Termination w.r.t. Q proof of /tmp/tmpQu6uKz/LengthOfFiniteLists_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 PoloCSRProof (⇔)

↳10 CSR

↳11 PoloCSRProof (⇔)

↳12 CSR

↳13 PoloCSRProof (⇔)

↳14 CSR

↳15 PoloCSRProof (⇔)

↳16 CSR

↳17 PoloCSRProof (⇔)

↳18 CSR

↳19 PoloCSRProof (⇔)

↳20 CSR

↳21 PoloCSRProof (⇔)

↳22 CSR

↳23 CSDependencyPairsProof (⇔)

↳24 QCSDP

↳25 QCSDependencyGraphProof (⇔)

↳26 AND

  ↳27 QCSDP

    ↳28 QCSUsableRulesProof (⇔)

    ↳29 QCSDP

    ↳30 QCSDPMuMonotonicPoloProof (⇔)

    ↳31 QCSDP

    ↳32 PIsEmptyProof (⇔)

    ↳33 TRUE

  ↳34 QCSDP

    ↳35 QCSDPSubtermProof (⇔)

    ↳36 QCSDP

    ↳37 QCSDependencyGraphProof (⇔)

    ↳38 TRUE

  ↳39 QCSDP

    ↳40 QCSUsableRulesProof (⇔)

    ↳41 QCSDP

    ↳42 QCSDPMuMonotonicPoloProof (⇔)

    ↳43 QCSDP

    ↳44 QCSDependencyGraphProof (⇔)

    ↳45 TRUE

  ↳46 QCSDP

    ↳47 QCSDPReductionPairProof (⇔)

    ↳48 QCSDP

    ↳49 QCSDependencyGraphProof (⇔)

    ↳50 TRUE

  ↳51 QCSDP

    ↳52 QCSUsableRulesProof (⇔)

    ↳53 QCSDP

    ↳54 QCSDPMuMonotonicPoloProof (⇔)

    ↳55 QCSDP

    ↳56 QCSDependencyGraphProof (⇔)

    ↳57 TRUE

(0) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U13(tt)) → mark(tt)
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)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt, V1, V2)) → mark(U82(isNatKind(V1), V1, V2))
active(U82(tt, V1, V2)) → mark(U83(isNatIListKind(V2), V1, V2))
active(U83(tt, V1, V2)) → mark(U84(isNatIListKind(V2), V1, V2))
active(U84(tt, V1, V2)) → mark(U85(isNat(V1), V2))
active(U85(tt, V2)) → mark(U86(isNatList(V2)))
active(U86(tt)) → mark(tt)
active(U91(tt, L, N)) → mark(U92(isNatIListKind(L), L, N))
active(U92(tt, L, N)) → mark(U93(isNat(N), L, N))
active(U93(tt, L, N)) → mark(U94(isNatKind(N), L))
active(U94(tt, L)) → mark(s(length(L)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(U61(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U71(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U81(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U91(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
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(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(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)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
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(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(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) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(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(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(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)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(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))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(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(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U13(tt)) → mark(tt)
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)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt, V1, V2)) → mark(U82(isNatKind(V1), V1, V2))
active(U82(tt, V1, V2)) → mark(U83(isNatIListKind(V2), V1, V2))
active(U83(tt, V1, V2)) → mark(U84(isNatIListKind(V2), V1, V2))
active(U84(tt, V1, V2)) → mark(U85(isNat(V1), V2))
active(U85(tt, V2)) → mark(U86(isNatList(V2)))
active(U86(tt)) → mark(tt)
active(U91(tt, L, N)) → mark(U92(isNatIListKind(L), L, N))
active(U92(tt, L, N)) → mark(U93(isNat(N), L, N))
active(U93(tt, L, N)) → mark(U94(isNatKind(N), L))
active(U94(tt, L)) → mark(s(length(L)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(U61(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U71(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U81(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U91(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
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(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(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)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
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(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(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) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(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(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(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)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(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))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(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(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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
U11: {1}
tt: empty set
U12: {1}
isNatIListKind: empty set
U13: {1}
isNatList: empty set
U21: {1}
U22: {1}
isNatKind: empty set
U23: {1}
isNat: empty set
U31: {1}
U32: {1}
U33: {1}
U41: {1}
U42: {1}
U43: {1}
U44: {1}
U45: {1}
U46: {1}
isNatIList: empty set
U51: {1}
U52: {1}
U61: {1}
U71: {1}
U81: {1}
U82: {1}
U83: {1}
U84: {1}
U85: {1}
U86: {1}
U91: {1}
U92: {1}
U93: {1}
U94: {1}
s: {1}
length: {1}
nil: empty set
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)
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
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) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(V1), V1, V2)
U82(tt, V1, V2) → U83(isNatIListKind(V2), V1, V2)
U83(tt, V1, V2) → U84(isNatIListKind(V2), V1, V2)
U84(tt, V1, V2) → U85(isNat(V1), V2)
U85(tt, V2) → U86(isNatList(V2))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(L), L, N)
U92(tt, L, N) → U93(isNat(N), L, N)
U93(tt, L, N) → U94(isNatKind(N), L)
U94(tt, L) → s(length(L))
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)
isNatKind(0) → tt
isNatKind(length(V1)) → U61(isNatIListKind(V1))
isNatKind(s(V1)) → U71(isNatKind(V1))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U81(isNatKind(V1), V1, V2)
length(nil) → 0
length(cons(N, L)) → U91(isNatList(L), L, 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(U11(x₁, x₂)) = 2·x₁
POL(U12(x₁, x₂)) = 2·x₁
POL(U13(x₁)) = 2·x₁
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)) = 2·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)) = 2·x₁
POL(U46(x₁)) = 2·x₁
POL(U51(x₁, x₂)) = 2·x₁
POL(U52(x₁)) = x₁
POL(U61(x₁)) = x₁
POL(U71(x₁)) = 2·x₁
POL(U81(x₁, x₂, x₃)) = 2·x₁
POL(U82(x₁, x₂, x₃)) = x₁
POL(U83(x₁, x₂, x₃)) = x₁
POL(U84(x₁, x₂, x₃)) = 2·x₁
POL(U85(x₁, x₂)) = 2·x₁
POL(U86(x₁)) = x₁
POL(U91(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + 2·x₃
POL(U92(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(U93(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(U94(x₁, x₂)) = 2·x₁ + 2·x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·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) = 1
POL(s(x₁)) = 2·x₁
POL(tt) = 0
POL(zeros) = 0

(4) Obligation:

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

zeros → cons(0, zeros)
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
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) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(V1), V1, V2)
U82(tt, V1, V2) → U83(isNatIListKind(V2), V1, V2)
U83(tt, V1, V2) → U84(isNatIListKind(V2), V1, V2)
U84(tt, V1, V2) → U85(isNat(V1), V2)
U85(tt, V2) → U86(isNatList(V2))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(L), L, N)
U92(tt, L, N) → U93(isNat(N), L, N)
U93(tt, L, N) → U94(isNatKind(N), L)
U94(tt, L) → s(length(L))
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)
isNatKind(0) → tt
isNatKind(length(V1)) → U61(isNatIListKind(V1))
isNatKind(s(V1)) → U71(isNatKind(V1))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U81(isNatKind(V1), V1, V2)
length(cons(N, L)) → U91(isNatList(L), L, 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:

isNatIList(V) → U31(isNatIListKind(V), V)
isNatIList(zeros) → tt

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(U11(x₁, x₂)) = x₁
POL(U12(x₁, x₂)) = 2·x₁
POL(U13(x₁)) = x₁
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)) = 2·x₁
POL(U41(x₁, x₂, x₃)) = 1 + 2·x₁
POL(U42(x₁, x₂, x₃)) = 1 + 2·x₁
POL(U43(x₁, x₂, x₃)) = 1 + 2·x₁
POL(U44(x₁, x₂, x₃)) = 1 + 2·x₁
POL(U45(x₁, x₂)) = 1 + 2·x₁
POL(U46(x₁)) = x₁
POL(U51(x₁, x₂)) = 2·x₁
POL(U52(x₁)) = x₁
POL(U61(x₁)) = 2·x₁
POL(U71(x₁)) = x₁
POL(U81(x₁, x₂, x₃)) = 2·x₁
POL(U82(x₁, x₂, x₃)) = 2·x₁
POL(U83(x₁, x₂, x₃)) = 2·x₁
POL(U84(x₁, x₂, x₃)) = 2·x₁
POL(U85(x₁, x₂)) = 2·x₁
POL(U86(x₁)) = 2·x₁
POL(U91(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + x₃
POL(U92(x₁, x₂, x₃)) = x₁ + 2·x₂ + x₃
POL(U93(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + x₃
POL(U94(x₁, x₂)) = 2·x₁ + 2·x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 1
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = 2·x₁
POL(tt) = 0
POL(zeros) = 0

(6) Obligation:

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

zeros → cons(0, zeros)
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
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) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(V1), V1, V2)
U82(tt, V1, V2) → U83(isNatIListKind(V2), V1, V2)
U83(tt, V1, V2) → U84(isNatIListKind(V2), V1, V2)
U84(tt, V1, V2) → U85(isNat(V1), V2)
U85(tt, V2) → U86(isNatList(V2))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(L), L, N)
U92(tt, L, N) → U93(isNat(N), L, N)
U93(tt, L, N) → U94(isNatKind(N), L)
U94(tt, L) → s(length(L))
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U61(isNatIListKind(V1))
isNatKind(s(V1)) → U71(isNatKind(V1))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U81(isNatKind(V1), V1, V2)
length(cons(N, L)) → U91(isNatList(L), L, 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:

U33(tt) → tt

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(U11(x₁, x₂)) = x₁
POL(U12(x₁, x₂)) = 2·x₁
POL(U13(x₁)) = 2·x₁
POL(U21(x₁, x₂)) = 2·x₁
POL(U22(x₁, x₂)) = 2·x₁
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = 1 + x₁ + x₂
POL(U32(x₁, x₂)) = 1 + x₁
POL(U33(x₁)) = 1 + x₁
POL(U41(x₁, x₂, x₃)) = x₁
POL(U42(x₁, x₂, x₃)) = 2·x₁
POL(U43(x₁, x₂, x₃)) = 2·x₁
POL(U44(x₁, x₂, x₃)) = x₁
POL(U45(x₁, x₂)) = x₁
POL(U46(x₁)) = 2·x₁
POL(U51(x₁, x₂)) = x₁
POL(U52(x₁)) = 2·x₁
POL(U61(x₁)) = x₁
POL(U71(x₁)) = x₁
POL(U81(x₁, x₂, x₃)) = 2·x₁
POL(U82(x₁, x₂, x₃)) = 2·x₁
POL(U83(x₁, x₂, x₃)) = 2·x₁
POL(U84(x₁, x₂, x₃)) = x₁
POL(U85(x₁, x₂)) = 2·x₁
POL(U86(x₁)) = 2·x₁
POL(U91(x₁, x₂, x₃)) = 2·x₁ + x₂ + 2·x₃
POL(U92(x₁, x₂, x₃)) = 2·x₁ + x₂ + x₃
POL(U93(x₁, x₂, x₃)) = 2·x₁ + x₂ + x₃
POL(U94(x₁, x₂)) = 2·x₁ + x₂
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) = 1
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

(8) Obligation:

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

zeros → cons(0, zeros)
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
U31(tt, V) → U32(isNatIListKind(V), V)
U32(tt, V) → U33(isNatList(V))
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) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(V1), V1, V2)
U82(tt, V1, V2) → U83(isNatIListKind(V2), V1, V2)
U83(tt, V1, V2) → U84(isNatIListKind(V2), V1, V2)
U84(tt, V1, V2) → U85(isNat(V1), V2)
U85(tt, V2) → U86(isNatList(V2))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(L), L, N)
U92(tt, L, N) → U93(isNat(N), L, N)
U93(tt, L, N) → U94(isNatKind(N), L)
U94(tt, L) → s(length(L))
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U61(isNatIListKind(V1))
isNatKind(s(V1)) → U71(isNatKind(V1))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U81(isNatKind(V1), V1, V2)
length(cons(N, L)) → U91(isNatList(L), L, N)

The replacement map contains the following entries:

(9) PoloCSRProof (EQUIVALENT transformation)

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

U32(tt, V) → U33(isNatList(V))

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(U11(x₁, x₂)) = x₁
POL(U12(x₁, x₂)) = 2·x₁
POL(U13(x₁)) = 2·x₁
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁, x₂)) = 2·x₁
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(U32(x₁, x₂)) = 1 + 2·x₁
POL(U33(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = 2·x₁ + x₂ + 2·x₃
POL(U42(x₁, x₂, x₃)) = 2·x₁ + x₂ + 2·x₃
POL(U43(x₁, x₂, x₃)) = x₁ + x₂ + 2·x₃
POL(U44(x₁, x₂, x₃)) = 2·x₁ + x₂ + 2·x₃
POL(U45(x₁, x₂)) = x₁ + x₂
POL(U46(x₁)) = x₁
POL(U51(x₁, x₂)) = x₁
POL(U52(x₁)) = 2·x₁
POL(U61(x₁)) = 2·x₁
POL(U71(x₁)) = x₁
POL(U81(x₁, x₂, x₃)) = 2·x₁
POL(U82(x₁, x₂, x₃)) = x₁
POL(U83(x₁, x₂, x₃)) = x₁
POL(U84(x₁, x₂, x₃)) = 2·x₁
POL(U85(x₁, x₂)) = x₁
POL(U86(x₁)) = 2·x₁
POL(U91(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + 2·x₃
POL(U92(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + x₃
POL(U93(x₁, x₂, x₃)) = 2·x₁ + 2·x₂
POL(U94(x₁, x₂)) = 2·x₁ + 2·x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = x₁
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = x₁
POL(nil) = 1
POL(s(x₁)) = 2·x₁
POL(tt) = 0
POL(zeros) = 0

(10) Obligation:

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

zeros → cons(0, zeros)
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
U31(tt, V) → U32(isNatIListKind(V), V)
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) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(V1), V1, V2)
U82(tt, V1, V2) → U83(isNatIListKind(V2), V1, V2)
U83(tt, V1, V2) → U84(isNatIListKind(V2), V1, V2)
U84(tt, V1, V2) → U85(isNat(V1), V2)
U85(tt, V2) → U86(isNatList(V2))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(L), L, N)
U92(tt, L, N) → U93(isNat(N), L, N)
U93(tt, L, N) → U94(isNatKind(N), L)
U94(tt, L) → s(length(L))
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U61(isNatIListKind(V1))
isNatKind(s(V1)) → U71(isNatKind(V1))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U81(isNatKind(V1), V1, V2)
length(cons(N, L)) → U91(isNatList(L), L, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U12: {1}
isNatIListKind: empty set
U13: {1}
isNatList: empty set
U21: {1}
U22: {1}
isNatKind: empty set
U23: {1}
isNat: empty set
U31: {1}
U32: {1}
U41: {1}
U42: {1}
U43: {1}
U44: {1}
U45: {1}
U46: {1}
isNatIList: empty set
U51: {1}
U52: {1}
U61: {1}
U71: {1}
U81: {1}
U82: {1}
U83: {1}
U84: {1}
U85: {1}
U86: {1}
U91: {1}
U92: {1}
U93: {1}
U94: {1}
s: {1}
length: {1}
nil: empty set

(11) PoloCSRProof (EQUIVALENT transformation)

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

U31(tt, V) → U32(isNatIListKind(V), V)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(U11(x₁, x₂)) = 2·x₁
POL(U12(x₁, x₂)) = 2·x₁
POL(U13(x₁)) = x₁
POL(U21(x₁, x₂)) = 2·x₁
POL(U22(x₁, x₂)) = x₁
POL(U23(x₁)) = 2·x₁
POL(U31(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(U32(x₁, x₂)) = 1 + 2·x₁ + x₂
POL(U41(x₁, x₂, x₃)) = 2·x₁
POL(U42(x₁, x₂, x₃)) = x₁
POL(U43(x₁, x₂, x₃)) = 2·x₁
POL(U44(x₁, x₂, x₃)) = 2·x₁
POL(U45(x₁, x₂)) = 2·x₁
POL(U46(x₁)) = 2·x₁
POL(U51(x₁, x₂)) = 2·x₁
POL(U52(x₁)) = 2·x₁
POL(U61(x₁)) = 2·x₁
POL(U71(x₁)) = 2·x₁
POL(U81(x₁, x₂, x₃)) = 2·x₁
POL(U82(x₁, x₂, x₃)) = 2·x₁
POL(U83(x₁, x₂, x₃)) = 2·x₁
POL(U84(x₁, x₂, x₃)) = 2·x₁
POL(U85(x₁, x₂)) = x₁
POL(U86(x₁)) = x₁
POL(U91(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + 2·x₃
POL(U92(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + x₃
POL(U93(x₁, x₂, x₃)) = 2·x₁ + x₂
POL(U94(x₁, x₂)) = 2·x₁ + x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·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) = 2
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

(12) Obligation:

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

zeros → cons(0, zeros)
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
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) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(V1), V1, V2)
U82(tt, V1, V2) → U83(isNatIListKind(V2), V1, V2)
U83(tt, V1, V2) → U84(isNatIListKind(V2), V1, V2)
U84(tt, V1, V2) → U85(isNat(V1), V2)
U85(tt, V2) → U86(isNatList(V2))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(L), L, N)
U92(tt, L, N) → U93(isNat(N), L, N)
U93(tt, L, N) → U94(isNatKind(N), L)
U94(tt, L) → s(length(L))
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U61(isNatIListKind(V1))
isNatKind(s(V1)) → U71(isNatKind(V1))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U81(isNatKind(V1), V1, V2)
length(cons(N, L)) → U91(isNatList(L), L, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U12: {1}
isNatIListKind: empty set
U13: {1}
isNatList: empty set
U21: {1}
U22: {1}
isNatKind: empty set
U23: {1}
isNat: empty set
U41: {1}
U42: {1}
U43: {1}
U44: {1}
U45: {1}
U46: {1}
isNatIList: empty set
U51: {1}
U52: {1}
U61: {1}
U71: {1}
U81: {1}
U82: {1}
U83: {1}
U84: {1}
U85: {1}
U86: {1}
U91: {1}
U92: {1}
U93: {1}
U94: {1}
s: {1}
length: {1}
nil: empty set

(13) PoloCSRProof (EQUIVALENT transformation)

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

isNatList(nil) → tt

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(U11(x₁, x₂)) = x₁ + x₂
POL(U12(x₁, x₂)) = x₁ + x₂
POL(U13(x₁)) = x₁
POL(U21(x₁, x₂)) = 2·x₁ + x₂
POL(U22(x₁, x₂)) = 2·x₁ + x₂
POL(U23(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(U42(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + 2·x₃
POL(U43(x₁, x₂, x₃)) = 2·x₁ + x₂ + 2·x₃
POL(U44(x₁, x₂, x₃)) = x₁ + x₂ + 2·x₃
POL(U45(x₁, x₂)) = x₁ + 2·x₂
POL(U46(x₁)) = 2·x₁
POL(U51(x₁, x₂)) = 2·x₁
POL(U52(x₁)) = x₁
POL(U61(x₁)) = x₁
POL(U71(x₁)) = x₁
POL(U81(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + 2·x₃
POL(U82(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + 2·x₃
POL(U83(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + 2·x₃
POL(U84(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(U85(x₁, x₂)) = x₁ + x₂
POL(U86(x₁)) = x₁
POL(U91(x₁, x₂, x₃)) = x₁ + x₂ + 2·x₃
POL(U92(x₁, x₂, x₃)) = 2·x₁ + x₂ + 2·x₃
POL(U93(x₁, x₂, x₃)) = 2·x₁ + x₂
POL(U94(x₁, x₂)) = 2·x₁ + x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = x₁
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = x₁
POL(nil) = 2
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

(14) Obligation:

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

zeros → cons(0, zeros)
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
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) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(V1), V1, V2)
U82(tt, V1, V2) → U83(isNatIListKind(V2), V1, V2)
U83(tt, V1, V2) → U84(isNatIListKind(V2), V1, V2)
U84(tt, V1, V2) → U85(isNat(V1), V2)
U85(tt, V2) → U86(isNatList(V2))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(L), L, N)
U92(tt, L, N) → U93(isNat(N), L, N)
U93(tt, L, N) → U94(isNatKind(N), L)
U94(tt, L) → s(length(L))
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U61(isNatIListKind(V1))
isNatKind(s(V1)) → U71(isNatKind(V1))
isNatList(cons(V1, V2)) → U81(isNatKind(V1), V1, V2)
length(cons(N, L)) → U91(isNatList(L), L, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U12: {1}
isNatIListKind: empty set
U13: {1}
isNatList: empty set
U21: {1}
U22: {1}
isNatKind: empty set
U23: {1}
isNat: empty set
U41: {1}
U42: {1}
U43: {1}
U44: {1}
U45: {1}
U46: {1}
isNatIList: empty set
U51: {1}
U52: {1}
U61: {1}
U71: {1}
U81: {1}
U82: {1}
U83: {1}
U84: {1}
U85: {1}
U86: {1}
U91: {1}
U92: {1}
U93: {1}
U94: {1}
s: {1}
length: {1}
nil: empty set

(15) PoloCSRProof (EQUIVALENT transformation)

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

U12(tt, V1) → U13(isNatList(V1))

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(U11(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(U12(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(U13(x₁)) = 2·x₁
POL(U21(x₁, x₂)) = 2·x₁ + x₂
POL(U22(x₁, x₂)) = x₁ + x₂
POL(U23(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + x₃
POL(U42(x₁, x₂, x₃)) = 2·x₁ + x₂ + x₃
POL(U43(x₁, x₂, x₃)) = 2·x₁ + x₂ + x₃
POL(U44(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U45(x₁, x₂)) = x₁ + x₂
POL(U46(x₁)) = x₁
POL(U51(x₁, x₂)) = x₁
POL(U52(x₁)) = 2·x₁
POL(U61(x₁)) = 2·x₁
POL(U71(x₁)) = x₁
POL(U81(x₁, x₂, x₃)) = 2·x₁ + x₂ + 2·x₃
POL(U82(x₁, x₂, x₃)) = 2·x₁ + x₂ + 2·x₃
POL(U83(x₁, x₂, x₃)) = 2·x₁ + x₂ + 2·x₃
POL(U84(x₁, x₂, x₃)) = x₁ + x₂ + 2·x₃
POL(U85(x₁, x₂)) = x₁ + x₂
POL(U86(x₁)) = x₁
POL(U91(x₁, x₂, x₃)) = 1 + x₁ + 2·x₂ + 2·x₃
POL(U92(x₁, x₂, x₃)) = 1 + 2·x₁ + 2·x₂ + 2·x₃
POL(U93(x₁, x₂, x₃)) = 1 + 2·x₁ + 2·x₂
POL(U94(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = x₁
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 1 + 2·x₁
POL(nil) = 1
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

(16) Obligation:

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

zeros → cons(0, zeros)
U11(tt, V1) → U12(isNatIListKind(V1), V1)
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(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) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(V1), V1, V2)
U82(tt, V1, V2) → U83(isNatIListKind(V2), V1, V2)
U83(tt, V1, V2) → U84(isNatIListKind(V2), V1, V2)
U84(tt, V1, V2) → U85(isNat(V1), V2)
U85(tt, V2) → U86(isNatList(V2))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(L), L, N)
U92(tt, L, N) → U93(isNat(N), L, N)
U93(tt, L, N) → U94(isNatKind(N), L)
U94(tt, L) → s(length(L))
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U61(isNatIListKind(V1))
isNatKind(s(V1)) → U71(isNatKind(V1))
isNatList(cons(V1, V2)) → U81(isNatKind(V1), V1, V2)
length(cons(N, L)) → U91(isNatList(L), L, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U12: {1}
isNatIListKind: empty set
U13: {1}
isNatList: empty set
U21: {1}
U22: {1}
isNatKind: empty set
U23: {1}
isNat: empty set
U41: {1}
U42: {1}
U43: {1}
U44: {1}
U45: {1}
U46: {1}
isNatIList: empty set
U51: {1}
U52: {1}
U61: {1}
U71: {1}
U81: {1}
U82: {1}
U83: {1}
U84: {1}
U85: {1}
U86: {1}
U91: {1}
U92: {1}
U93: {1}
U94: {1}
s: {1}
length: {1}
nil: empty set

(17) PoloCSRProof (EQUIVALENT transformation)

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

U13(tt) → tt

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(U11(x₁, x₂)) = x₁
POL(U12(x₁, x₂)) = 2·x₁
POL(U13(x₁)) = 2 + 2·x₁
POL(U21(x₁, x₂)) = 2·x₁
POL(U22(x₁, x₂)) = 2·x₁
POL(U23(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = x₁
POL(U42(x₁, x₂, x₃)) = 2·x₁
POL(U43(x₁, x₂, x₃)) = 2·x₁
POL(U44(x₁, x₂, x₃)) = x₁
POL(U45(x₁, x₂)) = 2·x₁
POL(U46(x₁)) = 2·x₁
POL(U51(x₁, x₂)) = x₁
POL(U52(x₁)) = 2·x₁
POL(U61(x₁)) = 2·x₁
POL(U71(x₁)) = 2·x₁
POL(U81(x₁, x₂, x₃)) = 2·x₁
POL(U82(x₁, x₂, x₃)) = 2·x₁
POL(U83(x₁, x₂, x₃)) = x₁
POL(U84(x₁, x₂, x₃)) = 2·x₁
POL(U85(x₁, x₂)) = 2·x₁
POL(U86(x₁)) = 2·x₁
POL(U91(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + 2·x₃
POL(U92(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + x₃
POL(U93(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + x₃
POL(U94(x₁, x₂)) = 2·x₁ + 2·x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·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) = 2
POL(s(x₁)) = 2·x₁
POL(tt) = 0
POL(zeros) = 0

(18) Obligation:

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

zeros → cons(0, zeros)
U11(tt, V1) → U12(isNatIListKind(V1), V1)
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(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) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(V1), V1, V2)
U82(tt, V1, V2) → U83(isNatIListKind(V2), V1, V2)
U83(tt, V1, V2) → U84(isNatIListKind(V2), V1, V2)
U84(tt, V1, V2) → U85(isNat(V1), V2)
U85(tt, V2) → U86(isNatList(V2))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(L), L, N)
U92(tt, L, N) → U93(isNat(N), L, N)
U93(tt, L, N) → U94(isNatKind(N), L)
U94(tt, L) → s(length(L))
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U61(isNatIListKind(V1))
isNatKind(s(V1)) → U71(isNatKind(V1))
isNatList(cons(V1, V2)) → U81(isNatKind(V1), V1, V2)
length(cons(N, L)) → U91(isNatList(L), L, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U12: {1}
isNatIListKind: empty set
isNatList: empty set
U21: {1}
U22: {1}
isNatKind: empty set
U23: {1}
isNat: empty set
U41: {1}
U42: {1}
U43: {1}
U44: {1}
U45: {1}
U46: {1}
isNatIList: empty set
U51: {1}
U52: {1}
U61: {1}
U71: {1}
U81: {1}
U82: {1}
U83: {1}
U84: {1}
U85: {1}
U86: {1}
U91: {1}
U92: {1}
U93: {1}
U94: {1}
s: {1}
length: {1}
nil: empty set

(19) PoloCSRProof (EQUIVALENT transformation)

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

isNat(length(V1)) → U11(isNatIListKind(V1), V1)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(U11(x₁, x₂)) = 2·x₁ + x₂
POL(U12(x₁, x₂)) = x₁
POL(U21(x₁, x₂)) = 2·x₁ + x₂
POL(U22(x₁, x₂)) = 2·x₁ + x₂
POL(U23(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + 2·x₃
POL(U42(x₁, x₂, x₃)) = 2·x₁ + x₂ + 2·x₃
POL(U43(x₁, x₂, x₃)) = 2·x₁ + x₂ + 2·x₃
POL(U44(x₁, x₂, x₃)) = 2·x₁ + x₂ + 2·x₃
POL(U45(x₁, x₂)) = x₁ + 2·x₂
POL(U46(x₁)) = x₁
POL(U51(x₁, x₂)) = 2·x₁
POL(U52(x₁)) = x₁
POL(U61(x₁)) = x₁
POL(U71(x₁)) = x₁
POL(U81(x₁, x₂, x₃)) = x₁ + x₂ + 2·x₃
POL(U82(x₁, x₂, x₃)) = 2·x₁ + x₂ + 2·x₃
POL(U83(x₁, x₂, x₃)) = x₁ + x₂ + 2·x₃
POL(U84(x₁, x₂, x₃)) = 2·x₁ + x₂ + 2·x₃
POL(U85(x₁, x₂)) = x₁ + 2·x₂
POL(U86(x₁)) = x₁
POL(U91(x₁, x₂, x₃)) = 1 + 2·x₁ + 2·x₂ + 2·x₃
POL(U92(x₁, x₂, x₃)) = 1 + 2·x₁ + 2·x₂ + x₃
POL(U93(x₁, x₂, x₃)) = 1 + x₁ + 2·x₂
POL(U94(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(cons(x₁, x₂)) = x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 2·x₁
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 1 + 2·x₁
POL(nil) = 2
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

(20) Obligation:

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

zeros → cons(0, zeros)
U11(tt, V1) → U12(isNatIListKind(V1), V1)
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(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) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(V1), V1, V2)
U82(tt, V1, V2) → U83(isNatIListKind(V2), V1, V2)
U83(tt, V1, V2) → U84(isNatIListKind(V2), V1, V2)
U84(tt, V1, V2) → U85(isNat(V1), V2)
U85(tt, V2) → U86(isNatList(V2))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(L), L, N)
U92(tt, L, N) → U93(isNat(N), L, N)
U93(tt, L, N) → U94(isNatKind(N), L)
U94(tt, L) → s(length(L))
isNat(0) → tt
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U61(isNatIListKind(V1))
isNatKind(s(V1)) → U71(isNatKind(V1))
isNatList(cons(V1, V2)) → U81(isNatKind(V1), V1, V2)
length(cons(N, L)) → U91(isNatList(L), L, N)

The replacement map contains the following entries:

(21) PoloCSRProof (EQUIVALENT transformation)

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

U11(tt, V1) → U12(isNatIListKind(V1), V1)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(U11(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(U12(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(U21(x₁, x₂)) = 2·x₁
POL(U22(x₁, x₂)) = 2·x₁
POL(U23(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + 2·x₃
POL(U42(x₁, x₂, x₃)) = 2·x₁ + x₂ + 2·x₃
POL(U43(x₁, x₂, x₃)) = x₁ + 2·x₃
POL(U44(x₁, x₂, x₃)) = 2·x₁ + 2·x₃
POL(U45(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U46(x₁)) = 2·x₁
POL(U51(x₁, x₂)) = 2·x₁
POL(U52(x₁)) = x₁
POL(U61(x₁)) = 2·x₁
POL(U71(x₁)) = 2·x₁
POL(U81(x₁, x₂, x₃)) = 2·x₁
POL(U82(x₁, x₂, x₃)) = x₁
POL(U83(x₁, x₂, x₃)) = 2·x₁
POL(U84(x₁, x₂, x₃)) = 2·x₁
POL(U85(x₁, x₂)) = x₁
POL(U86(x₁)) = 2·x₁
POL(U91(x₁, x₂, x₃)) = 2·x₁ + x₂ + 2·x₃
POL(U92(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U93(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U94(x₁, x₂)) = 2·x₁ + x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = x₁
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = x₁
POL(nil) = 1
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

(22) Obligation:

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

zeros → cons(0, zeros)
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(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) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(V1), V1, V2)
U82(tt, V1, V2) → U83(isNatIListKind(V2), V1, V2)
U83(tt, V1, V2) → U84(isNatIListKind(V2), V1, V2)
U84(tt, V1, V2) → U85(isNat(V1), V2)
U85(tt, V2) → U86(isNatList(V2))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(L), L, N)
U92(tt, L, N) → U93(isNat(N), L, N)
U93(tt, L, N) → U94(isNatKind(N), L)
U94(tt, L) → s(length(L))
isNat(0) → tt
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U61(isNatIListKind(V1))
isNatKind(s(V1)) → U71(isNatKind(V1))
isNatList(cons(V1, V2)) → U81(isNatKind(V1), V1, V2)
length(cons(N, L)) → U91(isNatList(L), L, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
tt: empty set
isNatIListKind: empty set
isNatList: empty set
U21: {1}
U22: {1}
isNatKind: empty set
U23: {1}
isNat: empty set
U41: {1}
U42: {1}
U43: {1}
U44: {1}
U45: {1}
U46: {1}
isNatIList: empty set
U51: {1}
U52: {1}
U61: {1}
U71: {1}
U81: {1}
U82: {1}
U83: {1}
U84: {1}
U85: {1}
U86: {1}
U91: {1}
U92: {1}
U93: {1}
U94: {1}
s: {1}
length: {1}
nil: empty set

(23) CSDependencyPairsProof (EQUIVALENT transformation)

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

(24) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {U23, U46, U52, U61, U71, U86, s, length, U23', U46', U52', U86', LENGTH, U61', U71'} are replacing on all positions.
For all symbols f in {cons, U21, U22, U41, U42, U43, U44, U45, U51, U81, U82, U83, U84, U85, U91, U92, U93, U94, U22', U21', U42', U41', U43', U44', U45', U51', U82', U81', U83', U84', U85', U92', U91', U93', U94'} we have µ(f) = {1}.
The symbols in {isNatKind, isNat, isNatIListKind, isNatIList, isNatList, ISNATKIND, ISNAT, ISNATILISTKIND, ISNATILIST, ISNATLIST, U} are not replacing on any position.

The ordinary context-sensitive dependency pairs DP_o are:

U21'(tt, V1) → U22'(isNatKind(V1), V1)
U21'(tt, V1) → ISNATKIND(V1)
U22'(tt, V1) → U23'(isNat(V1))
U22'(tt, V1) → ISNAT(V1)
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)
U81'(tt, V1, V2) → U82'(isNatKind(V1), V1, V2)
U81'(tt, V1, V2) → ISNATKIND(V1)
U82'(tt, V1, V2) → U83'(isNatIListKind(V2), V1, V2)
U82'(tt, V1, V2) → ISNATILISTKIND(V2)
U83'(tt, V1, V2) → U84'(isNatIListKind(V2), V1, V2)
U83'(tt, V1, V2) → ISNATILISTKIND(V2)
U84'(tt, V1, V2) → U85'(isNat(V1), V2)
U84'(tt, V1, V2) → ISNAT(V1)
U85'(tt, V2) → U86'(isNatList(V2))
U85'(tt, V2) → ISNATLIST(V2)
U91'(tt, L, N) → U92'(isNatIListKind(L), L, N)
U91'(tt, L, N) → ISNATILISTKIND(L)
U92'(tt, L, N) → U93'(isNat(N), L, N)
U92'(tt, L, N) → ISNAT(N)
U93'(tt, L, N) → U94'(isNatKind(N), L)
U93'(tt, L, N) → ISNATKIND(N)
U94'(tt, L) → LENGTH(L)
ISNAT(s(V1)) → U21'(isNatKind(V1), V1)
ISNAT(s(V1)) → ISNATKIND(V1)
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)
ISNATKIND(length(V1)) → U61'(isNatIListKind(V1))
ISNATKIND(length(V1)) → ISNATILISTKIND(V1)
ISNATKIND(s(V1)) → U71'(isNatKind(V1))
ISNATKIND(s(V1)) → ISNATKIND(V1)
ISNATLIST(cons(V1, V2)) → U81'(isNatKind(V1), V1, V2)
ISNATLIST(cons(V1, V2)) → ISNATKIND(V1)
LENGTH(cons(N, L)) → U91'(isNatList(L), L, N)
LENGTH(cons(N, L)) → ISNATLIST(L)

The collapsing dependency pairs are DP_c:

U94'(tt, L) → L

The hidden terms of R are:

zeros

Every hiding context is built from:none

Hence, the new unhiding pairs DP_u are :

U94'(tt, L) → U(L)
U(zeros) → ZEROS

The TRS R consists of the following rules:

zeros → cons(0, zeros)
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(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) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(V1), V1, V2)
U82(tt, V1, V2) → U83(isNatIListKind(V2), V1, V2)
U83(tt, V1, V2) → U84(isNatIListKind(V2), V1, V2)
U84(tt, V1, V2) → U85(isNat(V1), V2)
U85(tt, V2) → U86(isNatList(V2))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(L), L, N)
U92(tt, L, N) → U93(isNat(N), L, N)
U93(tt, L, N) → U94(isNatKind(N), L)
U94(tt, L) → s(length(L))
isNat(0) → tt
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U61(isNatIListKind(V1))
isNatKind(s(V1)) → U71(isNatKind(V1))
isNatList(cons(V1, V2)) → U81(isNatKind(V1), V1, V2)
length(cons(N, L)) → U91(isNatList(L), L, N)

Q is empty.

(25) QCSDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 5 SCCs with 24 less nodes.

(26) Complex Obligation (AND)

(27) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {U23, U46, U52, U61, U71, U86, s, length} are replacing on all positions.
For all symbols f in {cons, U21, U22, U41, U42, U43, U44, U45, U51, U81, U82, U83, U84, U85, U91, U92, U93, U94, U51'} we have µ(f) = {1}.
The symbols in {isNatKind, isNat, isNatIListKind, 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)

The TRS R consists of the following rules:

zeros → cons(0, zeros)
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(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) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(V1), V1, V2)
U82(tt, V1, V2) → U83(isNatIListKind(V2), V1, V2)
U83(tt, V1, V2) → U84(isNatIListKind(V2), V1, V2)
U84(tt, V1, V2) → U85(isNat(V1), V2)
U85(tt, V2) → U86(isNatList(V2))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(L), L, N)
U92(tt, L, N) → U93(isNat(N), L, N)
U93(tt, L, N) → U94(isNatKind(N), L)
U94(tt, L) → s(length(L))
isNat(0) → tt
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U61(isNatIListKind(V1))
isNatKind(s(V1)) → U71(isNatKind(V1))
isNatList(cons(V1, V2)) → U81(isNatKind(V1), V1, V2)
length(cons(N, L)) → U91(isNatList(L), L, N)

Q is empty.

(28) QCSUsableRulesProof (EQUIVALENT transformation)

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

zeros → cons(0, zeros)
U21(tt, x0) → U22(isNatKind(x0), x0)
U22(tt, x0) → U23(isNat(x0))
U23(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
U81(tt, x0, x1) → U82(isNatKind(x0), x0, x1)
U82(tt, x0, x1) → U83(isNatIListKind(x1), x0, x1)
U83(tt, x0, x1) → U84(isNatIListKind(x1), x0, x1)
U84(tt, x0, x1) → U85(isNat(x0), x1)
U85(tt, x0) → U86(isNatList(x0))
U86(tt) → tt
U91(tt, x0, x1) → U92(isNatIListKind(x0), x0, x1)
U92(tt, x0, x1) → U93(isNat(x1), x0, x1)
U93(tt, x0, x1) → U94(isNatKind(x1), x0)
U94(tt, x0) → s(length(x0))
isNat(0) → tt
isNat(s(x0)) → U21(isNatKind(x0), x0)
isNatIList(cons(x0, x1)) → U41(isNatKind(x0), x0, x1)
isNatList(cons(x0, x1)) → U81(isNatKind(x0), x0, x1)
length(cons(x0, x1)) → U91(isNatList(x1), x1, x0)

(29) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {length, U61, s, U71, U52} are replacing on all positions.
For all symbols f in {cons, U51, U51'} 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)

The TRS R consists of the following rules:

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

Q is empty.

(30) 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)
ISNATILISTKIND(cons(V1, V2)) → U51'(isNatKind(V1), V2)
U51'(tt, V2) → ISNATILISTKIND(V2)
ISNATILISTKIND(cons(V1, V2)) → ISNATKIND(V1)
ISNATKIND(s(V1)) → ISNATKIND(V1)

Strictly oriented rules of the TRS R:

isNatKind(0) → tt
isNatKind(length(V1)) → U61(isNatIListKind(V1))
U71(tt) → tt
U51(tt, V2) → U52(isNatIListKind(V2))
U52(tt) → tt
U61(tt) → tt

Used ordering: POLO with Polynomial interpretation [POLO]:

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

(31) Obligation:

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

The TRS P consists of the following rules:
none

The TRS R consists of the following rules:

isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatKind(s(V1)) → U71(isNatKind(V1))

Q is empty.

(32) PIsEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R,µ)-chain.

(33) TRUE

(34) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {U23, U46, U52, U61, U71, U86, s, length} are replacing on all positions.
For all symbols f in {cons, U21, U22, U41, U42, U43, U44, U45, U51, U81, U82, U83, U84, U85, U91, U92, U93, U94, U22', U21'} we have µ(f) = {1}.
The symbols in {isNatKind, isNat, isNatIListKind, isNatIList, isNatList, ISNAT} are not replacing on any position.

The TRS P consists of the following rules:

U22'(tt, V1) → ISNAT(V1)
ISNAT(s(V1)) → U21'(isNatKind(V1), V1)
U21'(tt, V1) → U22'(isNatKind(V1), V1)

The TRS R consists of the following rules:

zeros → cons(0, zeros)
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(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) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(V1), V1, V2)
U82(tt, V1, V2) → U83(isNatIListKind(V2), V1, V2)
U83(tt, V1, V2) → U84(isNatIListKind(V2), V1, V2)
U84(tt, V1, V2) → U85(isNat(V1), V2)
U85(tt, V2) → U86(isNatList(V2))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(L), L, N)
U92(tt, L, N) → U93(isNat(N), L, N)
U93(tt, L, N) → U94(isNatKind(N), L)
U94(tt, L) → s(length(L))
isNat(0) → tt
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U61(isNatIListKind(V1))
isNatKind(s(V1)) → U71(isNatKind(V1))
isNatList(cons(V1, V2)) → U81(isNatKind(V1), V1, V2)
length(cons(N, L)) → U91(isNatList(L), L, N)

Q is empty.

(35) QCSDPSubtermProof (EQUIVALENT transformation)

We use the subterm processor [DA_EMMES].

The following pairs can be oriented strictly and are deleted.

ISNAT(s(V1)) → U21'(isNatKind(V1), V1)

The remaining pairs can at least be oriented weakly.

U22'(tt, V1) → ISNAT(V1)
U21'(tt, V1) → U22'(isNatKind(V1), V1)

Used ordering: Combined order from the following AFS and order.
ISNAT(x1) = x1
U22'(x1, x2) = x2
U21'(x1, x2) = x2

Subterm Order

(36) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {U23, U46, U52, U61, U71, U86, s, length} are replacing on all positions.
For all symbols f in {cons, U21, U22, U41, U42, U43, U44, U45, U51, U81, U82, U83, U84, U85, U91, U92, U93, U94, U22', U21'} we have µ(f) = {1}.
The symbols in {isNatKind, isNat, isNatIListKind, isNatIList, isNatList, ISNAT} are not replacing on any position.

The TRS P consists of the following rules:

U22'(tt, V1) → ISNAT(V1)
U21'(tt, V1) → U22'(isNatKind(V1), V1)

The TRS R consists of the following rules:

zeros → cons(0, zeros)
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(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) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(V1), V1, V2)
U82(tt, V1, V2) → U83(isNatIListKind(V2), V1, V2)
U83(tt, V1, V2) → U84(isNatIListKind(V2), V1, V2)
U84(tt, V1, V2) → U85(isNat(V1), V2)
U85(tt, V2) → U86(isNatList(V2))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(L), L, N)
U92(tt, L, N) → U93(isNat(N), L, N)
U93(tt, L, N) → U94(isNatKind(N), L)
U94(tt, L) → s(length(L))
isNat(0) → tt
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U61(isNatIListKind(V1))
isNatKind(s(V1)) → U71(isNatKind(V1))
isNatList(cons(V1, V2)) → U81(isNatKind(V1), V1, V2)
length(cons(N, L)) → U91(isNatList(L), L, N)

Q is empty.

(37) QCSDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 2 less nodes.

(38) TRUE

(39) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {U23, U46, U52, U61, U71, U86, s, length} are replacing on all positions.
For all symbols f in {cons, U21, U22, U41, U42, U43, U44, U45, U51, U81, U82, U83, U84, U85, U91, U92, U93, U94, U85', U84', U81', U82', U83'} we have µ(f) = {1}.
The symbols in {isNatKind, isNat, isNatIListKind, isNatIList, isNatList, ISNATLIST} are not replacing on any position.

The TRS P consists of the following rules:

U84'(tt, V1, V2) → U85'(isNat(V1), V2)
U85'(tt, V2) → ISNATLIST(V2)
ISNATLIST(cons(V1, V2)) → U81'(isNatKind(V1), V1, V2)
U81'(tt, V1, V2) → U82'(isNatKind(V1), V1, V2)
U82'(tt, V1, V2) → U83'(isNatIListKind(V2), V1, V2)
U83'(tt, V1, V2) → U84'(isNatIListKind(V2), V1, V2)

The TRS R consists of the following rules:

zeros → cons(0, zeros)
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(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) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(V1), V1, V2)
U82(tt, V1, V2) → U83(isNatIListKind(V2), V1, V2)
U83(tt, V1, V2) → U84(isNatIListKind(V2), V1, V2)
U84(tt, V1, V2) → U85(isNat(V1), V2)
U85(tt, V2) → U86(isNatList(V2))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(L), L, N)
U92(tt, L, N) → U93(isNat(N), L, N)
U93(tt, L, N) → U94(isNatKind(N), L)
U94(tt, L) → s(length(L))
isNat(0) → tt
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U61(isNatIListKind(V1))
isNatKind(s(V1)) → U71(isNatKind(V1))
isNatList(cons(V1, V2)) → U81(isNatKind(V1), V1, V2)
length(cons(N, L)) → U91(isNatList(L), L, N)

Q is empty.

(40) QCSUsableRulesProof (EQUIVALENT transformation)

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

zeros → cons(0, zeros)
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
U81(tt, x0, x1) → U82(isNatKind(x0), x0, x1)
U82(tt, x0, x1) → U83(isNatIListKind(x1), x0, x1)
U83(tt, x0, x1) → U84(isNatIListKind(x1), x0, x1)
U84(tt, x0, x1) → U85(isNat(x0), x1)
U85(tt, x0) → U86(isNatList(x0))
U86(tt) → tt
U91(tt, x0, x1) → U92(isNatIListKind(x0), x0, x1)
U92(tt, x0, x1) → U93(isNat(x1), x0, x1)
U93(tt, x0, x1) → U94(isNatKind(x1), x0)
U94(tt, x0) → s(length(x0))
isNatIList(cons(x0, x1)) → U41(isNatKind(x0), x0, x1)
isNatList(cons(x0, x1)) → U81(isNatKind(x0), x0, x1)
length(cons(x0, x1)) → U91(isNatList(x1), x1, x0)

(41) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {s, length, U61, U71, U52, U23} are replacing on all positions.
For all symbols f in {U21, cons, U51, U22, U85', U84', U81', U82', U83'} we have µ(f) = {1}.
The symbols in {isNat, isNatKind, isNatIListKind, ISNATLIST} are not replacing on any position.

The TRS P consists of the following rules:

U84'(tt, V1, V2) → U85'(isNat(V1), V2)
U85'(tt, V2) → ISNATLIST(V2)
ISNATLIST(cons(V1, V2)) → U81'(isNatKind(V1), V1, V2)
U81'(tt, V1, V2) → U82'(isNatKind(V1), V1, V2)
U82'(tt, V1, V2) → U83'(isNatIListKind(V2), V1, V2)
U83'(tt, V1, V2) → U84'(isNatIListKind(V2), V1, V2)

The TRS R consists of the following rules:

isNat(0) → tt
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(length(V1)) → U61(isNatIListKind(V1))
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatKind(s(V1)) → U71(isNatKind(V1))
U71(tt) → tt
U51(tt, V2) → U52(isNatIListKind(V2))
U52(tt) → tt
U61(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt

Q is empty.

(42) QCSDPMuMonotonicPoloProof (EQUIVALENT transformation)

U81'(tt, V1, V2) → U82'(isNatKind(V1), V1, V2)

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(0) = 2
POL(ISNATLIST(x₁)) = x₁
POL(U21(x₁, x₂)) = 2·x₁
POL(U22(x₁, x₂)) = 2·x₁
POL(U23(x₁)) = 2·x₁
POL(U51(x₁, x₂)) = x₁
POL(U52(x₁)) = x₁
POL(U61(x₁)) = 2·x₁
POL(U71(x₁)) = x₁
POL(U81'(x₁, x₂, x₃)) = 2 + 2·x₁ + 2·x₃
POL(U82'(x₁, x₂, x₃)) = x₁ + 2·x₃
POL(U83'(x₁, x₂, x₃)) = 2·x₁ + 2·x₃
POL(U84'(x₁, x₂, x₃)) = x₁ + x₃
POL(U85'(x₁, x₂)) = 2·x₁ + x₂
POL(cons(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(isNat(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(length(x₁)) = 2 + 2·x₁
POL(nil) = 1
POL(s(x₁)) = 1 + 2·x₁
POL(tt) = 0
POL(zeros) = 2

(43) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {s, length, U61, U71, U52, U23} are replacing on all positions.
For all symbols f in {U21, cons, U51, U22, U85', U84', U81', U83', U82'} we have µ(f) = {1}.
The symbols in {isNat, isNatKind, isNatIListKind, ISNATLIST} are not replacing on any position.

The TRS P consists of the following rules:

U84'(tt, V1, V2) → U85'(isNat(V1), V2)
U85'(tt, V2) → ISNATLIST(V2)
ISNATLIST(cons(V1, V2)) → U81'(isNatKind(V1), V1, V2)
U82'(tt, V1, V2) → U83'(isNatIListKind(V2), V1, V2)
U83'(tt, V1, V2) → U84'(isNatIListKind(V2), V1, V2)

The TRS R consists of the following rules:

isNat(0) → tt
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(length(V1)) → U61(isNatIListKind(V1))
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatKind(s(V1)) → U71(isNatKind(V1))
U71(tt) → tt
U51(tt, V2) → U52(isNatIListKind(V2))
U52(tt) → tt
U61(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt

Q is empty.

(44) QCSDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 5 less nodes.

(45) TRUE

(46) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {U23, U46, U52, U61, U71, U86, s, length, LENGTH} are replacing on all positions.
For all symbols f in {cons, U21, U22, U41, U42, U43, U44, U45, U51, U81, U82, U83, U84, U85, U91, U92, U93, U94, U91', U92', U93', U94'} we have µ(f) = {1}.
The symbols in {isNatKind, isNat, isNatIListKind, isNatIList, isNatList} are not replacing on any position.

The TRS P consists of the following rules:

LENGTH(cons(N, L)) → U91'(isNatList(L), L, N)
U91'(tt, L, N) → U92'(isNatIListKind(L), L, N)
U92'(tt, L, N) → U93'(isNat(N), L, N)
U93'(tt, L, N) → U94'(isNatKind(N), L)
U94'(tt, L) → LENGTH(L)

The TRS R consists of the following rules:

zeros → cons(0, zeros)
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(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) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(V1), V1, V2)
U82(tt, V1, V2) → U83(isNatIListKind(V2), V1, V2)
U83(tt, V1, V2) → U84(isNatIListKind(V2), V1, V2)
U84(tt, V1, V2) → U85(isNat(V1), V2)
U85(tt, V2) → U86(isNatList(V2))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(L), L, N)
U92(tt, L, N) → U93(isNat(N), L, N)
U93(tt, L, N) → U94(isNatKind(N), L)
U94(tt, L) → s(length(L))
isNat(0) → tt
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U61(isNatIListKind(V1))
isNatKind(s(V1)) → U71(isNatKind(V1))
isNatList(cons(V1, V2)) → U81(isNatKind(V1), V1, V2)
length(cons(N, L)) → U91(isNatList(L), L, N)

Q is empty.

(47) QCSDPReductionPairProof (EQUIVALENT transformation)

Using the order
Polynomial interpretation [POLO]:

POL(0) = 0
POL(LENGTH(x₁)) = 1
POL(U21(x₁, x₂)) = 2
POL(U22(x₁, x₂)) = 2
POL(U23(x₁)) = 2
POL(U51(x₁, x₂)) = 2
POL(U52(x₁)) = x₁
POL(U61(x₁)) = 1 + x₁
POL(U71(x₁)) = 2
POL(U81(x₁, x₂, x₃)) = 0
POL(U82(x₁, x₂, x₃)) = 0
POL(U83(x₁, x₂, x₃)) = 0
POL(U84(x₁, x₂, x₃)) = 0
POL(U85(x₁, x₂)) = 0
POL(U86(x₁)) = x₁
POL(U91(x₁, x₂, x₃)) = 1
POL(U91'(x₁, x₂, x₃)) = 2·x₁
POL(U92(x₁, x₂, x₃)) = 0
POL(U92'(x₁, x₂, x₃)) = 2·x₁
POL(U93(x₁, x₂, x₃)) = 0
POL(U93'(x₁, x₂, x₃)) = 2·x₁
POL(U94(x₁, x₂)) = 0
POL(U94'(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 2
POL(isNatIListKind(x₁)) = 2
POL(isNatKind(x₁)) = 2 + x₁
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 2
POL(nil) = 2
POL(s(x₁)) = 0
POL(tt) = 2
POL(zeros) = 1

the following usable rules

isNatList(cons(V1, V2)) → U81(isNatKind(V1), V1, V2)
U81(tt, V1, V2) → U82(isNatKind(V1), V1, V2)
U82(tt, V1, V2) → U83(isNatIListKind(V2), V1, V2)
U83(tt, V1, V2) → U84(isNatIListKind(V2), V1, V2)
U84(tt, V1, V2) → U85(isNat(V1), V2)
U85(tt, V2) → U86(isNatList(V2))
U86(tt) → tt
isNat(0) → tt
isNat(s(V1)) → U21(isNatKind(V1), V1)
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
isNatKind(0) → tt
isNatKind(length(V1)) → U61(isNatIListKind(V1))
isNatKind(s(V1)) → U71(isNatKind(V1))
length(cons(N, L)) → U91(isNatList(L), L, N)
U91(tt, L, N) → U92(isNatIListKind(L), L, N)
U92(tt, L, N) → U93(isNat(N), L, N)
U93(tt, L, N) → U94(isNatKind(N), L)
U94(tt, L) → s(length(L))
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
zeros → cons(0, zeros)
U51(tt, V2) → U52(isNatIListKind(V2))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt

could all be oriented weakly.
Furthermore, the pairs

LENGTH(cons(N, L)) → U91'(isNatList(L), L, N)
U93'(tt, L, N) → U94'(isNatKind(N), L)

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

(48) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {U23, U46, U52, U61, U71, U86, s, length, LENGTH} are replacing on all positions.
For all symbols f in {cons, U21, U22, U41, U42, U43, U44, U45, U51, U81, U82, U83, U84, U85, U91, U92, U93, U94, U92', U91', U93', U94'} we have µ(f) = {1}.
The symbols in {isNatKind, isNat, isNatIListKind, isNatIList, isNatList} are not replacing on any position.

The TRS P consists of the following rules:

U91'(tt, L, N) → U92'(isNatIListKind(L), L, N)
U92'(tt, L, N) → U93'(isNat(N), L, N)
U94'(tt, L) → LENGTH(L)

The TRS R consists of the following rules:

zeros → cons(0, zeros)
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(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) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(V1), V1, V2)
U82(tt, V1, V2) → U83(isNatIListKind(V2), V1, V2)
U83(tt, V1, V2) → U84(isNatIListKind(V2), V1, V2)
U84(tt, V1, V2) → U85(isNat(V1), V2)
U85(tt, V2) → U86(isNatList(V2))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(L), L, N)
U92(tt, L, N) → U93(isNat(N), L, N)
U93(tt, L, N) → U94(isNatKind(N), L)
U94(tt, L) → s(length(L))
isNat(0) → tt
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U61(isNatIListKind(V1))
isNatKind(s(V1)) → U71(isNatKind(V1))
isNatList(cons(V1, V2)) → U81(isNatKind(V1), V1, V2)
length(cons(N, L)) → U91(isNatList(L), L, N)

Q is empty.

(49) QCSDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 2 less nodes.

(50) TRUE

(51) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {U23, U46, U52, U61, U71, U86, s, length} are replacing on all positions.
For all symbols f in {cons, U21, U22, U41, U42, U43, U44, U45, U51, U81, U82, U83, U84, U85, U91, U92, U93, U94, U45', U44', U41', U42', U43'} we have µ(f) = {1}.
The symbols in {isNatKind, isNat, isNatIListKind, isNatIList, isNatList, ISNATILIST} are not replacing on any position.

The TRS P consists of the following rules:

U44'(tt, V1, V2) → U45'(isNat(V1), V2)
U45'(tt, V2) → ISNATILIST(V2)
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)

The TRS R consists of the following rules:

zeros → cons(0, zeros)
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(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) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(V1), V1, V2)
U82(tt, V1, V2) → U83(isNatIListKind(V2), V1, V2)
U83(tt, V1, V2) → U84(isNatIListKind(V2), V1, V2)
U84(tt, V1, V2) → U85(isNat(V1), V2)
U85(tt, V2) → U86(isNatList(V2))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(L), L, N)
U92(tt, L, N) → U93(isNat(N), L, N)
U93(tt, L, N) → U94(isNatKind(N), L)
U94(tt, L) → s(length(L))
isNat(0) → tt
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(cons(V1, V2)) → U41(isNatKind(V1), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatKind(0) → tt
isNatKind(length(V1)) → U61(isNatIListKind(V1))
isNatKind(s(V1)) → U71(isNatKind(V1))
isNatList(cons(V1, V2)) → U81(isNatKind(V1), V1, V2)
length(cons(N, L)) → U91(isNatList(L), L, N)

Q is empty.

(52) QCSUsableRulesProof (EQUIVALENT transformation)

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

zeros → cons(0, zeros)
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
U81(tt, x0, x1) → U82(isNatKind(x0), x0, x1)
U82(tt, x0, x1) → U83(isNatIListKind(x1), x0, x1)
U83(tt, x0, x1) → U84(isNatIListKind(x1), x0, x1)
U84(tt, x0, x1) → U85(isNat(x0), x1)
U85(tt, x0) → U86(isNatList(x0))
U86(tt) → tt
U91(tt, x0, x1) → U92(isNatIListKind(x0), x0, x1)
U92(tt, x0, x1) → U93(isNat(x1), x0, x1)
U93(tt, x0, x1) → U94(isNatKind(x1), x0)
U94(tt, x0) → s(length(x0))
isNatIList(cons(x0, x1)) → U41(isNatKind(x0), x0, x1)
isNatList(cons(x0, x1)) → U81(isNatKind(x0), x0, x1)
length(cons(x0, x1)) → U91(isNatList(x1), x1, x0)

(53) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {s, length, U61, U71, U52, U23} are replacing on all positions.
For all symbols f in {U21, cons, U51, U22, U45', U44', U41', U42', U43'} we have µ(f) = {1}.
The symbols in {isNat, isNatKind, isNatIListKind, ISNATILIST} are not replacing on any position.

The TRS P consists of the following rules:

U44'(tt, V1, V2) → U45'(isNat(V1), V2)
U45'(tt, V2) → ISNATILIST(V2)
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)

The TRS R consists of the following rules:

isNat(0) → tt
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(length(V1)) → U61(isNatIListKind(V1))
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatKind(s(V1)) → U71(isNatKind(V1))
U71(tt) → tt
U51(tt, V2) → U52(isNatIListKind(V2))
U52(tt) → tt
U61(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt

Q is empty.

(54) QCSDPMuMonotonicPoloProof (EQUIVALENT transformation)

U41'(tt, V1, V2) → U42'(isNatKind(V1), V1, V2)

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(0) = 2
POL(ISNATILIST(x₁)) = x₁
POL(U21(x₁, x₂)) = 2·x₁
POL(U22(x₁, x₂)) = 2·x₁
POL(U23(x₁)) = 2·x₁
POL(U41'(x₁, x₂, x₃)) = 2 + 2·x₁ + 2·x₃
POL(U42'(x₁, x₂, x₃)) = x₁ + 2·x₃
POL(U43'(x₁, x₂, x₃)) = 2·x₁ + 2·x₃
POL(U44'(x₁, x₂, x₃)) = x₁ + x₃
POL(U45'(x₁, x₂)) = 2·x₁ + x₂
POL(U51(x₁, x₂)) = x₁
POL(U52(x₁)) = x₁
POL(U61(x₁)) = 2·x₁
POL(U71(x₁)) = x₁
POL(cons(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(isNat(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(length(x₁)) = 2 + 2·x₁
POL(nil) = 1
POL(s(x₁)) = 1 + 2·x₁
POL(tt) = 0
POL(zeros) = 2

(55) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {s, length, U61, U71, U52, U23} are replacing on all positions.
For all symbols f in {U21, cons, U51, U22, U45', U44', U41', U43', U42'} we have µ(f) = {1}.
The symbols in {isNat, isNatKind, isNatIListKind, ISNATILIST} are not replacing on any position.

The TRS P consists of the following rules:

U44'(tt, V1, V2) → U45'(isNat(V1), V2)
U45'(tt, V2) → ISNATILIST(V2)
ISNATILIST(cons(V1, V2)) → U41'(isNatKind(V1), V1, V2)
U42'(tt, V1, V2) → U43'(isNatIListKind(V2), V1, V2)
U43'(tt, V1, V2) → U44'(isNatIListKind(V2), V1, V2)

The TRS R consists of the following rules:

isNat(0) → tt
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(length(V1)) → U61(isNatIListKind(V1))
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → U51(isNatKind(V1), V2)
isNatKind(s(V1)) → U71(isNatKind(V1))
U71(tt) → tt
U51(tt, V2) → U52(isNatIListKind(V2))
U52(tt) → tt
U61(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt

Q is empty.

(56) QCSDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 5 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) PoloCSRProof (EQUIVALENT transformation)

(10) Obligation:

(11) PoloCSRProof (EQUIVALENT transformation)

(12) Obligation:

(13) PoloCSRProof (EQUIVALENT transformation)

(14) Obligation:

(15) PoloCSRProof (EQUIVALENT transformation)

(16) Obligation:

(17) PoloCSRProof (EQUIVALENT transformation)

(18) Obligation:

(19) PoloCSRProof (EQUIVALENT transformation)

(20) Obligation:

(21) PoloCSRProof (EQUIVALENT transformation)

(22) Obligation:

(23) CSDependencyPairsProof (EQUIVALENT transformation)

(24) Obligation:

(25) QCSDependencyGraphProof (EQUIVALENT transformation)

(26) Complex Obligation (AND)

(27) Obligation:

(28) QCSUsableRulesProof (EQUIVALENT transformation)

(29) Obligation:

(30) QCSDPMuMonotonicPoloProof (EQUIVALENT transformation)

(31) Obligation:

(32) PIsEmptyProof (EQUIVALENT transformation)

(33) TRUE

(34) Obligation:

(35) QCSDPSubtermProof (EQUIVALENT transformation)

(36) Obligation:

(37) QCSDependencyGraphProof (EQUIVALENT transformation)

(38) TRUE

(39) Obligation:

(40) QCSUsableRulesProof (EQUIVALENT transformation)

(41) Obligation:

(42) QCSDPMuMonotonicPoloProof (EQUIVALENT transformation)

(43) Obligation:

(44) QCSDependencyGraphProof (EQUIVALENT transformation)

(45) TRUE

(46) Obligation:

(47) QCSDPReductionPairProof (EQUIVALENT transformation)

(48) Obligation:

(49) QCSDependencyGraphProof (EQUIVALENT transformation)

(50) TRUE

(51) Obligation:

(52) QCSUsableRulesProof (EQUIVALENT transformation)

(53) Obligation:

(54) QCSDPMuMonotonicPoloProof (EQUIVALENT transformation)

(55) Obligation:

(56) QCSDependencyGraphProof (EQUIVALENT transformation)

(57) TRUE