive rewrite system:
The TRS R consists of the following rules:

zeros → cons(0, zeros)
U101(tt, V1, V2) → U102(isNatKind(V1), V1, V2)
U102(tt, V1, V2) → U103(isNatIListKind(V2), V1, V2)
U103(tt, V1, V2) → U104(isNatIListKind(V2), V1, V2)
U104(tt, V1, V2) → U105(isNat(V1), V2)
U105(tt, V2) → U106(isNatIList(V2))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(V1), V1)
U111(tt, L, N) → U112(isNatIListKind(L), L, N)
U112(tt, L, N) → U113(isNat(N), L, N)
U113(tt, L, N) → U114(isNatKind(N), L)
U114(tt, L) → s(length(L))
U12(tt, V1) → U13(isNatList(V1))
U121(tt, IL) → U122(isNatIListKind(IL))
U122(tt) → nil
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(IL), IL, M, N)
U132(tt, IL, M, N) → U133(isNat(M), IL, M, N)
U133(tt, IL, M, N) → U134(isNatKind(M), IL, M, N)
U134(tt, IL, M, N) → U135(isNat(N), IL, M, N)
U135(tt, IL, M, N) → U136(isNatKind(N), IL, M, N)
U136(tt, IL, M, N) → cons(N, take(M, IL))
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(V), V)
U32(tt, V) → U33(isNatList(V))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(V1), V1, V2)
U42(tt, V1, V2) → U43(isNatIListKind(V2), V1, V2)
U43(tt, V1, V2) → U44(isNatIListKind(V2), V1, V2)
U44(tt, V1, V2) → U45(isNat(V1), V2)
U45(tt, V2) → U46(isNatIList(V2))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(V2))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(V2))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(V1), V1, V2)
U92(tt, V1, V2) → U93(isNatIListKind(V2), V1, V2)
U93(tt, V1, V2) → U94(isNatIListKind(V2), V1, V2)
U94(tt, V1, V2) → U95(isNat(V1), V2)
U95(tt, V2) → U96(isNatList(V2))
U96(tt) → tt
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(V) → U31(isNatIListKind(V), V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatIListKind(take(V1, V2)) → U61(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U71(isNatIListKind(V1))
isNatKind(s(V1)) → U81(isNatKind(V1))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U91(isNatKind(V1), V1, V2)
isNatList(take(V1, V2)) → U101(isNatKind(V1), V1, V2)
length(cons(N, L)) → U111(isNatList(L), L, N)
take(s(M), cons(N, IL)) → U131(isNatIList(IL), IL, M, N)

The replacement map contains the following entries:

(7) PoloCSRProof (EQUIVALENT transformation)

The following rules can be removed because they are oriented strictly by a µ-monotonic polynomial ordering:

U121(tt, IL) → U122(isNatIListKind(IL))
U122(tt) → nil

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(U101(x₁, x₂, x₃)) = x₁
POL(U102(x₁, x₂, x₃)) = x₁
POL(U103(x₁, x₂, x₃)) = x₁
POL(U104(x₁, x₂, x₃)) = x₁
POL(U105(x₁, x₂)) = x₁
POL(U106(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁
POL(U111(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + 2·x₃
POL(U112(x₁, x₂, x₃)) = 2·x₁ + 2·x₂
POL(U113(x₁, x₂, x₃)) = 2·x₁ + 2·x₂
POL(U114(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U12(x₁, x₂)) = 2·x₁
POL(U121(x₁, x₂)) = 2 + x₁
POL(U122(x₁)) = 1 + 2·x₁
POL(U13(x₁)) = 2·x₁
POL(U131(x₁, x₂, x₃, x₄)) = 2·x₁ + x₂ + x₃ + 2·x₄
POL(U132(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + 2·x₄
POL(U133(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + 2·x₄
POL(U134(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + 2·x₄
POL(U135(x₁, x₂, x₃, x₄)) = 2·x₁ + x₂ + x₃ + 2·x₄
POL(U136(x₁, x₂, x₃, x₄)) = 2·x₁ + x₂ + x₃ + 2·x₄
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁, x₂)) = 2·x₁
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = 2·x₁
POL(U32(x₁, x₂)) = 2·x₁
POL(U33(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = 2·x₁
POL(U42(x₁, x₂, x₃)) = x₁
POL(U43(x₁, x₂, x₃)) = x₁
POL(U44(x₁, x₂, x₃)) = 2·x₁
POL(U45(x₁, x₂)) = x₁
POL(U46(x₁)) = 2·x₁
POL(U51(x₁, x₂)) = 2·x₁
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂)) = 2·x₁
POL(U62(x₁)) = x₁
POL(U71(x₁)) = x₁
POL(U81(x₁)) = x₁
POL(U91(x₁, x₂, x₃)) = 2·x₁
POL(U92(x₁, x₂, x₃)) = 2·x₁
POL(U93(x₁, x₂, x₃)) = x₁
POL(U94(x₁, x₂, x₃)) = 2·x₁
POL(U95(x₁, x₂)) = 2·x₁
POL(U96(x₁)) = x₁
POL(cons(x₁, x₂)) = 2·x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 2·x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = x₁ + x₂
POL(tt) = 0
POL(zeros) = 0

(8) Obligation:

Context-sensitive rewrite system:
The TRS R consists of the following rules:

zeros → cons(0, zeros)
U101(tt, V1, V2) → U102(isNatKind(V1), V1, V2)
U102(tt, V1, V2) → U103(isNatIListKind(V2), V1, V2)
U103(tt, V1, V2) → U104(isNatIListKind(V2), V1, V2)
U104(tt, V1, V2) → U105(isNat(V1), V2)
U105(tt, V2) → U106(isNatIList(V2))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(V1), V1)
U111(tt, L, N) → U112(isNatIListKind(L), L, N)
U112(tt, L, N) → U113(isNat(N), L, N)
U113(tt, L, N) → U114(isNatKind(N), L)
U114(tt, L) → s(length(L))
U12(tt, V1) → U13(isNatList(V1))
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(IL), IL, M, N)
U132(tt, IL, M, N) → U133(isNat(M), IL, M, N)
U133(tt, IL, M, N) → U134(isNatKind(M), IL, M, N)
U134(tt, IL, M, N) → U135(isNat(N), IL, M, N)
U135(tt, IL, M, N) → U136(isNatKind(N), IL, M, N)
U136(tt, IL, M, N) → cons(N, take(M, IL))
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(V), V)
U32(tt, V) → U33(isNatList(V))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(V1), V1, V2)
U42(tt, V1, V2) → U43(isNatIListKind(V2), V1, V2)
U43(tt, V1, V2) → U44(isNatIListKind(V2), V1, V2)
U44(tt, V1, V2) → U45(isNat(V1), V2)
U45(tt, V2) → U46(isNatIList(V2))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(V2))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(V2))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(V1), V1, V2)
U92(tt, V1, V2) → U93(isNatIListKind(V2), V1, V2)
U93(tt, V1, V2) → U94(isNatIListKind(V2), V1, V2)
U94(tt, V1, V2) → U95(isNat(V1), V2)
U95(tt, V2) → U96(isNatList(V2))
U96(tt) → tt
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(V) → U31(isNatIListKind(V), V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatIListKind(take(V1, V2)) → U61(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U71(isNatIListKind(V1))
isNatKind(s(V1)) → U81(isNatKind(V1))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U91(isNatKind(V1), V1, V2)
isNatList(take(V1, V2)) → U101(isNatKind(V1), V1, V2)
length(cons(N, L)) → U111(isNatList(L), L, N)
take(s(M), cons(N, IL)) → U131(isNatIList(IL), IL, M, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U101: {1}
tt: empty set
U102: {1}
isNatKind: empty set
U103: {1}
isNatIListKind: empty set
U104: {1}
U105: {1}
isNat: empty set
U106: {1}
isNatIList: empty set
U11: {1}
U12: {1}
U111: {1}
U112: {1}
U113: {1}
U114: {1}
s: {1}
length: {1}
U13: {1}
isNatList: empty set
nil: empty set
U131: {1}
U132: {1}
U133: {1}
U134: {1}
U135: {1}
U136: {1}
take: {1, 2}
U21: {1}
U22: {1}
U23: {1}
U31: {1}
U32: {1}
U33: {1}
U41: {1}
U42: {1}
U43: {1}
U44: {1}
U45: {1}
U46: {1}
U51: {1}
U52: {1}
U61: {1}
U62: {1}
U71: {1}
U81: {1}
U91: {1}
U92: {1}
U93: {1}
U94: {1}
U95: {1}
U96: {1}

(9) CSDependencyPairsProof (EQUIVALENT transformation)

Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.

(10) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {U106, s, length, U13, take, U23, U33, U46, U52, U62, U71, U81, U96, U106', LENGTH, U13', U23', U33', U46', U52', U62', U96', U71', U81', TAKE} are replacing on all positions.
For all symbols f in {cons, U101, U102, U103, U104, U105, U11, U12, U111, U112, U113, U114, U131, U132, U133, U134, U135, U136, U21, U22, U31, U32, U41, U42, U43, U44, U45, U51, U61, U91, U92, U93, U94, U95, U102', U101', U103', U104', U105', U12', U11', U112', U111', U113', U114', U132', U131', U133', U134', U135', U136', U22', U21', U32', U31', U42', U41', U43', U44', U45', U51', U61', U92', U91', U93', U94', U95'} we have µ(f) = {1}.
The symbols in {isNatKind, isNatIListKind, isNat, isNatIList, isNatList, ISNATKIND, ISNATILISTKIND, ISNAT, ISNATILIST, ISNATLIST, U} are not replacing on any position.

The ordinary context-sensitive dependency pairs DP_o are:

U101'(tt, V1, V2) → U102'(isNatKind(V1), V1, V2)
U101'(tt, V1, V2) → ISNATKIND(V1)
U102'(tt, V1, V2) → U103'(isNatIListKind(V2), V1, V2)
U102'(tt, V1, V2) → ISNATILISTKIND(V2)
U103'(tt, V1, V2) → U104'(isNatIListKind(V2), V1, V2)
U103'(tt, V1, V2) → ISNATILISTKIND(V2)
U104'(tt, V1, V2) → U105'(isNat(V1), V2)
U104'(tt, V1, V2) → ISNAT(V1)
U105'(tt, V2) → U106'(isNatIList(V2))
U105'(tt, V2) → ISNATILIST(V2)
U11'(tt, V1) → U12'(isNatIListKind(V1), V1)
U11'(tt, V1) → ISNATILISTKIND(V1)
U111'(tt, L, N) → U112'(isNatIListKind(L), L, N)
U111'(tt, L, N) → ISNATILISTKIND(L)
U112'(tt, L, N) → U113'(isNat(N), L, N)
U112'(tt, L, N) → ISNAT(N)
U113'(tt, L, N) → U114'(isNatKind(N), L)
U113'(tt, L, N) → ISNATKIND(N)
U114'(tt, L) → LENGTH(L)
U12'(tt, V1) → U13'(isNatList(V1))
U12'(tt, V1) → ISNATLIST(V1)
U131'(tt, IL, M, N) → U132'(isNatIListKind(IL), IL, M, N)
U131'(tt, IL, M, N) → ISNATILISTKIND(IL)
U132'(tt, IL, M, N) → U133'(isNat(M), IL, M, N)
U132'(tt, IL, M, N) → ISNAT(M)
U133'(tt, IL, M, N) → U134'(isNatKind(M), IL, M, N)
U133'(tt, IL, M, N) → ISNATKIND(M)
U134'(tt, IL, M, N) → U135'(isNat(N), IL, M, N)
U134'(tt, IL, M, N) → ISNAT(N)
U135'(tt, IL, M, N) → U136'(isNatKind(N), IL, M, N)
U135'(tt, IL, M, N) → ISNATKIND(N)
U21'(tt, V1) → U22'(isNatKind(V1), V1)
U21'(tt, V1) → ISNATKIND(V1)
U22'(tt, V1) → U23'(isNat(V1))
U22'(tt, V1) → ISNAT(V1)
U31'(tt, V) → U32'(isNatIListKind(V), V)
U31'(tt, V) → ISNATILISTKIND(V)
U32'(tt, V) → U33'(isNatList(V))
U32'(tt, V) → ISNATLIST(V)
U41'(tt, V1, V2) → U42'(isNatKind(V1), V1, V2)
U41'(tt, V1, V2) → ISNATKIND(V1)
U42'(tt, V1, V2) → U43'(isNatIListKind(V2), V1, V2)
U42'(tt, V1, V2) → ISNATILISTKIND(V2)
U43'(tt, V1, V2) → U44'(isNatIListKind(V2), V1, V2)
U43'(tt, V1, V2) → ISNATILISTKIND(V2)
U44'(tt, V1, V2) → U45'(isNat(V1), V2)
U44'(tt, V1, V2) → ISNAT(V1)
U45'(tt, V2) → U46'(isNatIList(V2))
U45'(tt, V2) → ISNATILIST(V2)
U51'(tt, V2) → U52'(isNatIListKind(V2))
U51'(tt, V2) → ISNATILISTKIND(V2)
U61'(tt, V2) → U62'(isNatIListKind(V2))
U61'(tt, V2) → ISNATILISTKIND(V2)
U91'(tt, V1, V2) → U92'(isNatKind(V1), V1, V2)
U91'(tt, V1, V2) → ISNATKIND(V1)
U92'(tt, V1, V2) → U93'(isNatIListKind(V2), V1, V2)
U92'(tt, V1, V2) → ISNATILISTKIND(V2)
U93'(tt, V1, V2) → U94'(isNatIListKind(V2), V1, V2)
U93'(tt, V1, V2) → ISNATILISTKIND(V2)
U94'(tt, V1, V2) → U95'(isNat(V1), V2)
U94'(tt, V1, V2) → ISNAT(V1)
U95'(tt, V2) → U96'(isNatList(V2))
U95'(tt, V2) → ISNATLIST(V2)
ISNAT(length(V1)) → U11'(isNatIListKind(V1), V1)
ISNAT(length(V1)) → ISNATILISTKIND(V1)
ISNAT(s(V1)) → U21'(isNatKind(V1), V1)
ISNAT(s(V1)) → ISNATKIND(V1)
ISNATILIST(V) → U31'(isNatIListKind(V), V)
ISNATILIST(V) → ISNATILISTKIND(V)
ISNATILIST(cons(V1, V2)) → U41'(isNatKind(V1), V1, V2)
ISNATILIST(cons(V1, V2)) → ISNATKIND(V1)
ISNATILISTKIND(cons(V1, V2)) → U51'(isNatKind(V1), V2)
ISNATILISTKIND(cons(V1, V2)) → ISNATKIND(V1)
ISNATILISTKIND(take(V1, V2)) → U61'(isNatKind(V1), V2)
ISNATILISTKIND(take(V1, V2)) → ISNATKIND(V1)
ISNATKIND(length(V1)) → U71'(isNatIListKind(V1))
ISNATKIND(length(V1)) → ISNATILISTKIND(V1)
ISNATKIND(s(V1)) → U81'(isNatKind(V1))
ISNATKIND(s(V1)) → ISNATKIND(V1)
ISNATLIST(cons(V1, V2)) → U91'(isNatKind(V1), V1, V2)
ISNATLIST(cons(V1, V2)) → ISNATKIND(V1)
ISNATLIST(take(V1, V2)) → U101'(isNatKind(V1), V1, V2)
ISNATLIST(take(V1, V2)) → ISNATKIND(V1)
LENGTH(cons(N, L)) → U111'(isNatList(L), L, N)
LENGTH(cons(N, L)) → ISNATLIST(L)
TAKE(s(M), cons(N, IL)) → U131'(isNatIList(IL), IL, M, N)
TAKE(s(M), cons(N, IL)) → ISNATILIST(IL)

The collapsing dependency pairs are DP_c:

U114'(tt, L) → L
U136'(tt, IL, M, N) → N

The hidden terms of R are:

zeros
take(x0, x1)

Every hiding context is built from:

take on positions {1, 2}

Hence, the new unhiding pairs DP_u are :

U114'(tt, L) → U(L)
U136'(tt, IL, M, N) → U(N)
U(take(x_0, x_1)) → U(x_0)
U(take(x_0, x_1)) → U(x_1)
U(zeros) → ZEROS
U(take(x0, x1)) → TAKE(x0, x1)

The TRS R consists of the following rules:

zeros → cons(0, zeros)
U101(tt, V1, V2) → U102(isNatKind(V1), V1, V2)
U102(tt, V1, V2) → U103(isNatIListKind(V2), V1, V2)
U103(tt, V1, V2) → U104(isNatIListKind(V2), V1, V2)
U104(tt, V1, V2) → U105(isNat(V1), V2)
U105(tt, V2) → U106(isNatIList(V2))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(V1), V1)
U111(tt, L, N) → U112(isNatIListKind(L), L, N)
U112(tt, L, N) → U113(isNat(N), L, N)
U113(tt, L, N) → U114(isNatKind(N), L)
U114(tt, L) → s(length(L))
U12(tt, V1) → U13(isNatList(V1))
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(IL), IL, M, N)
U132(tt, IL, M, N) → U133(isNat(M), IL, M, N)
U133(tt, IL, M, N) → U134(isNatKind(M), IL, M, N)
U134(tt, IL, M, N) → U135(isNat(N), IL, M, N)
U135(tt, IL, M, N) → U136(isNatKind(N), IL, M, N)
U136(tt, IL, M, N) → cons(N, take(M, IL))
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(V), V)
U32(tt, V) → U33(isNatList(V))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(V1), V1, V2)
U42(tt, V1, V2) → U43(isNatIListKind(V2), V1, V2)
U43(tt, V1, V2) → U44(isNatIListKind(V2), V1, V2)
U44(tt, V1, V2) → U45(isNat(V1), V2)
U45(tt, V2) → U46(isNatIList(V2))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(V2))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(V2))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(V1), V1, V2)
U92(tt, V1, V2) → U93(isNatIListKind(V2), V1, V2)
U93(tt, V1, V2) → U94(isNatIListKind(V2), V1, V2)
U94(tt, V1, V2) → U95(isNat(V1), V2)
U95(tt, V2) → U96(isNatList(V2))
U96(tt) → tt
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(V) → U31(isNatIListKind(V), V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatIListKind(take(V1, V2)) → U61(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U71(isNatIListKind(V1))
isNatKind(s(V1)) → U81(isNatKind(V1))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U91(isNatKind(V1), V1, V2)
isNatList(take(V1, V2)) → U101(isNatKind(V1), V1, V2)
length(cons(N, L)) → U111(isNatList(L), L, N)
take(s(M), cons(N, IL)) → U131(isNatIList(IL), IL, M, N)

Q is empty.

(11) QCSDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 4 SCCs with 40 less nodes.

(12) Complex Obligation (AND)

(13) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {U106, s, length, U13, take, U23, U33, U46, U52, U62, U71, U81, U96} are replacing on all positions.
For all symbols f in {cons, U101, U102, U103, U104, U105, U11, U12, U111, U112, U113, U114, U131, U132, U133, U134, U135, U136, U21, U22, U31, U32, U41, U42, U43, U44, U45, U51, U61, U91, U92, U93, U94, U95, U51', U61'} we have µ(f) = {1}.
The symbols in {isNatKind, isNatIListKind, isNat, isNatIList, isNatList, ISNATILISTKIND, ISNATKIND} are not replacing on any position.

The TRS P consists of the following rules:

ISNATKIND(length(V1)) → ISNATILISTKIND(V1)
ISNATILISTKIND(cons(V1, V2)) → U51'(isNatKind(V1), V2)
U51'(tt, V2) → ISNATILISTKIND(V2)
ISNATILISTKIND(cons(V1, V2)) → ISNATKIND(V1)
ISNATKIND(s(V1)) → ISNATKIND(V1)
ISNATILISTKIND(take(V1, V2)) → U61'(isNatKind(V1), V2)
U61'(tt, V2) → ISNATILISTKIND(V2)
ISNATILISTKIND(take(V1, V2)) → ISNATKIND(V1)

The TRS R consists of the following rules:

zeros → cons(0, zeros)
U101(tt, V1, V2) → U102(isNatKind(V1), V1, V2)
U102(tt, V1, V2) → U103(isNatIListKind(V2), V1, V2)
U103(tt, V1, V2) → U104(isNatIListKind(V2), V1, V2)
U104(tt, V1, V2) → U105(isNat(V1), V2)
U105(tt, V2) → U106(isNatIList(V2))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(V1), V1)
U111(tt, L, N) → U112(isNatIListKind(L), L, N)
U112(tt, L, N) → U113(isNat(N), L, N)
U113(tt, L, N) → U114(isNatKind(N), L)
U114(tt, L) → s(length(L))
U12(tt, V1) → U13(isNatList(V1))
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(IL), IL, M, N)
U132(tt, IL, M, N) → U133(isNat(M), IL, M, N)
U133(tt, IL, M, N) → U134(isNatKind(M), IL, M, N)
U134(tt, IL, M, N) → U135(isNat(N), IL, M, N)
U135(tt, IL, M, N) → U136(isNatKind(N), IL, M, N)
U136(tt, IL, M, N) → cons(N, take(M, IL))
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(V), V)
U32(tt, V) → U33(isNatList(V))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(V1), V1, V2)
U42(tt, V1, V2) → U43(isNatIListKind(V2), V1, V2)
U43(tt, V1, V2) → U44(isNatIListKind(V2), V1, V2)
U44(tt, V1, V2) → U45(isNat(V1), V2)
U45(tt, V2) → U46(isNatIList(V2))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(V2))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(V2))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(V1), V1, V2)
U92(tt, V1, V2) → U93(isNatIListKind(V2), V1, V2)
U93(tt, V1, V2) → U94(isNatIListKind(V2), V1, V2)
U94(tt, V1, V2) → U95(isNat(V1), V2)
U95(tt, V2) → U96(isNatList(V2))
U96(tt) → tt
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(V) → U31(isNatIListKind(V), V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatIListKind(take(V1, V2)) → U61(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U71(isNatIListKind(V1))
isNatKind(s(V1)) → U81(isNatKind(V1))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U91(isNatKind(V1), V1, V2)
isNatList(take(V1, V2)) → U101(isNatKind(V1), V1, V2)
length(cons(N, L)) → U111(isNatList(L), L, N)
take(s(M), cons(N, IL)) → U131(isNatIList(IL), IL, M, N)

Q is empty.

(14) QCSUsableRulesProof (EQUIVALENT transformation)

The following rules are not useable [DA_EMMES] and can be deleted:

zeros → cons(0, zeros)
U101(tt, x0, x1) → U102(isNatKind(x0), x0, x1)
U102(tt, x0, x1) → U103(isNatIListKind(x1), x0, x1)
U103(tt, x0, x1) → U104(isNatIListKind(x1), x0, x1)
U104(tt, x0, x1) → U105(isNat(x0), x1)
U105(tt, x0) → U106(isNatIList(x0))
U106(tt) → tt
U11(tt, x0) → U12(isNatIListKind(x0), x0)
U111(tt, x0, x1) → U112(isNatIListKind(x0), x0, x1)
U112(tt, x0, x1) → U113(isNat(x1), x0, x1)
U113(tt, x0, x1) → U114(isNatKind(x1), x0)
U114(tt, x0) → s(length(x0))
U12(tt, x0) → U13(isNatList(x0))
U13(tt) → tt
U131(tt, x0, x1, x2) → U132(isNatIListKind(x0), x0, x1, x2)
U132(tt, x0, x1, x2) → U133(isNat(x1), x0, x1, x2)
U133(tt, x0, x1, x2) → U134(isNatKind(x1), x0, x1, x2)
U134(tt, x0, x1, x2) → U135(isNat(x2), x0, x1, x2)
U135(tt, x0, x1, x2) → U136(isNatKind(x2), x0, x1, x2)
U136(tt, x0, x1, x2) → cons(x2, take(x1, x0))
U21(tt, x0) → U22(isNatKind(x0), x0)
U22(tt, x0) → U23(isNat(x0))
U23(tt) → tt
U31(tt, x0) → U32(isNatIListKind(x0), x0)
U32(tt, x0) → U33(isNatList(x0))
U33(tt) → tt
U41(tt, x0, x1) → U42(isNatKind(x0), x0, x1)
U42(tt, x0, x1) → U43(isNatIListKind(x1), x0, x1)
U43(tt, x0, x1) → U44(isNatIListKind(x1), x0, x1)
U44(tt, x0, x1) → U45(isNat(x0), x1)
U45(tt, x0) → U46(isNatIList(x0))
U46(tt) → tt
U91(tt, x0, x1) → U92(isNatKind(x0), x0, x1)
U92(tt, x0, x1) → U93(isNatIListKind(x1), x0, x1)
U93(tt, x0, x1) → U94(isNatIListKind(x1), x0, x1)
U94(tt, x0, x1) → U95(isNat(x0), x1)
U95(tt, x0) → U96(isNatList(x0))
U96(tt) → tt
isNat(0) → tt
isNat(length(x0)) → U11(isNatIListKind(x0), x0)
isNat(s(x0)) → U21(isNatKind(x0), x0)
isNatIList(x0) → U31(isNatIListKind(x0), x0)
isNatIList(zeros) → tt
isNatIList(cons(x0, x1)) → U41(isNatKind(x0), x0, x1)
isNatList(nil) → tt
isNatList(cons(x0, x1)) → U91(isNatKind(x0), x0, x1)
isNatList(take(x0, x1)) → U101(isNatKind(x0), x0, x1)
length(cons(x0, x1)) → U111(isNatList(x1), x1, x0)
take(s(x0), cons(x1, x2)) → U131(isNatIList(x2), x2, x0, x1)

(15) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {length, U71, s, U81, U52, take, U62} are replacing on all positions.
For all symbols f in {cons, U51, U61, U51', U61'} we have µ(f) = {1}.
The symbols in {isNatKind, isNatIListKind, ISNATILISTKIND, ISNATKIND} are not replacing on any position.

The TRS P consists of the following rules:

ISNATKIND(length(V1)) → ISNATILISTKIND(V1)
ISNATILISTKIND(cons(V1, V2)) → U51'(isNatKind(V1), V2)
U51'(tt, V2) → ISNATILISTKIND(V2)
ISNATILISTKIND(cons(V1, V2)) → ISNATKIND(V1)
ISNATKIND(s(V1)) → ISNATKIND(V1)
ISNATILISTKIND(take(V1, V2)) → U61'(isNatKind(V1), V2)
U61'(tt, V2) → ISNATILISTKIND(V2)
ISNATILISTKIND(take(V1, V2)) → ISNATKIND(V1)

The TRS R consists of the following rules:

isNatKind(0) → tt
isNatKind(length(V1)) → U71(isNatIListKind(V1))
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatKind(s(V1)) → U81(isNatKind(V1))
U81(tt) → tt
U51(tt, V2) → U52(isNatIListKind(V2))
isNatIListKind(take(V1, V2)) → U61(isNatKind(V1), V2)
U61(tt, V2) → U62(isNatIListKind(V2))
U62(tt) → tt
U52(tt) → tt
U71(tt) → tt

Q is empty.

(16) QCSDPMuMonotonicPoloProof (EQUIVALENT transformation)

By using the following µ-monotonic polynomial ordering [POLO], at least one Dependency Pair or term rewrite system rule of this Q-CSDP problem can be strictly oriented and thus deleted.
Strictly oriented dependency pairs:

ISNATKIND(length(V1)) → ISNATILISTKIND(V1)
U51'(tt, V2) → ISNATILISTKIND(V2)
ISNATKIND(s(V1)) → ISNATKIND(V1)
ISNATILISTKIND(take(V1, V2)) → U61'(isNatKind(V1), V2)
U61'(tt, V2) → ISNATILISTKIND(V2)
ISNATILISTKIND(take(V1, V2)) → ISNATKIND(V1)

Strictly oriented rules of the TRS R:

isNatKind(0) → tt
isNatKind(length(V1)) → U71(isNatIListKind(V1))
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
U81(tt) → tt
U51(tt, V2) → U52(isNatIListKind(V2))
isNatIListKind(take(V1, V2)) → U61(isNatKind(V1), V2)
U62(tt) → tt

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(0) = 1
POL(ISNATILISTKIND(x₁)) = x₁
POL(ISNATKIND(x₁)) = 1 + 2·x₁
POL(U51(x₁, x₂)) = x₁ + 2·x₂
POL(U51'(x₁, x₂)) = x₁ + x₂
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂)) = x₁ + x₂
POL(U61'(x₁, x₂)) = x₁ + x₂
POL(U62(x₁)) = 1 + x₁
POL(U71(x₁)) = x₁
POL(U81(x₁)) = 2·x₁
POL(cons(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(isNatIListKind(x₁)) = x₁
POL(isNatKind(x₁)) = 1 + x₁
POL(length(x₁)) = 2·x₁
POL(nil) = 2
POL(s(x₁)) = 1 + 2·x₁
POL(take(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(tt) = 1
POL(zeros) = 2

(17) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {s, U81, U62, U52, U71} are replacing on all positions.
For all symbols f in {cons, U51, U61, U51'} we have µ(f) = {1}.
The symbols in {isNatIListKind, isNatKind, ISNATILISTKIND, ISNATKIND} are not replacing on any position.

The TRS P consists of the following rules:

ISNATILISTKIND(cons(V1, V2)) → U51'(isNatKind(V1), V2)
ISNATILISTKIND(cons(V1, V2)) → ISNATKIND(V1)

The TRS R consists of the following rules:

isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatKind(s(V1)) → U81(isNatKind(V1))
U61(tt, V2) → U62(isNatIListKind(V2))
U52(tt) → tt
U71(tt) → tt

Q is empty.

(18) QCSDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs.

(19) TRUE

(20) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {U106, s, length, U13, take, U23, U33, U46, U52, U62, U71, U81, U96} are replacing on all positions.
For all symbols f in {cons, U101, U102, U103, U104, U105, U11, U12, U111, U112, U113, U114, U131, U132, U133, U134, U135, U136, U21, U22, U31, U32, U41, U42, U43, U44, U45, U51, U61, U91, U92, U93, U94, U95, U103', U102', U104', U105', U31', U32', U91', U92', U93', U94', U95', U101', U11', U12', U21', U22', U41', U42', U43', U44', U45'} we have µ(f) = {1}.
The symbols in {isNatKind, isNatIListKind, isNat, isNatIList, isNatList, ISNATILIST, ISNATLIST, ISNAT} are not replacing on any position.

The TRS P consists of the following rules:

U102'(tt, V1, V2) → U103'(isNatIListKind(V2), V1, V2)
U103'(tt, V1, V2) → U104'(isNatIListKind(V2), V1, V2)
U104'(tt, V1, V2) → U105'(isNat(V1), V2)
U105'(tt, V2) → ISNATILIST(V2)
ISNATILIST(V) → U31'(isNatIListKind(V), V)
U31'(tt, V) → U32'(isNatIListKind(V), V)
U32'(tt, V) → ISNATLIST(V)
ISNATLIST(cons(V1, V2)) → U91'(isNatKind(V1), V1, V2)
U91'(tt, V1, V2) → U92'(isNatKind(V1), V1, V2)
U92'(tt, V1, V2) → U93'(isNatIListKind(V2), V1, V2)
U93'(tt, V1, V2) → U94'(isNatIListKind(V2), V1, V2)
U94'(tt, V1, V2) → U95'(isNat(V1), V2)
U95'(tt, V2) → ISNATLIST(V2)
ISNATLIST(take(V1, V2)) → U101'(isNatKind(V1), V1, V2)
U101'(tt, V1, V2) → U102'(isNatKind(V1), V1, V2)
U94'(tt, V1, V2) → ISNAT(V1)
ISNAT(length(V1)) → U11'(isNatIListKind(V1), V1)
U11'(tt, V1) → U12'(isNatIListKind(V1), V1)
U12'(tt, V1) → ISNATLIST(V1)
ISNAT(s(V1)) → U21'(isNatKind(V1), V1)
U21'(tt, V1) → U22'(isNatKind(V1), V1)
U22'(tt, V1) → ISNAT(V1)
ISNATILIST(cons(V1, V2)) → U41'(isNatKind(V1), V1, V2)
U41'(tt, V1, V2) → U42'(isNatKind(V1), V1, V2)
U42'(tt, V1, V2) → U43'(isNatIListKind(V2), V1, V2)
U43'(tt, V1, V2) → U44'(isNatIListKind(V2), V1, V2)
U44'(tt, V1, V2) → U45'(isNat(V1), V2)
U45'(tt, V2) → ISNATILIST(V2)
U44'(tt, V1, V2) → ISNAT(V1)
U104'(tt, V1, V2) → ISNAT(V1)

The TRS R consists of the following rules:

zeros → cons(0, zeros)
U101(tt, V1, V2) → U102(isNatKind(V1), V1, V2)
U102(tt, V1, V2) → U103(isNatIListKind(V2), V1, V2)
U103(tt, V1, V2) → U104(isNatIListKind(V2), V1, V2)
U104(tt, V1, V2) → U105(isNat(V1), V2)
U105(tt, V2) → U106(isNatIList(V2))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(V1), V1)
U111(tt, L, N) → U112(isNatIListKind(L), L, N)
U112(tt, L, N) → U113(isNat(N), L, N)
U113(tt, L, N) → U114(isNatKind(N), L)
U114(tt, L) → s(length(L))
U12(tt, V1) → U13(isNatList(V1))
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(IL), IL, M, N)
U132(tt, IL, M, N) → U133(isNat(M), IL, M, N)
U133(tt, IL, M, N) → U134(isNatKind(M), IL, M, N)
U134(tt, IL, M, N) → U135(isNat(N), IL, M, N)
U135(tt, IL, M, N) → U136(isNatKind(N), IL, M, N)
U136(tt, IL, M, N) → cons(N, take(M, IL))
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(V), V)
U32(tt, V) → U33(isNatList(V))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(V1), V1, V2)
U42(tt, V1, V2) → U43(isNatIListKind(V2), V1, V2)
U43(tt, V1, V2) → U44(isNatIListKind(V2), V1, V2)
U44(tt, V1, V2) → U45(isNat(V1), V2)
U45(tt, V2) → U46(isNatIList(V2))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(V2))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(V2))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(V1), V1, V2)
U92(tt, V1, V2) → U93(isNatIListKind(V2), V1, V2)
U93(tt, V1, V2) → U94(isNatIListKind(V2), V1, V2)
U94(tt, V1, V2) → U95(isNat(V1), V2)
U95(tt, V2) → U96(isNatList(V2))
U96(tt) → tt
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(V) → U31(isNatIListKind(V), V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatIListKind(take(V1, V2)) → U61(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U71(isNatIListKind(V1))
isNatKind(s(V1)) → U81(isNatKind(V1))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U91(isNatKind(V1), V1, V2)
isNatList(take(V1, V2)) → U101(isNatKind(V1), V1, V2)
length(cons(N, L)) → U111(isNatList(L), L, N)
take(s(M), cons(N, IL)) → U131(isNatIList(IL), IL, M, N)

Q is empty.

(21) QCSUsableRulesProof (EQUIVALENT transformation)

The following rules are not useable [DA_EMMES] and can be deleted:

zeros → cons(0, zeros)
U111(tt, x0, x1) → U112(isNatIListKind(x0), x0, x1)
U112(tt, x0, x1) → U113(isNat(x1), x0, x1)
U113(tt, x0, x1) → U114(isNatKind(x1), x0)
U114(tt, x0) → s(length(x0))
U131(tt, x0, x1, x2) → U132(isNatIListKind(x0), x0, x1, x2)
U132(tt, x0, x1, x2) → U133(isNat(x1), x0, x1, x2)
U133(tt, x0, x1, x2) → U134(isNatKind(x1), x0, x1, x2)
U134(tt, x0, x1, x2) → U135(isNat(x2), x0, x1, x2)
U135(tt, x0, x1, x2) → U136(isNatKind(x2), x0, x1, x2)
U136(tt, x0, x1, x2) → cons(x2, take(x1, x0))
length(cons(x0, x1)) → U111(isNatList(x1), x1, x0)
take(s(x0), cons(x1, x2)) → U131(isNatIList(x2), x2, x0, x1)

(22) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {length, U71, take, s, U81, U62, U52, U13, U23, U96, U106, U33, U46} are replacing on all positions.
For all symbols f in {cons, U51, U61, U11, U12, U91, U92, U93, U94, U95, U21, U22, U101, U102, U103, U104, U105, U31, U32, U41, U42, U43, U44, U45, U103', U102', U104', U105', U31', U32', U91', U92', U93', U94', U95', U101', U11', U12', U21', U22', U41', U42', U43', U44', U45'} we have µ(f) = {1}.
The symbols in {isNatIListKind, isNatKind, isNat, isNatList, isNatIList, ISNATILIST, ISNATLIST, ISNAT} are not replacing on any position.

The TRS P consists of the following rules:

U102'(tt, V1, V2) → U103'(isNatIListKind(V2), V1, V2)
U103'(tt, V1, V2) → U104'(isNatIListKind(V2), V1, V2)
U104'(tt, V1, V2) → U105'(isNat(V1), V2)
U105'(tt, V2) → ISNATILIST(V2)
ISNATILIST(V) → U31'(isNatIListKind(V), V)
U31'(tt, V) → U32'(isNatIListKind(V), V)
U32'(tt, V) → ISNATLIST(V)
ISNATLIST(cons(V1, V2)) → U91'(isNatKind(V1), V1, V2)
U91'(tt, V1, V2) → U92'(isNatKind(V1), V1, V2)
U92'(tt, V1, V2) → U93'(isNatIListKind(V2), V1, V2)
U93'(tt, V1, V2) → U94'(isNatIListKind(V2), V1, V2)
U94'(tt, V1, V2) → U95'(isNat(V1), V2)
U95'(tt, V2) → ISNATLIST(V2)
ISNATLIST(take(V1, V2)) → U101'(isNatKind(V1), V1, V2)
U101'(tt, V1, V2) → U102'(isNatKind(V1), V1, V2)
U94'(tt, V1, V2) → ISNAT(V1)
ISNAT(length(V1)) → U11'(isNatIListKind(V1), V1)
U11'(tt, V1) → U12'(isNatIListKind(V1), V1)
U12'(tt, V1) → ISNATLIST(V1)
ISNAT(s(V1)) → U21'(isNatKind(V1), V1)
U21'(tt, V1) → U22'(isNatKind(V1), V1)
U22'(tt, V1) → ISNAT(V1)
ISNATILIST(cons(V1, V2)) → U41'(isNatKind(V1), V1, V2)
U41'(tt, V1, V2) → U42'(isNatKind(V1), V1, V2)
U42'(tt, V1, V2) → U43'(isNatIListKind(V2), V1, V2)
U43'(tt, V1, V2) → U44'(isNatIListKind(V2), V1, V2)
U44'(tt, V1, V2) → U45'(isNat(V1), V2)
U45'(tt, V2) → ISNATILIST(V2)
U44'(tt, V1, V2) → ISNAT(V1)
U104'(tt, V1, V2) → ISNAT(V1)

The TRS R consists of the following rules:

isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U71(isNatIListKind(V1))
isNatIListKind(take(V1, V2)) → U61(isNatKind(V1), V2)
isNatKind(s(V1)) → U81(isNatKind(V1))
U81(tt) → tt
U61(tt, V2) → U62(isNatIListKind(V2))
U62(tt) → tt
U71(tt) → tt
U51(tt, V2) → U52(isNatIListKind(V2))
U52(tt) → tt
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
U11(tt, V1) → U12(isNatIListKind(V1), V1)
U12(tt, V1) → U13(isNatList(V1))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U91(isNatKind(V1), V1, V2)
U91(tt, V1, V2) → U92(isNatKind(V1), V1, V2)
U92(tt, V1, V2) → U93(isNatIListKind(V2), V1, V2)
U93(tt, V1, V2) → U94(isNatIListKind(V2), V1, V2)
U94(tt, V1, V2) → U95(isNat(V1), V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U95(tt, V2) → U96(isNatList(V2))
isNatList(take(V1, V2)) → U101(isNatKind(V1), V1, V2)
U101(tt, V1, V2) → U102(isNatKind(V1), V1, V2)
U102(tt, V1, V2) → U103(isNatIListKind(V2), V1, V2)
U103(tt, V1, V2) → U104(isNatIListKind(V2), V1, V2)
U104(tt, V1, V2) → U105(isNat(V1), V2)
U105(tt, V2) → U106(isNatIList(V2))
isNatIList(V) → U31(isNatIListKind(V), V)
U31(tt, V) → U32(isNatIListKind(V), V)
U32(tt, V) → U33(isNatList(V))
U33(tt) → tt
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
U41(tt, V1, V2) → U42(isNatKind(V1), V1, V2)
U42(tt, V1, V2) → U43(isNatIListKind(V2), V1, V2)
U43(tt, V1, V2) → U44(isNatIListKind(V2), V1, V2)
U44(tt, V1, V2) → U45(isNat(V1), V2)
U45(tt, V2) → U46(isNatIList(V2))
U46(tt) → tt
U106(tt) → tt
U96(tt) → tt
U13(tt) → tt

Q is empty.

(23) QCSDPMuMonotonicPoloProof (EQUIVALENT transformation)

U93'(tt, V1, V2) → U94'(isNatIListKind(V2), V1, V2)
U94'(tt, V1, V2) → U95'(isNat(V1), V2)
ISNATLIST(take(V1, V2)) → U101'(isNatKind(V1), V1, V2)
U94'(tt, V1, V2) → ISNAT(V1)
ISNAT(length(V1)) → U11'(isNatIListKind(V1), V1)
U11'(tt, V1) → U12'(isNatIListKind(V1), V1)
ISNAT(s(V1)) → U21'(isNatKind(V1), V1)
U22'(tt, V1) → ISNAT(V1)
ISNATILIST(cons(V1, V2)) → U41'(isNatKind(V1), V1, V2)
U41'(tt, V1, V2) → U42'(isNatKind(V1), V1, V2)

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(0) = 1
POL(ISNAT(x₁)) = x₁
POL(ISNATILIST(x₁)) = 2·x₁
POL(ISNATLIST(x₁)) = x₁
POL(U101(x₁, x₂, x₃)) = 2·x₁
POL(U101'(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + 2·x₃
POL(U102(x₁, x₂, x₃)) = x₁
POL(U102'(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + 2·x₃
POL(U103(x₁, x₂, x₃)) = x₁
POL(U103'(x₁, x₂, x₃)) = 2·x₁ + x₂ + 2·x₃
POL(U104(x₁, x₂, x₃)) = x₁
POL(U104'(x₁, x₂, x₃)) = x₁ + x₂ + 2·x₃
POL(U105(x₁, x₂)) = 2·x₁
POL(U105'(x₁, x₂)) = x₁ + 2·x₂
POL(U106(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁
POL(U11'(x₁, x₂)) = 1 + 2·x₁ + x₂
POL(U12(x₁, x₂)) = 2·x₁
POL(U12'(x₁, x₂)) = 2·x₁ + x₂
POL(U13(x₁)) = 2·x₁
POL(U21(x₁, x₂)) = 2·x₁
POL(U21'(x₁, x₂)) = 1 + 2·x₁ + x₂
POL(U22(x₁, x₂)) = x₁
POL(U22'(x₁, x₂)) = 1 + x₁ + x₂
POL(U23(x₁)) = 2·x₁
POL(U31(x₁, x₂)) = 2·x₁
POL(U31'(x₁, x₂)) = 2·x₁ + x₂
POL(U32(x₁, x₂)) = 2·x₁
POL(U32'(x₁, x₂)) = 2·x₁ + x₂
POL(U33(x₁)) = 2·x₁
POL(U41(x₁, x₂, x₃)) = x₁
POL(U41'(x₁, x₂, x₃)) = 1 + x₁ + 2·x₂ + 2·x₃
POL(U42(x₁, x₂, x₃)) = 2·x₁
POL(U42'(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + 2·x₃
POL(U43(x₁, x₂, x₃)) = x₁
POL(U43'(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + 2·x₃
POL(U44(x₁, x₂, x₃)) = x₁
POL(U44'(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(U45(x₁, x₂)) = x₁
POL(U45'(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U46(x₁)) = x₁
POL(U51(x₁, x₂)) = x₁
POL(U52(x₁)) = 2·x₁
POL(U61(x₁, x₂)) = 2·x₁
POL(U62(x₁)) = 2·x₁
POL(U71(x₁)) = 2·x₁
POL(U81(x₁)) = 2·x₁
POL(U91(x₁, x₂, x₃)) = 2·x₁
POL(U91'(x₁, x₂, x₃)) = 2 + x₁ + 2·x₂ + x₃
POL(U92(x₁, x₂, x₃)) = 2·x₁
POL(U92'(x₁, x₂, x₃)) = 2 + 2·x₁ + 2·x₂ + x₃
POL(U93(x₁, x₂, x₃)) = x₁
POL(U93'(x₁, x₂, x₃)) = 2 + 2·x₁ + 2·x₂ + x₃
POL(U94(x₁, x₂, x₃)) = x₁
POL(U94'(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U95(x₁, x₂)) = x₁
POL(U95'(x₁, x₂)) = 2·x₁ + x₂
POL(U96(x₁)) = x₁
POL(cons(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 2 + 2·x₁
POL(nil) = 2
POL(s(x₁)) = 2 + 2·x₁
POL(take(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(tt) = 0
POL(zeros) = 0

(24) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {length, U71, take, s, U81, U62, U52, U13, U23, U96, U106, U33, U46} are replacing on all positions.
For all symbols f in {cons, U51, U61, U11, U12, U91, U92, U93, U94, U95, U21, U22, U101, U102, U103, U104, U105, U31, U32, U41, U42, U43, U44, U45, U103', U102', U104', U105', U31', U32', U91', U92', U93', U95', U101', U12', U22', U21', U43', U42', U44', U45'} we have µ(f) = {1}.
The symbols in {isNatIListKind, isNatKind, isNat, isNatList, isNatIList, ISNATILIST, ISNATLIST, ISNAT} are not replacing on any position.

The TRS P consists of the following rules:

U102'(tt, V1, V2) → U103'(isNatIListKind(V2), V1, V2)
U103'(tt, V1, V2) → U104'(isNatIListKind(V2), V1, V2)
U104'(tt, V1, V2) → U105'(isNat(V1), V2)
U105'(tt, V2) → ISNATILIST(V2)
ISNATILIST(V) → U31'(isNatIListKind(V), V)
U31'(tt, V) → U32'(isNatIListKind(V), V)
U32'(tt, V) → ISNATLIST(V)
ISNATLIST(cons(V1, V2)) → U91'(isNatKind(V1), V1, V2)
U91'(tt, V1, V2) → U92'(isNatKind(V1), V1, V2)
U92'(tt, V1, V2) → U93'(isNatIListKind(V2), V1, V2)
U95'(tt, V2) → ISNATLIST(V2)
U101'(tt, V1, V2) → U102'(isNatKind(V1), V1, V2)
U12'(tt, V1) → ISNATLIST(V1)
U21'(tt, V1) → U22'(isNatKind(V1), V1)
U42'(tt, V1, V2) → U43'(isNatIListKind(V2), V1, V2)
U43'(tt, V1, V2) → U44'(isNatIListKind(V2), V1, V2)
U44'(tt, V1, V2) → U45'(isNat(V1), V2)
U45'(tt, V2) → ISNATILIST(V2)
U44'(tt, V1, V2) → ISNAT(V1)
U104'(tt, V1, V2) → ISNAT(V1)

The TRS R consists of the following rules:

isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U71(isNatIListKind(V1))
isNatIListKind(take(V1, V2)) → U61(isNatKind(V1), V2)
isNatKind(s(V1)) → U81(isNatKind(V1))
U81(tt) → tt
U61(tt, V2) → U62(isNatIListKind(V2))
U62(tt) → tt
U71(tt) → tt
U51(tt, V2) → U52(isNatIListKind(V2))
U52(tt) → tt
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
U11(tt, V1) → U12(isNatIListKind(V1), V1)
U12(tt, V1) → U13(isNatList(V1))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U91(isNatKind(V1), V1, V2)
U91(tt, V1, V2) → U92(isNatKind(V1), V1, V2)
U92(tt, V1, V2) → U93(isNatIListKind(V2), V1, V2)
U93(tt, V1, V2) → U94(isNatIListKind(V2), V1, V2)
U94(tt, V1, V2) → U95(isNat(V1), V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U95(tt, V2) → U96(isNatList(V2))
isNatList(take(V1, V2)) → U101(isNatKind(V1), V1, V2)
U101(tt, V1, V2) → U102(isNatKind(V1), V1, V2)
U102(tt, V1, V2) → U103(isNatIListKind(V2), V1, V2)
U103(tt, V1, V2) → U104(isNatIListKind(V2), V1, V2)
U104(tt, V1, V2) → U105(isNat(V1), V2)
U105(tt, V2) → U106(isNatIList(V2))
isNatIList(V) → U31(isNatIListKind(V), V)
U31(tt, V) → U32(isNatIListKind(V), V)
U32(tt, V) → U33(isNatList(V))
U33(tt) → tt
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
U41(tt, V1, V2) → U42(isNatKind(V1), V1, V2)
U42(tt, V1, V2) → U43(isNatIListKind(V2), V1, V2)
U43(tt, V1, V2) → U44(isNatIListKind(V2), V1, V2)
U44(tt, V1, V2) → U45(isNat(V1), V2)
U45(tt, V2) → U46(isNatIList(V2))
U46(tt) → tt
U106(tt) → tt
U96(tt) → tt
U13(tt) → tt

Q is empty.

(25) QCSDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 19 less nodes.

(26) TRUE

(27) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {U106, s, length, U13, take, U23, U33, U46, U52, U62, U71, U81, U96, TAKE} are replacing on all positions.
For all symbols f in {cons, U101, U102, U103, U104, U105, U11, U12, U111, U112, U113, U114, U131, U132, U133, U134, U135, U136, U21, U22, U31, U32, U41, U42, U43, U44, U45, U51, U61, U91, U92, U93, U94, U95, U133', U132', U134', U135', U136', U131'} we have µ(f) = {1}.
The symbols in {isNatKind, isNatIListKind, isNat, isNatIList, isNatList, U} are not replacing on any position.

The TRS P consists of the following rules:

U132'(tt, IL, M, N) → U133'(isNat(M), IL, M, N)
U133'(tt, IL, M, N) → U134'(isNatKind(M), IL, M, N)
U134'(tt, IL, M, N) → U135'(isNat(N), IL, M, N)
U135'(tt, IL, M, N) → U136'(isNatKind(N), IL, M, N)
U136'(tt, IL, M, N) → U(N)
U(take(x_0, x_1)) → U(x_0)
U(take(x_0, x_1)) → U(x_1)
U(take(x0, x1)) → TAKE(x0, x1)
TAKE(s(M), cons(N, IL)) → U131'(isNatIList(IL), IL, M, N)
U131'(tt, IL, M, N) → U132'(isNatIListKind(IL), IL, M, N)

The TRS R consists of the following rules:

zeros → cons(0, zeros)
U101(tt, V1, V2) → U102(isNatKind(V1), V1, V2)
U102(tt, V1, V2) → U103(isNatIListKind(V2), V1, V2)
U103(tt, V1, V2) → U104(isNatIListKind(V2), V1, V2)
U104(tt, V1, V2) → U105(isNat(V1), V2)
U105(tt, V2) → U106(isNatIList(V2))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(V1), V1)
U111(tt, L, N) → U112(isNatIListKind(L), L, N)
U112(tt, L, N) → U113(isNat(N), L, N)
U113(tt, L, N) → U114(isNatKind(N), L)
U114(tt, L) → s(length(L))
U12(tt, V1) → U13(isNatList(V1))
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(IL), IL, M, N)
U132(tt, IL, M, N) → U133(isNat(M), IL, M, N)
U133(tt, IL, M, N) → U134(isNatKind(M), IL, M, N)
U134(tt, IL, M, N) → U135(isNat(N), IL, M, N)
U135(tt, IL, M, N) → U136(isNatKind(N), IL, M, N)
U136(tt, IL, M, N) → cons(N, take(M, IL))
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(V), V)
U32(tt, V) → U33(isNatList(V))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(V1), V1, V2)
U42(tt, V1, V2) → U43(isNatIListKind(V2), V1, V2)
U43(tt, V1, V2) → U44(isNatIListKind(V2), V1, V2)
U44(tt, V1, V2) → U45(isNat(V1), V2)
U45(tt, V2) → U46(isNatIList(V2))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(V2))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(V2))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(V1), V1, V2)
U92(tt, V1, V2) → U93(isNatIListKind(V2), V1, V2)
U93(tt, V1, V2) → U94(isNatIListKind(V2), V1, V2)
U94(tt, V1, V2) → U95(isNat(V1), V2)
U95(tt, V2) → U96(isNatList(V2))
U96(tt) → tt
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(V) → U31(isNatIListKind(V), V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatIListKind(take(V1, V2)) → U61(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U71(isNatIListKind(V1))
isNatKind(s(V1)) → U81(isNatKind(V1))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U91(isNatKind(V1), V1, V2)
isNatList(take(V1, V2)) → U101(isNatKind(V1), V1, V2)
length(cons(N, L)) → U111(isNatList(L), L, N)
take(s(M), cons(N, IL)) → U131(isNatIList(IL), IL, M, N)

Q is empty.

(28) QCSDPSubtermProof (EQUIVALENT transformation)

We use the subterm processor [DA_EMMES].

The following pairs can be oriented strictly and are deleted.

U(take(x_0, x_1)) → U(x_0)
U(take(x_0, x_1)) → U(x_1)
U(take(x0, x1)) → TAKE(x0, x1)
TAKE(s(M), cons(N, IL)) → U131'(isNatIList(IL), IL, M, N)

The remaining pairs can at least be oriented weakly.

U132'(tt, IL, M, N) → U133'(isNat(M), IL, M, N)
U133'(tt, IL, M, N) → U134'(isNatKind(M), IL, M, N)
U134'(tt, IL, M, N) → U135'(isNat(N), IL, M, N)
U135'(tt, IL, M, N) → U136'(isNatKind(N), IL, M, N)
U136'(tt, IL, M, N) → U(N)
U131'(tt, IL, M, N) → U132'(isNatIListKind(IL), IL, M, N)

Used ordering: Combined order from the following AFS and order.
U133'(x1, x2, x3, x4) = x4
U132'(x1, x2, x3, x4) = x4
U134'(x1, x2, x3, x4) = x4
U135'(x1, x2, x3, x4) = x4
U136'(x1, x2, x3, x4) = x4
U(x1) = x1
TAKE(x1, x2) = x2
U131'(x1, x2, x3, x4) = x4

Subterm Order

(29) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {U106, s, length, U13, take, U23, U33, U46, U52, U62, U71, U81, U96} are replacing on all positions.
For all symbols f in {cons, U101, U102, U103, U104, U105, U11, U12, U111, U112, U113, U114, U131, U132, U133, U134, U135, U136, U21, U22, U31, U32, U41, U42, U43, U44, U45, U51, U61, U91, U92, U93, U94, U95, U133', U132', U134', U135', U136', U131'} we have µ(f) = {1}.
The symbols in {isNatKind, isNatIListKind, isNat, isNatIList, isNatList, U} are not replacing on any position.

The TRS P consists of the following rules:

U132'(tt, IL, M, N) → U133'(isNat(M), IL, M, N)
U133'(tt, IL, M, N) → U134'(isNatKind(M), IL, M, N)
U134'(tt, IL, M, N) → U135'(isNat(N), IL, M, N)
U135'(tt, IL, M, N) → U136'(isNatKind(N), IL, M, N)
U136'(tt, IL, M, N) → U(N)
U131'(tt, IL, M, N) → U132'(isNatIListKind(IL), IL, M, N)

The TRS R consists of the following rules:

zeros → cons(0, zeros)
U101(tt, V1, V2) → U102(isNatKind(V1), V1, V2)
U102(tt, V1, V2) → U103(isNatIListKind(V2), V1, V2)
U103(tt, V1, V2) → U104(isNatIListKind(V2), V1, V2)
U104(tt, V1, V2) → U105(isNat(V1), V2)
U105(tt, V2) → U106(isNatIList(V2))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(V1), V1)
U111(tt, L, N) → U112(isNatIListKind(L), L, N)
U112(tt, L, N) → U113(isNat(N), L, N)
U113(tt, L, N) → U114(isNatKind(N), L)
U114(tt, L) → s(length(L))
U12(tt, V1) → U13(isNatList(V1))
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(IL), IL, M, N)
U132(tt, IL, M, N) → U133(isNat(M), IL, M, N)
U133(tt, IL, M, N) → U134(isNatKind(M), IL, M, N)
U134(tt, IL, M, N) → U135(isNat(N), IL, M, N)
U135(tt, IL, M, N) → U136(isNatKind(N), IL, M, N)
U136(tt, IL, M, N) → cons(N, take(M, IL))
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(V), V)
U32(tt, V) → U33(isNatList(V))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(V1), V1, V2)
U42(tt, V1, V2) → U43(isNatIListKind(V2), V1, V2)
U43(tt, V1, V2) → U44(isNatIListKind(V2), V1, V2)
U44(tt, V1, V2) → U45(isNat(V1), V2)
U45(tt, V2) → U46(isNatIList(V2))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(V2))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(V2))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(V1), V1, V2)
U92(tt, V1, V2) → U93(isNatIListKind(V2), V1, V2)
U93(tt, V1, V2) → U94(isNatIListKind(V2), V1, V2)
U94(tt, V1, V2) → U95(isNat(V1), V2)
U95(tt, V2) → U96(isNatList(V2))
U96(tt) → tt
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(V) → U31(isNatIListKind(V), V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatIListKind(take(V1, V2)) → U61(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U71(isNatIListKind(V1))
isNatKind(s(V1)) → U81(isNatKind(V1))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U91(isNatKind(V1), V1, V2)
isNatList(take(V1, V2)) → U101(isNatKind(V1), V1, V2)
length(cons(N, L)) → U111(isNatList(L), L, N)
take(s(M), cons(N, IL)) → U131(isNatIList(IL), IL, M, N)

Q is empty.

(30) QCSDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 6 less nodes.

(31) TRUE

(32) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {U106, s, length, U13, take, U23, U33, U46, U52, U62, U71, U81, U96, LENGTH} are replacing on all positions.
For all symbols f in {cons, U101, U102, U103, U104, U105, U11, U12, U111, U112, U113, U114, U131, U132, U133, U134, U135, U136, U21, U22, U31, U32, U41, U42, U43, U44, U45, U51, U61, U91, U92, U93, U94, U95, U113', U112', U114', U111'} we have µ(f) = {1}.
The symbols in {isNatKind, isNatIListKind, isNat, isNatIList, isNatList} are not replacing on any position.

The TRS P consists of the following rules:

U112'(tt, L, N) → U113'(isNat(N), L, N)
U113'(tt, L, N) → U114'(isNatKind(N), L)
U114'(tt, L) → LENGTH(L)
LENGTH(cons(N, L)) → U111'(isNatList(L), L, N)
U111'(tt, L, N) → U112'(isNatIListKind(L), L, N)

The TRS R consists of the following rules:

zeros → cons(0, zeros)
U101(tt, V1, V2) → U102(isNatKind(V1), V1, V2)
U102(tt, V1, V2) → U103(isNatIListKind(V2), V1, V2)
U103(tt, V1, V2) → U104(isNatIListKind(V2), V1, V2)
U104(tt, V1, V2) → U105(isNat(V1), V2)
U105(tt, V2) → U106(isNatIList(V2))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(V1), V1)
U111(tt, L, N) → U112(isNatIListKind(L), L, N)
U112(tt, L, N) → U113(isNat(N), L, N)
U113(tt, L, N) → U114(isNatKind(N), L)
U114(tt, L) → s(length(L))
U12(tt, V1) → U13(isNatList(V1))
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(IL), IL, M, N)
U132(tt, IL, M, N) → U133(isNat(M), IL, M, N)
U133(tt, IL, M, N) → U134(isNatKind(M), IL, M, N)
U134(tt, IL, M, N) → U135(isNat(N), IL, M, N)
U135(tt, IL, M, N) → U136(isNatKind(N), IL, M, N)
U136(tt, IL, M, N) → cons(N, take(M, IL))
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(V), V)
U32(tt, V) → U33(isNatList(V))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(V1), V1, V2)
U42(tt, V1, V2) → U43(isNatIListKind(V2), V1, V2)
U43(tt, V1, V2) → U44(isNatIListKind(V2), V1, V2)
U44(tt, V1, V2) → U45(isNat(V1), V2)
U45(tt, V2) → U46(isNatIList(V2))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(V2))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(V2))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(V1), V1, V2)
U92(tt, V1, V2) → U93(isNatIListKind(V2), V1, V2)
U93(tt, V1, V2) → U94(isNatIListKind(V2), V1, V2)
U94(tt, V1, V2) → U95(isNat(V1), V2)
U95(tt, V2) → U96(isNatList(V2))
U96(tt) → tt
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(V) → U31(isNatIListKind(V), V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatIListKind(take(V1, V2)) → U61(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U71(isNatIListKind(V1))
isNatKind(s(V1)) → U81(isNatKind(V1))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U91(isNatKind(V1), V1, V2)
isNatList(take(V1, V2)) → U101(isNatKind(V1), V1, V2)
length(cons(N, L)) → U111(isNatList(L), L, N)
take(s(M), cons(N, IL)) → U131(isNatIList(IL), IL, M, N)

Q is empty.

(33) QCSDPReductionPairProof (EQUIVALENT transformation)

Using the order
Polynomial interpretation [POLO]:

POL(0) = 2
POL(LENGTH(x₁)) = 2·x₁
POL(U101(x₁, x₂, x₃)) = 2·x₂
POL(U102(x₁, x₂, x₃)) = 2·x₂
POL(U103(x₁, x₂, x₃)) = 2·x₂
POL(U104(x₁, x₂, x₃)) = 2·x₂
POL(U105(x₁, x₂)) = x₁
POL(U106(x₁)) = 2
POL(U11(x₁, x₂)) = 2·x₂
POL(U111(x₁, x₂, x₃)) = 2·x₂
POL(U111'(x₁, x₂, x₃)) = x₁ + 2·x₂
POL(U112(x₁, x₂, x₃)) = 2·x₂
POL(U112'(x₁, x₂, x₃)) = 2·x₂
POL(U113(x₁, x₂, x₃)) = 2·x₂
POL(U113'(x₁, x₂, x₃)) = 2·x₂
POL(U114(x₁, x₂)) = 2·x₂
POL(U114'(x₁, x₂)) = 2·x₂
POL(U12(x₁, x₂)) = 2·x₂
POL(U13(x₁)) = x₁
POL(U131(x₁, x₂, x₃, x₄)) = 2·x₃
POL(U132(x₁, x₂, x₃, x₄)) = 2·x₃
POL(U133(x₁, x₂, x₃, x₄)) = 2·x₃
POL(U134(x₁, x₂, x₃, x₄)) = 2·x₃
POL(U135(x₁, x₂, x₃, x₄)) = 2·x₃
POL(U136(x₁, x₂, x₃, x₄)) = 2·x₃
POL(U21(x₁, x₂)) = 2·x₂
POL(U22(x₁, x₂)) = 2·x₂
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = 2·x₂
POL(U32(x₁, x₂)) = 2·x₂
POL(U33(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = 2 + x₃
POL(U42(x₁, x₂, x₃)) = 2 + x₃
POL(U43(x₁, x₂, x₃)) = 2
POL(U44(x₁, x₂, x₃)) = 2
POL(U45(x₁, x₂)) = 2
POL(U46(x₁)) = 2
POL(U51(x₁, x₂)) = x₁
POL(U52(x₁)) = 2
POL(U61(x₁, x₂)) = 2
POL(U62(x₁)) = 2
POL(U71(x₁)) = 2
POL(U81(x₁)) = 2
POL(U91(x₁, x₂, x₃)) = 2·x₃
POL(U92(x₁, x₂, x₃)) = 2·x₃
POL(U93(x₁, x₂, x₃)) = 2·x₃
POL(U94(x₁, x₂, x₃)) = 2·x₃
POL(U95(x₁, x₂)) = 2·x₂
POL(U96(x₁)) = x₁
POL(cons(x₁, x₂)) = 2·x₂
POL(isNat(x₁)) = 2·x₁
POL(isNatIList(x₁)) = 2 + 2·x₁
POL(isNatIListKind(x₁)) = 2
POL(isNatKind(x₁)) = 2
POL(isNatList(x₁)) = 2·x₁
POL(length(x₁)) = x₁
POL(nil) = 1
POL(s(x₁)) = 2·x₁
POL(take(x₁, x₂)) = x₁
POL(tt) = 2
POL(zeros) = 0

the following usable rules

isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
length(cons(N, L)) → U111(isNatList(L), L, N)
U111(tt, L, N) → U112(isNatIListKind(L), L, N)
U112(tt, L, N) → U113(isNat(N), L, N)
U113(tt, L, N) → U114(isNatKind(N), L)
U114(tt, L) → s(length(L))
isNatKind(0) → tt
isNatKind(length(V1)) → U71(isNatIListKind(V1))
isNatKind(s(V1)) → U81(isNatKind(V1))
U71(tt) → tt
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatIListKind(take(V1, V2)) → U61(isNatKind(V1), V2)
zeros → cons(0, zeros)
U51(tt, V2) → U52(isNatIListKind(V2))
U52(tt) → tt
take(s(M), cons(N, IL)) → U131(isNatIList(IL), IL, M, N)
U131(tt, IL, M, N) → U132(isNatIListKind(IL), IL, M, N)
U132(tt, IL, M, N) → U133(isNat(M), IL, M, N)
U133(tt, IL, M, N) → U134(isNatKind(M), IL, M, N)
U134(tt, IL, M, N) → U135(isNat(N), IL, M, N)
U135(tt, IL, M, N) → U136(isNatKind(N), IL, M, N)
U136(tt, IL, M, N) → cons(N, take(M, IL))
isNatIList(V) → U31(isNatIListKind(V), V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
U31(tt, V) → U32(isNatIListKind(V), V)
U32(tt, V) → U33(isNatList(V))
U33(tt) → tt
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U91(isNatKind(V1), V1, V2)
isNatList(take(V1, V2)) → U101(isNatKind(V1), V1, V2)
U91(tt, V1, V2) → U92(isNatKind(V1), V1, V2)
U92(tt, V1, V2) → U93(isNatIListKind(V2), V1, V2)
U93(tt, V1, V2) → U94(isNatIListKind(V2), V1, V2)
U94(tt, V1, V2) → U95(isNat(V1), V2)
U95(tt, V2) → U96(isNatList(V2))
U96(tt) → tt
U101(tt, V1, V2) → U102(isNatKind(V1), V1, V2)
U102(tt, V1, V2) → U103(isNatIListKind(V2), V1, V2)
U103(tt, V1, V2) → U104(isNatIListKind(V2), V1, V2)
U104(tt, V1, V2) → U105(isNat(V1), V2)
U105(tt, V2) → U106(isNatIList(V2))
U106(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(V1), V1, V2)
U42(tt, V1, V2) → U43(isNatIListKind(V2), V1, V2)
U43(tt, V1, V2) → U44(isNatIListKind(V2), V1, V2)
U44(tt, V1, V2) → U45(isNat(V1), V2)
U45(tt, V2) → U46(isNatIList(V2))
U46(tt) → tt
U61(tt, V2) → U62(isNatIListKind(V2))
U62(tt) → tt
U81(tt) → tt
U11(tt, V1) → U12(isNatIListKind(V1), V1)
U12(tt, V1) → U13(isNatList(V1))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt

could all be oriented weakly.
Furthermore, the pairs

U111'(tt, L, N) → U112'(isNatIListKind(L), L, N)

could be oriented strictly and thus removed by the CS-Reduction Pair Processor [LPAR08,DA_EMMES].

(34) Obligation:

U112'(tt, L, N) → U113'(isNat(N), L, N)
U113'(tt, L, N) → U114'(isNatKind(N), L)
U114'(tt, L) → LENGTH(L)
LENGTH(cons(N, L)) → U111'(isNatList(L), L, N)

The TRS R consists of the following rules:

zeros → cons(0, zeros)
U101(tt, V1, V2) → U102(isNatKind(V1), V1, V2)
U102(tt, V1, V2) → U103(isNatIListKind(V2), V1, V2)
U103(tt, V1, V2) → U104(isNatIListKind(V2), V1, V2)
U104(tt, V1, V2) → U105(isNat(V1), V2)
U105(tt, V2) → U106(isNatIList(V2))
U106(tt) → tt
U11(tt, V1) → U12(isNatIListKind(V1), V1)
U111(tt, L, N) → U112(isNatIListKind(L), L, N)
U112(tt, L, N) → U113(isNat(N), L, N)
U113(tt, L, N) → U114(isNatKind(N), L)
U114(tt, L) → s(length(L))
U12(tt, V1) → U13(isNatList(V1))
U13(tt) → tt
U131(tt, IL, M, N) → U132(isNatIListKind(IL), IL, M, N)
U132(tt, IL, M, N) → U133(isNat(M), IL, M, N)
U133(tt, IL, M, N) → U134(isNatKind(M), IL, M, N)
U134(tt, IL, M, N) → U135(isNat(N), IL, M, N)
U135(tt, IL, M, N) → U136(isNatKind(N), IL, M, N)
U136(tt, IL, M, N) → cons(N, take(M, IL))
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(V), V)
U32(tt, V) → U33(isNatList(V))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(V1), V1, V2)
U42(tt, V1, V2) → U43(isNatIListKind(V2), V1, V2)
U43(tt, V1, V2) → U44(isNatIListKind(V2), V1, V2)
U44(tt, V1, V2) → U45(isNat(V1), V2)
U45(tt, V2) → U46(isNatIList(V2))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(V2))
U52(tt) → tt
U61(tt, V2) → U62(isNatIListKind(V2))
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt, V1, V2) → U92(isNatKind(V1), V1, V2)
U92(tt, V1, V2) → U93(isNatIListKind(V2), V1, V2)
U93(tt, V1, V2) → U94(isNatIListKind(V2), V1, V2)
U94(tt, V1, V2) → U95(isNat(V1), V2)
U95(tt, V2) → U96(isNatList(V2))
U96(tt) → tt
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(V) → U31(isNatIListKind(V), V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatIListKind(take(V1, V2)) → U61(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U71(isNatIListKind(V1))
isNatKind(s(V1)) → U81(isNatKind(V1))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U91(isNatKind(V1), V1, V2)
isNatList(take(V1, V2)) → U101(isNatKind(V1), V1, V2)
length(cons(N, L)) → U111(isNatList(L), L, N)
take(s(M), cons(N, IL)) → U131(isNatIList(IL), IL, M, N)

Q is empty.

(35) QCSDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 4 less nodes.

(0) Obligation:

(1) QTRSToCSRProof (EQUIVALENT transformation)

(2) Obligation:

(3) PoloCSRProof (EQUIVALENT transformation)

(4) Obligation:

(5) PoloCSRProof (EQUIVALENT transformation)

(6) Obligation:

(7) PoloCSRProof (EQUIVALENT transformation)

(8) Obligation:

(9) CSDependencyPairsProof (EQUIVALENT transformation)

(10) Obligation:

(11) QCSDependencyGraphProof (EQUIVALENT transformation)

(12) Complex Obligation (AND)

(13) Obligation:

(14) QCSUsableRulesProof (EQUIVALENT transformation)

(15) Obligation:

(16) QCSDPMuMonotonicPoloProof (EQUIVALENT transformation)

(17) Obligation:

(18) QCSDependencyGraphProof (EQUIVALENT transformation)

(19) TRUE

(20) Obligation:

(21) QCSUsableRulesProof (EQUIVALENT transformation)

(22) Obligation:

(23) QCSDPMuMonotonicPoloProof (EQUIVALENT transformation)

(24) Obligation:

(25) QCSDependencyGraphProof (EQUIVALENT transformation)

(26) TRUE

(27) Obligation:

(28) QCSDPSubtermProof (EQUIVALENT transformation)

(29) Obligation:

(30) QCSDependencyGraphProof (EQUIVALENT transformation)

(31) TRUE

(32) Obligation:

(33) QCSDPReductionPairProof (EQUIVALENT transformation)

(34) Obligation:

(35) QCSDependencyGraphProof (EQUIVALENT transformation)

(36) TRUE