Termination w.r.t. Q proof of /tmp/tmpWJM6Ji/OvConsOS_complete_noand

Termination w.r.t. Q of the given QTRS could not be shown:

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

  ↳5 QDP

    ↳6 QDPOrderProof (⇔)

    ↳7 QDP

    ↳8 DependencyGraphProof (⇔)

    ↳9 TRUE

  ↳10 QDP

    ↳11 QDPOrderProof (⇔)

    ↳12 QDP

    ↳13 DependencyGraphProof (⇔)

    ↳14 QDP

    ↳15 QDPOrderProof (⇔)

    ↳16 QDP

    ↳17 DependencyGraphProof (⇔)

    ↳18 TRUE

  ↳19 QDP

(0) Obligation:

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

a__zeros → cons(0, zeros)
a__U101(tt, V1, V2) → a__U102(a__isNatKind(V1), V1, V2)
a__U102(tt, V1, V2) → a__U103(a__isNatIListKind(V2), V1, V2)
a__U103(tt, V1, V2) → a__U104(a__isNatIListKind(V2), V1, V2)
a__U104(tt, V1, V2) → a__U105(a__isNat(V1), V2)
a__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__U111(tt, L, N) → a__U112(a__isNatIListKind(L), L, N)
a__U112(tt, L, N) → a__U113(a__isNat(N), L, N)
a__U113(tt, L, N) → a__U114(a__isNatKind(N), L)
a__U114(tt, L) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
a__U131(tt, IL, M, N) → a__U132(a__isNatIListKind(IL), IL, M, N)
a__U132(tt, IL, M, N) → a__U133(a__isNat(M), IL, M, N)
a__U133(tt, IL, M, N) → a__U134(a__isNatKind(M), IL, M, N)
a__U134(tt, IL, M, N) → a__U135(a__isNat(N), IL, M, N)
a__U135(tt, IL, M, N) → a__U136(a__isNatKind(N), IL, M, N)
a__U136(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
a__U91(tt, V1, V2) → a__U92(a__isNatKind(V1), V1, V2)
a__U92(tt, V1, V2) → a__U93(a__isNatIListKind(V2), V1, V2)
a__U93(tt, V1, V2) → a__U94(a__isNatIListKind(V2), V1, V2)
a__U94(tt, V1, V2) → a__U95(a__isNat(V1), V2)
a__U95(tt, V2) → a__U96(a__isNatList(V2))
a__U96(tt) → tt
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
a__take(0, IL) → a__U121(a__isNatIList(IL), IL)
a__take(s(M), cons(N, IL)) → a__U131(a__isNatIList(IL), IL, M, N)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U111(X1, X2, X3)) → a__U111(mark(X1), X2, X3)
mark(U112(X1, X2, X3)) → a__U112(mark(X1), X2, X3)
mark(U113(X1, X2, X3)) → a__U113(mark(X1), X2, X3)
mark(U114(X1, X2)) → a__U114(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(mark(X))
mark(U131(X1, X2, X3, X4)) → a__U131(mark(X1), X2, X3, X4)
mark(U132(X1, X2, X3, X4)) → a__U132(mark(X1), X2, X3, X4)
mark(U133(X1, X2, X3, X4)) → a__U133(mark(X1), X2, X3, X4)
mark(U134(X1, X2, X3, X4)) → a__U134(mark(X1), X2, X3, X4)
mark(U135(X1, X2, X3, X4)) → a__U135(mark(X1), X2, X3, X4)
mark(U136(X1, X2, X3, X4)) → a__U136(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U94(X1, X2, X3)) → a__U94(mark(X1), X2, X3)
mark(U95(X1, X2)) → a__U95(mark(X1), X2)
mark(U96(X)) → a__U96(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__U101(tt, V1, V2) → A__U102(a__isNatKind(V1), V1, V2)
A__U101(tt, V1, V2) → A__ISNATKIND(V1)
A__U102(tt, V1, V2) → A__U103(a__isNatIListKind(V2), V1, V2)
A__U102(tt, V1, V2) → A__ISNATILISTKIND(V2)
A__U103(tt, V1, V2) → A__U104(a__isNatIListKind(V2), V1, V2)
A__U103(tt, V1, V2) → A__ISNATILISTKIND(V2)
A__U104(tt, V1, V2) → A__U105(a__isNat(V1), V2)
A__U104(tt, V1, V2) → A__ISNAT(V1)
A__U105(tt, V2) → A__U106(a__isNatIList(V2))
A__U105(tt, V2) → A__ISNATILIST(V2)
A__U11(tt, V1) → A__U12(a__isNatIListKind(V1), V1)
A__U11(tt, V1) → A__ISNATILISTKIND(V1)
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__U111(tt, L, N) → A__ISNATILISTKIND(L)
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)
A__U112(tt, L, N) → A__ISNAT(N)
A__U113(tt, L, N) → A__U114(a__isNatKind(N), L)
A__U113(tt, L, N) → A__ISNATKIND(N)
A__U114(tt, L) → A__LENGTH(mark(L))
A__U114(tt, L) → MARK(L)
A__U12(tt, V1) → A__U13(a__isNatList(V1))
A__U12(tt, V1) → A__ISNATLIST(V1)
A__U121(tt, IL) → A__U122(a__isNatIListKind(IL))
A__U121(tt, IL) → A__ISNATILISTKIND(IL)
A__U131(tt, IL, M, N) → A__U132(a__isNatIListKind(IL), IL, M, N)
A__U131(tt, IL, M, N) → A__ISNATILISTKIND(IL)
A__U132(tt, IL, M, N) → A__U133(a__isNat(M), IL, M, N)
A__U132(tt, IL, M, N) → A__ISNAT(M)
A__U133(tt, IL, M, N) → A__U134(a__isNatKind(M), IL, M, N)
A__U133(tt, IL, M, N) → A__ISNATKIND(M)
A__U134(tt, IL, M, N) → A__U135(a__isNat(N), IL, M, N)
A__U134(tt, IL, M, N) → A__ISNAT(N)
A__U135(tt, IL, M, N) → A__U136(a__isNatKind(N), IL, M, N)
A__U135(tt, IL, M, N) → A__ISNATKIND(N)
A__U136(tt, IL, M, N) → MARK(N)
A__U21(tt, V1) → A__U22(a__isNatKind(V1), V1)
A__U21(tt, V1) → A__ISNATKIND(V1)
A__U22(tt, V1) → A__U23(a__isNat(V1))
A__U22(tt, V1) → A__ISNAT(V1)
A__U31(tt, V) → A__U32(a__isNatIListKind(V), V)
A__U31(tt, V) → A__ISNATILISTKIND(V)
A__U32(tt, V) → A__U33(a__isNatList(V))
A__U32(tt, V) → A__ISNATLIST(V)
A__U41(tt, V1, V2) → A__U42(a__isNatKind(V1), V1, V2)
A__U41(tt, V1, V2) → A__ISNATKIND(V1)
A__U42(tt, V1, V2) → A__U43(a__isNatIListKind(V2), V1, V2)
A__U42(tt, V1, V2) → A__ISNATILISTKIND(V2)
A__U43(tt, V1, V2) → A__U44(a__isNatIListKind(V2), V1, V2)
A__U43(tt, V1, V2) → A__ISNATILISTKIND(V2)
A__U44(tt, V1, V2) → A__U45(a__isNat(V1), V2)
A__U44(tt, V1, V2) → A__ISNAT(V1)
A__U45(tt, V2) → A__U46(a__isNatIList(V2))
A__U45(tt, V2) → A__ISNATILIST(V2)
A__U51(tt, V2) → A__U52(a__isNatIListKind(V2))
A__U51(tt, V2) → A__ISNATILISTKIND(V2)
A__U61(tt, V2) → A__U62(a__isNatIListKind(V2))
A__U61(tt, V2) → A__ISNATILISTKIND(V2)
A__U91(tt, V1, V2) → A__U92(a__isNatKind(V1), V1, V2)
A__U91(tt, V1, V2) → A__ISNATKIND(V1)
A__U92(tt, V1, V2) → A__U93(a__isNatIListKind(V2), V1, V2)
A__U92(tt, V1, V2) → A__ISNATILISTKIND(V2)
A__U93(tt, V1, V2) → A__U94(a__isNatIListKind(V2), V1, V2)
A__U93(tt, V1, V2) → A__ISNATILISTKIND(V2)
A__U94(tt, V1, V2) → A__U95(a__isNat(V1), V2)
A__U94(tt, V1, V2) → A__ISNAT(V1)
A__U95(tt, V2) → A__U96(a__isNatList(V2))
A__U95(tt, V2) → A__ISNATLIST(V2)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
A__ISNATILIST(V) → A__ISNATILISTKIND(V)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNatKind(V1), V1, V2)
A__ISNATILIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__U51(a__isNatKind(V1), V2)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNATILISTKIND(take(V1, V2)) → A__U61(a__isNatKind(V1), V2)
A__ISNATILISTKIND(take(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(length(V1)) → A__U71(a__isNatIListKind(V1))
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATKIND(s(V1)) → A__U81(a__isNatKind(V1))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNATLIST(cons(V1, V2)) → A__U91(a__isNatKind(V1), V1, V2)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNATLIST(take(V1, V2)) → A__U101(a__isNatKind(V1), V1, V2)
A__ISNATLIST(take(V1, V2)) → A__ISNATKIND(V1)
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__TAKE(0, IL) → A__U121(a__isNatIList(IL), IL)
A__TAKE(0, IL) → A__ISNATILIST(IL)
A__TAKE(s(M), cons(N, IL)) → A__U131(a__isNatIList(IL), IL, M, N)
A__TAKE(s(M), cons(N, IL)) → A__ISNATILIST(IL)
MARK(zeros) → A__ZEROS
MARK(U101(X1, X2, X3)) → A__U101(mark(X1), X2, X3)
MARK(U101(X1, X2, X3)) → MARK(X1)
MARK(U102(X1, X2, X3)) → A__U102(mark(X1), X2, X3)
MARK(U102(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U103(X1, X2, X3)) → A__U103(mark(X1), X2, X3)
MARK(U103(X1, X2, X3)) → MARK(X1)
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
MARK(U104(X1, X2, X3)) → A__U104(mark(X1), X2, X3)
MARK(U104(X1, X2, X3)) → MARK(X1)
MARK(U105(X1, X2)) → A__U105(mark(X1), X2)
MARK(U105(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U106(X)) → A__U106(mark(X))
MARK(U106(X)) → MARK(X)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X1, X2)) → A__U12(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U111(X1, X2, X3)) → A__U111(mark(X1), X2, X3)
MARK(U111(X1, X2, X3)) → MARK(X1)
MARK(U112(X1, X2, X3)) → A__U112(mark(X1), X2, X3)
MARK(U112(X1, X2, X3)) → MARK(X1)
MARK(U113(X1, X2, X3)) → A__U113(mark(X1), X2, X3)
MARK(U113(X1, X2, X3)) → MARK(X1)
MARK(U114(X1, X2)) → A__U114(mark(X1), X2)
MARK(U114(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(length(X)) → MARK(X)
MARK(U13(X)) → A__U13(mark(X))
MARK(U13(X)) → MARK(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U121(X1, X2)) → A__U121(mark(X1), X2)
MARK(U121(X1, X2)) → MARK(X1)
MARK(U122(X)) → A__U122(mark(X))
MARK(U122(X)) → MARK(X)
MARK(U131(X1, X2, X3, X4)) → A__U131(mark(X1), X2, X3, X4)
MARK(U131(X1, X2, X3, X4)) → MARK(X1)
MARK(U132(X1, X2, X3, X4)) → A__U132(mark(X1), X2, X3, X4)
MARK(U132(X1, X2, X3, X4)) → MARK(X1)
MARK(U133(X1, X2, X3, X4)) → A__U133(mark(X1), X2, X3, X4)
MARK(U133(X1, X2, X3, X4)) → MARK(X1)
MARK(U134(X1, X2, X3, X4)) → A__U134(mark(X1), X2, X3, X4)
MARK(U134(X1, X2, X3, X4)) → MARK(X1)
MARK(U135(X1, X2, X3, X4)) → A__U135(mark(X1), X2, X3, X4)
MARK(U135(X1, X2, X3, X4)) → MARK(X1)
MARK(U136(X1, X2, X3, X4)) → A__U136(mark(X1), X2, X3, X4)
MARK(U136(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → A__U22(mark(X1), X2)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → A__U23(mark(X))
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U33(X)) → A__U33(mark(X))
MARK(U33(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2, X3)) → A__U42(mark(X1), X2, X3)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(U43(X1, X2, X3)) → A__U43(mark(X1), X2, X3)
MARK(U43(X1, X2, X3)) → MARK(X1)
MARK(U44(X1, X2, X3)) → A__U44(mark(X1), X2, X3)
MARK(U44(X1, X2, X3)) → MARK(X1)
MARK(U45(X1, X2)) → A__U45(mark(X1), X2)
MARK(U45(X1, X2)) → MARK(X1)
MARK(U46(X)) → A__U46(mark(X))
MARK(U46(X)) → MARK(X)
MARK(U51(X1, X2)) → A__U51(mark(X1), X2)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X)) → A__U52(mark(X))
MARK(U52(X)) → MARK(X)
MARK(U61(X1, X2)) → A__U61(mark(X1), X2)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U62(X)) → A__U62(mark(X))
MARK(U62(X)) → MARK(X)
MARK(U71(X)) → A__U71(mark(X))
MARK(U71(X)) → MARK(X)
MARK(U81(X)) → A__U81(mark(X))
MARK(U81(X)) → MARK(X)
MARK(U91(X1, X2, X3)) → A__U91(mark(X1), X2, X3)
MARK(U91(X1, X2, X3)) → MARK(X1)
MARK(U92(X1, X2, X3)) → A__U92(mark(X1), X2, X3)
MARK(U92(X1, X2, X3)) → MARK(X1)
MARK(U93(X1, X2, X3)) → A__U93(mark(X1), X2, X3)
MARK(U93(X1, X2, X3)) → MARK(X1)
MARK(U94(X1, X2, X3)) → A__U94(mark(X1), X2, X3)
MARK(U94(X1, X2, X3)) → MARK(X1)
MARK(U95(X1, X2)) → A__U95(mark(X1), X2)
MARK(U95(X1, X2)) → MARK(X1)
MARK(U96(X)) → A__U96(mark(X))
MARK(U96(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U101(tt, V1, V2) → a__U102(a__isNatKind(V1), V1, V2)
a__U102(tt, V1, V2) → a__U103(a__isNatIListKind(V2), V1, V2)
a__U103(tt, V1, V2) → a__U104(a__isNatIListKind(V2), V1, V2)
a__U104(tt, V1, V2) → a__U105(a__isNat(V1), V2)
a__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__U111(tt, L, N) → a__U112(a__isNatIListKind(L), L, N)
a__U112(tt, L, N) → a__U113(a__isNat(N), L, N)
a__U113(tt, L, N) → a__U114(a__isNatKind(N), L)
a__U114(tt, L) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
a__U131(tt, IL, M, N) → a__U132(a__isNatIListKind(IL), IL, M, N)
a__U132(tt, IL, M, N) → a__U133(a__isNat(M), IL, M, N)
a__U133(tt, IL, M, N) → a__U134(a__isNatKind(M), IL, M, N)
a__U134(tt, IL, M, N) → a__U135(a__isNat(N), IL, M, N)
a__U135(tt, IL, M, N) → a__U136(a__isNatKind(N), IL, M, N)
a__U136(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
a__U91(tt, V1, V2) → a__U92(a__isNatKind(V1), V1, V2)
a__U92(tt, V1, V2) → a__U93(a__isNatIListKind(V2), V1, V2)
a__U93(tt, V1, V2) → a__U94(a__isNatIListKind(V2), V1, V2)
a__U94(tt, V1, V2) → a__U95(a__isNat(V1), V2)
a__U95(tt, V2) → a__U96(a__isNatList(V2))
a__U96(tt) → tt
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
a__take(0, IL) → a__U121(a__isNatIList(IL), IL)
a__take(s(M), cons(N, IL)) → a__U131(a__isNatIList(IL), IL, M, N)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U111(X1, X2, X3)) → a__U111(mark(X1), X2, X3)
mark(U112(X1, X2, X3)) → a__U112(mark(X1), X2, X3)
mark(U113(X1, X2, X3)) → a__U113(mark(X1), X2, X3)
mark(U114(X1, X2)) → a__U114(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(mark(X))
mark(U131(X1, X2, X3, X4)) → a__U131(mark(X1), X2, X3, X4)
mark(U132(X1, X2, X3, X4)) → a__U132(mark(X1), X2, X3, X4)
mark(U133(X1, X2, X3, X4)) → a__U133(mark(X1), X2, X3, X4)
mark(U134(X1, X2, X3, X4)) → a__U134(mark(X1), X2, X3, X4)
mark(U135(X1, X2, X3, X4)) → a__U135(mark(X1), X2, X3, X4)
mark(U136(X1, X2, X3, X4)) → a__U136(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U94(X1, X2, X3)) → a__U94(mark(X1), X2, X3)
mark(U95(X1, X2)) → a__U95(mark(X1), X2)
mark(U96(X)) → a__U96(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 83 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__U51(tt, V2) → A__ISNATILISTKIND(V2)
A__ISNATILISTKIND(cons(V1, V2)) → A__U51(a__isNatKind(V1), V2)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATILISTKIND(take(V1, V2)) → A__U61(a__isNatKind(V1), V2)
A__U61(tt, V2) → A__ISNATILISTKIND(V2)
A__ISNATILISTKIND(take(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U101(tt, V1, V2) → a__U102(a__isNatKind(V1), V1, V2)
a__U102(tt, V1, V2) → a__U103(a__isNatIListKind(V2), V1, V2)
a__U103(tt, V1, V2) → a__U104(a__isNatIListKind(V2), V1, V2)
a__U104(tt, V1, V2) → a__U105(a__isNat(V1), V2)
a__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__U111(tt, L, N) → a__U112(a__isNatIListKind(L), L, N)
a__U112(tt, L, N) → a__U113(a__isNat(N), L, N)
a__U113(tt, L, N) → a__U114(a__isNatKind(N), L)
a__U114(tt, L) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
a__U131(tt, IL, M, N) → a__U132(a__isNatIListKind(IL), IL, M, N)
a__U132(tt, IL, M, N) → a__U133(a__isNat(M), IL, M, N)
a__U133(tt, IL, M, N) → a__U134(a__isNatKind(M), IL, M, N)
a__U134(tt, IL, M, N) → a__U135(a__isNat(N), IL, M, N)
a__U135(tt, IL, M, N) → a__U136(a__isNatKind(N), IL, M, N)
a__U136(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
a__U91(tt, V1, V2) → a__U92(a__isNatKind(V1), V1, V2)
a__U92(tt, V1, V2) → a__U93(a__isNatIListKind(V2), V1, V2)
a__U93(tt, V1, V2) → a__U94(a__isNatIListKind(V2), V1, V2)
a__U94(tt, V1, V2) → a__U95(a__isNat(V1), V2)
a__U95(tt, V2) → a__U96(a__isNatList(V2))
a__U96(tt) → tt
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
a__take(0, IL) → a__U121(a__isNatIList(IL), IL)
a__take(s(M), cons(N, IL)) → a__U131(a__isNatIList(IL), IL, M, N)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U111(X1, X2, X3)) → a__U111(mark(X1), X2, X3)
mark(U112(X1, X2, X3)) → a__U112(mark(X1), X2, X3)
mark(U113(X1, X2, X3)) → a__U113(mark(X1), X2, X3)
mark(U114(X1, X2)) → a__U114(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(mark(X))
mark(U131(X1, X2, X3, X4)) → a__U131(mark(X1), X2, X3, X4)
mark(U132(X1, X2, X3, X4)) → a__U132(mark(X1), X2, X3, X4)
mark(U133(X1, X2, X3, X4)) → a__U133(mark(X1), X2, X3, X4)
mark(U134(X1, X2, X3, X4)) → a__U134(mark(X1), X2, X3, X4)
mark(U135(X1, X2, X3, X4)) → a__U135(mark(X1), X2, X3, X4)
mark(U136(X1, X2, X3, X4)) → a__U136(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U94(X1, X2, X3)) → a__U94(mark(X1), X2, X3)
mark(U95(X1, X2)) → a__U95(mark(X1), X2)
mark(U96(X)) → a__U96(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__U51(tt, V2) → A__ISNATILISTKIND(V2)
A__ISNATILISTKIND(cons(V1, V2)) → A__U51(a__isNatKind(V1), V2)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATILISTKIND(take(V1, V2)) → A__U61(a__isNatKind(V1), V2)
A__ISNATILISTKIND(take(V1, V2)) → A__ISNATKIND(V1)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__U51(x1, x2) = A__U51(x2)
tt = tt
A__ISNATILISTKIND(x1) = x1
cons(x1, x2) = cons(x1, x2)
a__isNatKind(x1) = a__isNatKind
A__ISNATKIND(x1) = A__ISNATKIND(x1)
length(x1) = x1
take(x1, x2) = take(x1, x2)
A__U61(x1, x2) = x2
s(x1) = s(x1)
a__isNatIListKind(x1) = a__isNatIListKind
a__U61(x1, x2) = a__U61(x1)
0 = 0
zeros = zeros
a__U51(x1, x2) = x2
nil = nil
a__U81(x1) = x1
U81(x1) = U81
a__U71(x1) = a__U71(x1)
U71(x1) = U71(x1)
U51(x1, x2) = U51(x2)
a__U52(x1) = a__U52(x1)
U52(x1) = U52
U61(x1, x2) = U61(x1, x2)
isNatIListKind(x1) = isNatIListKind
a__U62(x1) = a__U62(x1)
U62(x1) = U62(x1)
isNatKind(x1) = x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

cons₂ > A_{_U51₁}
cons₂ > [A_{_ISNATKIND₁}, take₂]
[a_{_{isNatIListKind}}, a_{_U61₁}] > a_{_isNatKind} > [tt, zeros, a_{_U71₁}, a_{_U52₁}, a_{_U62₁}]
[a_{_{isNatIListKind}}, a_{_U61₁}] > U61₂
[a_{_{isNatIListKind}}, a_{_U61₁}] > isNatIListKind
0 > [tt, zeros, a_{_U71₁}, a_{_U52₁}, a_{_U62₁}]
nil > [tt, zeros, a_{_U71₁}, a_{_U52₁}, a_{_U62₁}]

Status:

a_{_U62₁}: [1]
a_{_U52₁}: [1]
A_{_U51₁}: [1]
A_{_ISNATKIND₁}: [1]
U52: []
a_{_{isNatIListKind}}: []
take₂: [1,2]
0: []
U62₁: [1]
cons₂: [2,1]
a_{_U61₁}: [1]
tt: []
U61₂: [1,2]
zeros: []
s₁: [1]
U81: []
U71₁: [1]
nil: []
a_{_isNatKind}: []
a_{_U71₁}: [1]
U51₁: [1]
isNatIListKind: []

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__U61(tt, V2) → A__ISNATILISTKIND(V2)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U101(tt, V1, V2) → a__U102(a__isNatKind(V1), V1, V2)
a__U102(tt, V1, V2) → a__U103(a__isNatIListKind(V2), V1, V2)
a__U103(tt, V1, V2) → a__U104(a__isNatIListKind(V2), V1, V2)
a__U104(tt, V1, V2) → a__U105(a__isNat(V1), V2)
a__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__U111(tt, L, N) → a__U112(a__isNatIListKind(L), L, N)
a__U112(tt, L, N) → a__U113(a__isNat(N), L, N)
a__U113(tt, L, N) → a__U114(a__isNatKind(N), L)
a__U114(tt, L) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
a__U131(tt, IL, M, N) → a__U132(a__isNatIListKind(IL), IL, M, N)
a__U132(tt, IL, M, N) → a__U133(a__isNat(M), IL, M, N)
a__U133(tt, IL, M, N) → a__U134(a__isNatKind(M), IL, M, N)
a__U134(tt, IL, M, N) → a__U135(a__isNat(N), IL, M, N)
a__U135(tt, IL, M, N) → a__U136(a__isNatKind(N), IL, M, N)
a__U136(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
a__U91(tt, V1, V2) → a__U92(a__isNatKind(V1), V1, V2)
a__U92(tt, V1, V2) → a__U93(a__isNatIListKind(V2), V1, V2)
a__U93(tt, V1, V2) → a__U94(a__isNatIListKind(V2), V1, V2)
a__U94(tt, V1, V2) → a__U95(a__isNat(V1), V2)
a__U95(tt, V2) → a__U96(a__isNatList(V2))
a__U96(tt) → tt
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
a__take(0, IL) → a__U121(a__isNatIList(IL), IL)
a__take(s(M), cons(N, IL)) → a__U131(a__isNatIList(IL), IL, M, N)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U111(X1, X2, X3)) → a__U111(mark(X1), X2, X3)
mark(U112(X1, X2, X3)) → a__U112(mark(X1), X2, X3)
mark(U113(X1, X2, X3)) → a__U113(mark(X1), X2, X3)
mark(U114(X1, X2)) → a__U114(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(mark(X))
mark(U131(X1, X2, X3, X4)) → a__U131(mark(X1), X2, X3, X4)
mark(U132(X1, X2, X3, X4)) → a__U132(mark(X1), X2, X3, X4)
mark(U133(X1, X2, X3, X4)) → a__U133(mark(X1), X2, X3, X4)
mark(U134(X1, X2, X3, X4)) → a__U134(mark(X1), X2, X3, X4)
mark(U135(X1, X2, X3, X4)) → a__U135(mark(X1), X2, X3, X4)
mark(U136(X1, X2, X3, X4)) → a__U136(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U94(X1, X2, X3)) → a__U94(mark(X1), X2, X3)
mark(U95(X1, X2)) → a__U95(mark(X1), X2)
mark(U96(X)) → a__U96(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(8) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(9) TRUE

(10) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__U102(tt, V1, V2) → A__U103(a__isNatIListKind(V2), V1, V2)
A__U103(tt, V1, V2) → A__U104(a__isNatIListKind(V2), V1, V2)
A__U104(tt, V1, V2) → A__U105(a__isNat(V1), V2)
A__U105(tt, V2) → A__ISNATILIST(V2)
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
A__U31(tt, V) → A__U32(a__isNatIListKind(V), V)
A__U32(tt, V) → A__ISNATLIST(V)
A__ISNATLIST(cons(V1, V2)) → A__U91(a__isNatKind(V1), V1, V2)
A__U91(tt, V1, V2) → A__U92(a__isNatKind(V1), V1, V2)
A__U92(tt, V1, V2) → A__U93(a__isNatIListKind(V2), V1, V2)
A__U93(tt, V1, V2) → A__U94(a__isNatIListKind(V2), V1, V2)
A__U94(tt, V1, V2) → A__U95(a__isNat(V1), V2)
A__U95(tt, V2) → A__ISNATLIST(V2)
A__ISNATLIST(take(V1, V2)) → A__U101(a__isNatKind(V1), V1, V2)
A__U101(tt, V1, V2) → A__U102(a__isNatKind(V1), V1, V2)
A__U94(tt, V1, V2) → A__ISNAT(V1)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__U11(tt, V1) → A__U12(a__isNatIListKind(V1), V1)
A__U12(tt, V1) → A__ISNATLIST(V1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U21(tt, V1) → A__U22(a__isNatKind(V1), V1)
A__U22(tt, V1) → A__ISNAT(V1)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNatKind(V1), V1, V2)
A__U41(tt, V1, V2) → A__U42(a__isNatKind(V1), V1, V2)
A__U42(tt, V1, V2) → A__U43(a__isNatIListKind(V2), V1, V2)
A__U43(tt, V1, V2) → A__U44(a__isNatIListKind(V2), V1, V2)
A__U44(tt, V1, V2) → A__U45(a__isNat(V1), V2)
A__U45(tt, V2) → A__ISNATILIST(V2)
A__U44(tt, V1, V2) → A__ISNAT(V1)
A__U104(tt, V1, V2) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U101(tt, V1, V2) → a__U102(a__isNatKind(V1), V1, V2)
a__U102(tt, V1, V2) → a__U103(a__isNatIListKind(V2), V1, V2)
a__U103(tt, V1, V2) → a__U104(a__isNatIListKind(V2), V1, V2)
a__U104(tt, V1, V2) → a__U105(a__isNat(V1), V2)
a__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__U111(tt, L, N) → a__U112(a__isNatIListKind(L), L, N)
a__U112(tt, L, N) → a__U113(a__isNat(N), L, N)
a__U113(tt, L, N) → a__U114(a__isNatKind(N), L)
a__U114(tt, L) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
a__U131(tt, IL, M, N) → a__U132(a__isNatIListKind(IL), IL, M, N)
a__U132(tt, IL, M, N) → a__U133(a__isNat(M), IL, M, N)
a__U133(tt, IL, M, N) → a__U134(a__isNatKind(M), IL, M, N)
a__U134(tt, IL, M, N) → a__U135(a__isNat(N), IL, M, N)
a__U135(tt, IL, M, N) → a__U136(a__isNatKind(N), IL, M, N)
a__U136(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
a__U91(tt, V1, V2) → a__U92(a__isNatKind(V1), V1, V2)
a__U92(tt, V1, V2) → a__U93(a__isNatIListKind(V2), V1, V2)
a__U93(tt, V1, V2) → a__U94(a__isNatIListKind(V2), V1, V2)
a__U94(tt, V1, V2) → a__U95(a__isNat(V1), V2)
a__U95(tt, V2) → a__U96(a__isNatList(V2))
a__U96(tt) → tt
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
a__take(0, IL) → a__U121(a__isNatIList(IL), IL)
a__take(s(M), cons(N, IL)) → a__U131(a__isNatIList(IL), IL, M, N)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U111(X1, X2, X3)) → a__U111(mark(X1), X2, X3)
mark(U112(X1, X2, X3)) → a__U112(mark(X1), X2, X3)
mark(U113(X1, X2, X3)) → a__U113(mark(X1), X2, X3)
mark(U114(X1, X2)) → a__U114(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(mark(X))
mark(U131(X1, X2, X3, X4)) → a__U131(mark(X1), X2, X3, X4)
mark(U132(X1, X2, X3, X4)) → a__U132(mark(X1), X2, X3, X4)
mark(U133(X1, X2, X3, X4)) → a__U133(mark(X1), X2, X3, X4)
mark(U134(X1, X2, X3, X4)) → a__U134(mark(X1), X2, X3, X4)
mark(U135(X1, X2, X3, X4)) → a__U135(mark(X1), X2, X3, X4)
mark(U136(X1, X2, X3, X4)) → a__U136(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U94(X1, X2, X3)) → a__U94(mark(X1), X2, X3)
mark(U95(X1, X2)) → a__U95(mark(X1), X2)
mark(U96(X)) → a__U96(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__U102(tt, V1, V2) → A__U103(a__isNatIListKind(V2), V1, V2)
A__U103(tt, V1, V2) → A__U104(a__isNatIListKind(V2), V1, V2)
A__U104(tt, V1, V2) → A__U105(a__isNat(V1), V2)
A__U92(tt, V1, V2) → A__U93(a__isNatIListKind(V2), V1, V2)
A__U93(tt, V1, V2) → A__U94(a__isNatIListKind(V2), V1, V2)
A__U94(tt, V1, V2) → A__U95(a__isNat(V1), V2)
A__U101(tt, V1, V2) → A__U102(a__isNatKind(V1), V1, V2)
A__U94(tt, V1, V2) → A__ISNAT(V1)
A__U11(tt, V1) → A__U12(a__isNatIListKind(V1), V1)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNatKind(V1), V1, V2)
A__U41(tt, V1, V2) → A__U42(a__isNatKind(V1), V1, V2)
A__U42(tt, V1, V2) → A__U43(a__isNatIListKind(V2), V1, V2)
A__U43(tt, V1, V2) → A__U44(a__isNatIListKind(V2), V1, V2)
A__U44(tt, V1, V2) → A__U45(a__isNat(V1), V2)
A__U45(tt, V2) → A__ISNATILIST(V2)
A__U44(tt, V1, V2) → A__ISNAT(V1)
A__U104(tt, V1, V2) → A__ISNAT(V1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__U102(x1, x2, x3) = A__U102(x1, x2, x3)
tt = tt
A__U103(x1, x2, x3) = A__U103(x1, x2, x3)
a__isNatIListKind(x1) = x1
A__U104(x1, x2, x3) = A__U104(x2, x3)
A__U105(x1, x2) = x2
a__isNat(x1) = a__isNat
A__ISNATILIST(x1) = x1
A__U31(x1, x2) = x2
A__U32(x1, x2) = x2
A__ISNATLIST(x1) = x1
cons(x1, x2) = cons(x1, x2)
A__U91(x1, x2, x3) = A__U91(x2, x3)
a__isNatKind(x1) = a__isNatKind(x1)
A__U92(x1, x2, x3) = A__U92(x2, x3)
A__U93(x1, x2, x3) = A__U93(x2, x3)
A__U94(x1, x2, x3) = A__U94(x2, x3)
A__U95(x1, x2) = x2
take(x1, x2) = take(x1, x2)
A__U101(x1, x2, x3) = A__U101(x2, x3)
A__ISNAT(x1) = x1
length(x1) = length(x1)
A__U11(x1, x2) = A__U11(x2)
A__U12(x1, x2) = x2
s(x1) = x1
A__U21(x1, x2) = x2
A__U22(x1, x2) = x2
A__U41(x1, x2, x3) = A__U41(x1, x2, x3)
A__U42(x1, x2, x3) = A__U42(x2, x3)
A__U43(x1, x2, x3) = A__U43(x1, x2, x3)
A__U44(x1, x2, x3) = A__U44(x2, x3)
A__U45(x1, x2) = A__U45(x2)
a__U11(x1, x2) = a__U11(x1)
a__U21(x1, x2) = a__U21
a__U96(x1) = x1
0 = 0
a__U94(x1, x2, x3) = a__U94(x2, x3)
a__U95(x1, x2) = x2
a__isNatList(x1) = a__isNatList
a__U92(x1, x2, x3) = a__U92(x2)
a__U93(x1, x2, x3) = a__U93(x1, x2, x3)
a__U61(x1, x2) = x2
U96(x1) = U96
U95(x1, x2) = U95
zeros = zeros
U94(x1, x2, x3) = U94
a__U51(x1, x2) = a__U51(x1)
U93(x1, x2, x3) = U93(x1, x2, x3)
a__isNatIList(x1) = x1
a__U41(x1, x2, x3) = a__U41(x2)
U92(x1, x2, x3) = U92(x1, x2, x3)
nil = nil
a__U91(x1, x2, x3) = a__U91(x2, x3)
U91(x1, x2, x3) = U91
a__U81(x1) = x1
U81(x1) = x1
a__U31(x1, x2) = x2
a__U71(x1) = x1
U71(x1) = x1
a__U101(x1, x2, x3) = x2
a__U13(x1) = x1
a__U12(x1, x2) = a__U12
U13(x1) = U13(x1)
a__U22(x1, x2) = a__U22
a__U23(x1) = a__U23
isNatList(x1) = x1
U51(x1, x2) = U51
a__U43(x1, x2, x3) = a__U43(x3)
a__U44(x1, x2, x3) = a__U44(x3)
a__U52(x1) = a__U52
U52(x1) = U52
a__U42(x1, x2, x3) = a__U42
U61(x1, x2) = x2
a__U45(x1, x2) = a__U45
a__U46(x1) = a__U46
a__U62(x1) = x1
U62(x1) = U62
U43(x1, x2, x3) = U43
a__U32(x1, x2) = x1
a__U33(x1) = x1
U44(x1, x2, x3) = U44(x1, x2, x3)
U45(x1, x2) = U45
U46(x1) = U46(x1)
U32(x1, x2) = U32
U33(x1) = U33
U41(x1, x2, x3) = U41(x1, x2, x3)
U42(x1, x2, x3) = U42(x1, x2, x3)
U21(x1, x2) = U21(x1)
U22(x1, x2) = U22
U23(x1) = U23
U31(x1, x2) = U31(x1, x2)
a__U105(x1, x2) = a__U105(x1, x2)
U105(x1, x2) = U105(x1, x2)
isNat(x1) = isNat(x1)
isNatIListKind(x1) = x1
a__U104(x1, x2, x3) = a__U104(x2, x3)
U104(x1, x2, x3) = U104(x1, x2, x3)
U11(x1, x2) = U11(x1, x2)
U12(x1, x2) = U12(x1)
a__U106(x1) = a__U106(x1)
U106(x1) = x1
isNatIList(x1) = isNatIList(x1)
a__U103(x1, x2, x3) = a__U103
a__U102(x1, x2, x3) = a__U102(x1, x2, x3)
isNatKind(x1) = x1
U103(x1, x2, x3) = U103(x1, x2, x3)
U101(x1, x2, x3) = U101(x3)
U102(x1, x2, x3) = U102(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

[take₂, A_{_U101₂}] > A_{_U102₃} > A_{_U103₃} > A_{_U104₂} > a_{_isNat} > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
[take₂, A_{_U101₂}] > [a_{_isNatKind₁}, a_{_isNatList}] > a_{_U91₂} > U91 > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
[length₁, A_{_U11₁}] > a_{_U11₁} > a_{_U12} > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
[length₁, A_{_U11₁}] > a_{_U11₁} > U11₂ > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
0 > [tt, zeros, nil, a_{_U52}, U52] > A_{_U102₃} > A_{_U103₃} > A_{_U104₂} > a_{_isNat} > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
0 > [tt, zeros, nil, a_{_U52}, U52] > [cons₂, A_{_U91₂}, A_{_U92₂}] > [a_{_isNatKind₁}, a_{_isNatList}] > a_{_U91₂} > U91 > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
0 > [tt, zeros, nil, a_{_U52}, U52] > [cons₂, A_{_U91₂}, A_{_U92₂}] > A_{_U93₂} > A_{_U94₂} > a_{_isNat} > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
0 > [tt, zeros, nil, a_{_U52}, U52] > [cons₂, A_{_U91₂}, A_{_U92₂}] > A_{_U41₃} > A_{_U42₂} > A_{_U43₃} > [A_{_U44₂}, A_{_U45₁}] > a_{_isNat} > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
0 > [tt, zeros, nil, a_{_U52}, U52] > [cons₂, A_{_U91₂}, A_{_U92₂}] > a_{_U51₁} > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
0 > [tt, zeros, nil, a_{_U52}, U52] > [cons₂, A_{_U91₂}, A_{_U92₂}] > a_{_U41₁} > [a_{_U42}, U42₃] > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
0 > [tt, zeros, nil, a_{_U52}, U52] > [cons₂, A_{_U91₂}, A_{_U92₂}] > a_{_U41₁} > U41₃ > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
0 > [tt, zeros, nil, a_{_U52}, U52] > a_{_U92₁} > [a_{_U93₃}, U93₃] > [a_{_U94₂}, U94] > a_{_isNat} > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
0 > [tt, zeros, nil, a_{_U52}, U52] > a_{_U92₁} > U92₃ > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
0 > [tt, zeros, nil, a_{_U52}, U52] > a_{_U12} > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
0 > [tt, zeros, nil, a_{_U52}, U52] > [a_{_U23}, U23] > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
0 > [tt, zeros, nil, a_{_U52}, U52] > a_{_U44₁} > a_{_isNat} > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
0 > [tt, zeros, nil, a_{_U52}, U52] > a_{_U45} > U45 > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
0 > [tt, zeros, nil, a_{_U52}, U52] > a_{_U46} > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
0 > [tt, zeros, nil, a_{_U52}, U52] > [a_{_U105₂}, U105₂] > a_{_U106₁} > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
0 > [tt, zeros, nil, a_{_U52}, U52] > a_{_U103} > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
0 > [tt, zeros, nil, a_{_U52}, U52] > a_{_U102₃} > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
U96 > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
U95 > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
U13₁ > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
a_{_U43₁} > a_{_U44₁} > a_{_isNat} > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
a_{_U43₁} > U43 > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
U44₃ > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
U46₁ > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
U32 > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
U22 > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
U31₂ > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
isNat₁ > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
a_{_U104₂} > a_{_isNat} > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
a_{_U104₂} > [a_{_U105₂}, U105₂] > a_{_U106₁} > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
a_{_U104₂} > U104₃ > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
U12₁ > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
isNatIList₁ > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
U103₃ > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
U101₁ > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]
U102₁ > [a_{_U21}, a_{_U22}, U51, U62, U33, U21₁]

Status:

a_{_U44₁}: [1]
U22: []
A_{_U91₂}: [2,1]
U41₃: [3,2,1]
A_{_U92₂}: [2,1]
U33: []
A_{_U43₃}: [3,2,1]
take₂: [2,1]
U103₃: [3,2,1]
a_{_U105₂}: [1,2]
a_{_U102₃}: [3,1,2]
tt: []
a_{_U41₁}: [1]
a_{_U45}: []
U93₃: [2,3,1]
A_{_U42₂}: [2,1]
U105₂: [1,2]
a_{_U106₁}: [1]
a_{_U92₁}: [1]
A_{_U93₂}: [2,1]
nil: []
U51: []
a_{_U46}: []
U31₂: [1,2]
A_{_U44₂}: [2,1]
U101₁: [1]
a_{_U104₂}: [2,1]
A_{_U45₁}: [1]
a_{_U103}: []
U44₃: [3,2,1]
U45: []
A_{_U104₂}: [2,1]
length₁: [1]
a_{_U21}: []
U102₁: [1]
a_{_U51₁}: [1]
U96: []
U104₃: [1,2,3]
a_{_U52}: []
U42₃: [2,3,1]
a_{_U11₁}: [1]
zeros: []
U94: []
isNatIList₁: [1]
U12₁: [1]
a_{_U12}: []
a_{_U43₁}: [1]
U43: []
U23: []
isNat₁: [1]
A_{_U101₂}: [2,1]
U95: []
a_{_U93₃}: [1,3,2]
U62: []
U32: []
a_{_U42}: []
U91: []
U52: []
A_{_U102₃}: [1,2,3]
a_{_isNat}: []
U11₂: [1,2]
a_{_isNatList}: []
0: []
a_{_U22}: []
A_{_U11₁}: [1]
cons₂: [2,1]
a_{_isNatKind₁}: [1]
U46₁: [1]
U13₁: [1]
U92₃: [3,2,1]
a_{_U91₂}: [1,2]
A_{_U41₃}: [2,3,1]
A_{_U103₃}: [3,1,2]
A_{_U94₂}: [2,1]
a_{_U23}: []
a_{_U94₂}: [2,1]
U21₁: [1]

The following usable rules [FROCOS05] were oriented:

a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(nil) → tt
a__U81(X) → U81(X)
a__U71(X) → U71(X)
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(tt) → tt
a__U62(tt) → tt
a__U81(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U52(tt) → tt
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

(12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__U105(tt, V2) → A__ISNATILIST(V2)
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
A__U31(tt, V) → A__U32(a__isNatIListKind(V), V)
A__U32(tt, V) → A__ISNATLIST(V)
A__ISNATLIST(cons(V1, V2)) → A__U91(a__isNatKind(V1), V1, V2)
A__U91(tt, V1, V2) → A__U92(a__isNatKind(V1), V1, V2)
A__U95(tt, V2) → A__ISNATLIST(V2)
A__ISNATLIST(take(V1, V2)) → A__U101(a__isNatKind(V1), V1, V2)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__U12(tt, V1) → A__ISNATLIST(V1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U21(tt, V1) → A__U22(a__isNatKind(V1), V1)
A__U22(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U101(tt, V1, V2) → a__U102(a__isNatKind(V1), V1, V2)
a__U102(tt, V1, V2) → a__U103(a__isNatIListKind(V2), V1, V2)
a__U103(tt, V1, V2) → a__U104(a__isNatIListKind(V2), V1, V2)
a__U104(tt, V1, V2) → a__U105(a__isNat(V1), V2)
a__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__U111(tt, L, N) → a__U112(a__isNatIListKind(L), L, N)
a__U112(tt, L, N) → a__U113(a__isNat(N), L, N)
a__U113(tt, L, N) → a__U114(a__isNatKind(N), L)
a__U114(tt, L) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
a__U131(tt, IL, M, N) → a__U132(a__isNatIListKind(IL), IL, M, N)
a__U132(tt, IL, M, N) → a__U133(a__isNat(M), IL, M, N)
a__U133(tt, IL, M, N) → a__U134(a__isNatKind(M), IL, M, N)
a__U134(tt, IL, M, N) → a__U135(a__isNat(N), IL, M, N)
a__U135(tt, IL, M, N) → a__U136(a__isNatKind(N), IL, M, N)
a__U136(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
a__U91(tt, V1, V2) → a__U92(a__isNatKind(V1), V1, V2)
a__U92(tt, V1, V2) → a__U93(a__isNatIListKind(V2), V1, V2)
a__U93(tt, V1, V2) → a__U94(a__isNatIListKind(V2), V1, V2)
a__U94(tt, V1, V2) → a__U95(a__isNat(V1), V2)
a__U95(tt, V2) → a__U96(a__isNatList(V2))
a__U96(tt) → tt
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
a__take(0, IL) → a__U121(a__isNatIList(IL), IL)
a__take(s(M), cons(N, IL)) → a__U131(a__isNatIList(IL), IL, M, N)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U111(X1, X2, X3)) → a__U111(mark(X1), X2, X3)
mark(U112(X1, X2, X3)) → a__U112(mark(X1), X2, X3)
mark(U113(X1, X2, X3)) → a__U113(mark(X1), X2, X3)
mark(U114(X1, X2)) → a__U114(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(mark(X))
mark(U131(X1, X2, X3, X4)) → a__U131(mark(X1), X2, X3, X4)
mark(U132(X1, X2, X3, X4)) → a__U132(mark(X1), X2, X3, X4)
mark(U133(X1, X2, X3, X4)) → a__U133(mark(X1), X2, X3, X4)
mark(U134(X1, X2, X3, X4)) → a__U134(mark(X1), X2, X3, X4)
mark(U135(X1, X2, X3, X4)) → a__U135(mark(X1), X2, X3, X4)
mark(U136(X1, X2, X3, X4)) → a__U136(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U94(X1, X2, X3)) → a__U94(mark(X1), X2, X3)
mark(U95(X1, X2)) → a__U95(mark(X1), X2)
mark(U96(X)) → a__U96(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(13) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 10 less nodes.

(14) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U21(tt, V1) → A__U22(a__isNatKind(V1), V1)
A__U22(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U101(tt, V1, V2) → a__U102(a__isNatKind(V1), V1, V2)
a__U102(tt, V1, V2) → a__U103(a__isNatIListKind(V2), V1, V2)
a__U103(tt, V1, V2) → a__U104(a__isNatIListKind(V2), V1, V2)
a__U104(tt, V1, V2) → a__U105(a__isNat(V1), V2)
a__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__U111(tt, L, N) → a__U112(a__isNatIListKind(L), L, N)
a__U112(tt, L, N) → a__U113(a__isNat(N), L, N)
a__U113(tt, L, N) → a__U114(a__isNatKind(N), L)
a__U114(tt, L) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
a__U131(tt, IL, M, N) → a__U132(a__isNatIListKind(IL), IL, M, N)
a__U132(tt, IL, M, N) → a__U133(a__isNat(M), IL, M, N)
a__U133(tt, IL, M, N) → a__U134(a__isNatKind(M), IL, M, N)
a__U134(tt, IL, M, N) → a__U135(a__isNat(N), IL, M, N)
a__U135(tt, IL, M, N) → a__U136(a__isNatKind(N), IL, M, N)
a__U136(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
a__U91(tt, V1, V2) → a__U92(a__isNatKind(V1), V1, V2)
a__U92(tt, V1, V2) → a__U93(a__isNatIListKind(V2), V1, V2)
a__U93(tt, V1, V2) → a__U94(a__isNatIListKind(V2), V1, V2)
a__U94(tt, V1, V2) → a__U95(a__isNat(V1), V2)
a__U95(tt, V2) → a__U96(a__isNatList(V2))
a__U96(tt) → tt
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
a__take(0, IL) → a__U121(a__isNatIList(IL), IL)
a__take(s(M), cons(N, IL)) → a__U131(a__isNatIList(IL), IL, M, N)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U111(X1, X2, X3)) → a__U111(mark(X1), X2, X3)
mark(U112(X1, X2, X3)) → a__U112(mark(X1), X2, X3)
mark(U113(X1, X2, X3)) → a__U113(mark(X1), X2, X3)
mark(U114(X1, X2)) → a__U114(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(mark(X))
mark(U131(X1, X2, X3, X4)) → a__U131(mark(X1), X2, X3, X4)
mark(U132(X1, X2, X3, X4)) → a__U132(mark(X1), X2, X3, X4)
mark(U133(X1, X2, X3, X4)) → a__U133(mark(X1), X2, X3, X4)
mark(U134(X1, X2, X3, X4)) → a__U134(mark(X1), X2, X3, X4)
mark(U135(X1, X2, X3, X4)) → a__U135(mark(X1), X2, X3, X4)
mark(U136(X1, X2, X3, X4)) → a__U136(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U94(X1, X2, X3)) → a__U94(mark(X1), X2, X3)
mark(U95(X1, X2)) → a__U95(mark(X1), X2)
mark(U96(X)) → a__U96(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__ISNAT(x1) = x1
s(x1) = s(x1)
A__U21(x1, x2) = x2
a__isNatKind(x1) = x1
tt = tt
A__U22(x1, x2) = x2
a__isNatIListKind(x1) = a__isNatIListKind(x1)
take(x1, x2) = x1
a__U61(x1, x2) = a__U61(x1)
0 = 0
zeros = zeros
cons(x1, x2) = cons
a__U51(x1, x2) = a__U51(x1, x2)
nil = nil
a__U81(x1) = a__U81(x1)
U81(x1) = U81
a__U71(x1) = x1
U71(x1) = U71
U51(x1, x2) = U51(x1, x2)
a__U52(x1) = x1
U52(x1) = U52(x1)
isNatIListKind(x1) = isNatIListKind(x1)
U61(x1, x2) = U61
a__U62(x1) = x1
U62(x1) = U62
length(x1) = length
isNatKind(x1) = isNatKind(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

a_{_U61₁} > a_{_{isNatIListKind₁}} > tt
0 > tt
zeros > tt
cons > a_{_U51₂} > a_{_{isNatIListKind₁}} > tt
nil > tt
a_{_U81₁} > tt
a_{_U81₁} > U81
length > a_{_{isNatIListKind₁}} > tt

Status:

U62: []
isNatKind₁: [1]
U71: []
length: []
0: []
U51₂: [2,1]
isNatIListKind₁: [1]
a_{_U81₁}: [1]
a_{_U61₁}: [1]
U61: []
tt: []
cons: []
zeros: []
U52₁: [1]
s₁: [1]
a_{_{isNatIListKind₁}}: [1]
U81: []
nil: []
a_{_U51₂}: [1,2]

The following usable rules [FROCOS05] were oriented: none

(16) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A__U21(tt, V1) → A__U22(a__isNatKind(V1), V1)
A__U22(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U101(tt, V1, V2) → a__U102(a__isNatKind(V1), V1, V2)
a__U102(tt, V1, V2) → a__U103(a__isNatIListKind(V2), V1, V2)
a__U103(tt, V1, V2) → a__U104(a__isNatIListKind(V2), V1, V2)
a__U104(tt, V1, V2) → a__U105(a__isNat(V1), V2)
a__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__U111(tt, L, N) → a__U112(a__isNatIListKind(L), L, N)
a__U112(tt, L, N) → a__U113(a__isNat(N), L, N)
a__U113(tt, L, N) → a__U114(a__isNatKind(N), L)
a__U114(tt, L) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
a__U131(tt, IL, M, N) → a__U132(a__isNatIListKind(IL), IL, M, N)
a__U132(tt, IL, M, N) → a__U133(a__isNat(M), IL, M, N)
a__U133(tt, IL, M, N) → a__U134(a__isNatKind(M), IL, M, N)
a__U134(tt, IL, M, N) → a__U135(a__isNat(N), IL, M, N)
a__U135(tt, IL, M, N) → a__U136(a__isNatKind(N), IL, M, N)
a__U136(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
a__U91(tt, V1, V2) → a__U92(a__isNatKind(V1), V1, V2)
a__U92(tt, V1, V2) → a__U93(a__isNatIListKind(V2), V1, V2)
a__U93(tt, V1, V2) → a__U94(a__isNatIListKind(V2), V1, V2)
a__U94(tt, V1, V2) → a__U95(a__isNat(V1), V2)
a__U95(tt, V2) → a__U96(a__isNatList(V2))
a__U96(tt) → tt
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
a__take(0, IL) → a__U121(a__isNatIList(IL), IL)
a__take(s(M), cons(N, IL)) → a__U131(a__isNatIList(IL), IL, M, N)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U111(X1, X2, X3)) → a__U111(mark(X1), X2, X3)
mark(U112(X1, X2, X3)) → a__U112(mark(X1), X2, X3)
mark(U113(X1, X2, X3)) → a__U113(mark(X1), X2, X3)
mark(U114(X1, X2)) → a__U114(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(mark(X))
mark(U131(X1, X2, X3, X4)) → a__U131(mark(X1), X2, X3, X4)
mark(U132(X1, X2, X3, X4)) → a__U132(mark(X1), X2, X3, X4)
mark(U133(X1, X2, X3, X4)) → a__U133(mark(X1), X2, X3, X4)
mark(U134(X1, X2, X3, X4)) → a__U134(mark(X1), X2, X3, X4)
mark(U135(X1, X2, X3, X4)) → a__U135(mark(X1), X2, X3, X4)
mark(U136(X1, X2, X3, X4)) → a__U136(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U94(X1, X2, X3)) → a__U94(mark(X1), X2, X3)
mark(U95(X1, X2)) → a__U95(mark(X1), X2)
mark(U96(X)) → a__U96(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(17) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(18) TRUE

(19) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(U101(X1, X2, X3)) → MARK(X1)
MARK(U102(X1, X2, X3)) → MARK(X1)
MARK(U103(X1, X2, X3)) → MARK(X1)
MARK(U104(X1, X2, X3)) → MARK(X1)
MARK(U105(X1, X2)) → MARK(X1)
MARK(U106(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U111(X1, X2, X3)) → A__U111(mark(X1), X2, X3)
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)
A__U113(tt, L, N) → A__U114(a__isNatKind(N), L)
A__U114(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
A__U114(tt, L) → MARK(L)
MARK(U111(X1, X2, X3)) → MARK(X1)
MARK(U112(X1, X2, X3)) → A__U112(mark(X1), X2, X3)
MARK(U112(X1, X2, X3)) → MARK(X1)
MARK(U113(X1, X2, X3)) → A__U113(mark(X1), X2, X3)
MARK(U113(X1, X2, X3)) → MARK(X1)
MARK(U114(X1, X2)) → A__U114(mark(X1), X2)
MARK(U114(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(length(X)) → MARK(X)
MARK(U13(X)) → MARK(X)
MARK(U121(X1, X2)) → MARK(X1)
MARK(U122(X)) → MARK(X)
MARK(U131(X1, X2, X3, X4)) → A__U131(mark(X1), X2, X3, X4)
A__U131(tt, IL, M, N) → A__U132(a__isNatIListKind(IL), IL, M, N)
A__U132(tt, IL, M, N) → A__U133(a__isNat(M), IL, M, N)
A__U133(tt, IL, M, N) → A__U134(a__isNatKind(M), IL, M, N)
A__U134(tt, IL, M, N) → A__U135(a__isNat(N), IL, M, N)
A__U135(tt, IL, M, N) → A__U136(a__isNatKind(N), IL, M, N)
A__U136(tt, IL, M, N) → MARK(N)
MARK(U131(X1, X2, X3, X4)) → MARK(X1)
MARK(U132(X1, X2, X3, X4)) → A__U132(mark(X1), X2, X3, X4)
MARK(U132(X1, X2, X3, X4)) → MARK(X1)
MARK(U133(X1, X2, X3, X4)) → A__U133(mark(X1), X2, X3, X4)
MARK(U133(X1, X2, X3, X4)) → MARK(X1)
MARK(U134(X1, X2, X3, X4)) → A__U134(mark(X1), X2, X3, X4)
MARK(U134(X1, X2, X3, X4)) → MARK(X1)
MARK(U135(X1, X2, X3, X4)) → A__U135(mark(X1), X2, X3, X4)
MARK(U135(X1, X2, X3, X4)) → MARK(X1)
MARK(U136(X1, X2, X3, X4)) → A__U136(mark(X1), X2, X3, X4)
MARK(U136(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__TAKE(s(M), cons(N, IL)) → A__U131(a__isNatIList(IL), IL, M, N)
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U33(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(U43(X1, X2, X3)) → MARK(X1)
MARK(U44(X1, X2, X3)) → MARK(X1)
MARK(U45(X1, X2)) → MARK(X1)
MARK(U46(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U62(X)) → MARK(X)
MARK(U71(X)) → MARK(X)
MARK(U81(X)) → MARK(X)
MARK(U91(X1, X2, X3)) → MARK(X1)
MARK(U92(X1, X2, X3)) → MARK(X1)
MARK(U93(X1, X2, X3)) → MARK(X1)
MARK(U94(X1, X2, X3)) → MARK(X1)
MARK(U95(X1, X2)) → MARK(X1)
MARK(U96(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U101(tt, V1, V2) → a__U102(a__isNatKind(V1), V1, V2)
a__U102(tt, V1, V2) → a__U103(a__isNatIListKind(V2), V1, V2)
a__U103(tt, V1, V2) → a__U104(a__isNatIListKind(V2), V1, V2)
a__U104(tt, V1, V2) → a__U105(a__isNat(V1), V2)
a__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__U111(tt, L, N) → a__U112(a__isNatIListKind(L), L, N)
a__U112(tt, L, N) → a__U113(a__isNat(N), L, N)
a__U113(tt, L, N) → a__U114(a__isNatKind(N), L)
a__U114(tt, L) → s(a__length(mark(L)))
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U122(tt) → nil
a__U13(tt) → tt
a__U131(tt, IL, M, N) → a__U132(a__isNatIListKind(IL), IL, M, N)
a__U132(tt, IL, M, N) → a__U133(a__isNat(M), IL, M, N)
a__U133(tt, IL, M, N) → a__U134(a__isNatKind(M), IL, M, N)
a__U134(tt, IL, M, N) → a__U135(a__isNat(N), IL, M, N)
a__U135(tt, IL, M, N) → a__U136(a__isNatKind(N), IL, M, N)
a__U136(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U62(tt) → tt
a__U71(tt) → tt
a__U81(tt) → tt
a__U91(tt, V1, V2) → a__U92(a__isNatKind(V1), V1, V2)
a__U92(tt, V1, V2) → a__U93(a__isNatIListKind(V2), V1, V2)
a__U93(tt, V1, V2) → a__U94(a__isNatIListKind(V2), V1, V2)
a__U94(tt, V1, V2) → a__U95(a__isNat(V1), V2)
a__U95(tt, V2) → a__U96(a__isNatList(V2))
a__U96(tt) → tt
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
a__take(0, IL) → a__U121(a__isNatIList(IL), IL)
a__take(s(M), cons(N, IL)) → a__U131(a__isNatIList(IL), IL, M, N)
mark(zeros) → a__zeros
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
mark(isNatKind(X)) → a__isNatKind(X)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U111(X1, X2, X3)) → a__U111(mark(X1), X2, X3)
mark(U112(X1, X2, X3)) → a__U112(mark(X1), X2, X3)
mark(U113(X1, X2, X3)) → a__U113(mark(X1), X2, X3)
mark(U114(X1, X2)) → a__U114(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U122(X)) → a__U122(mark(X))
mark(U131(X1, X2, X3, X4)) → a__U131(mark(X1), X2, X3, X4)
mark(U132(X1, X2, X3, X4)) → a__U132(mark(X1), X2, X3, X4)
mark(U133(X1, X2, X3, X4)) → a__U133(mark(X1), X2, X3, X4)
mark(U134(X1, X2, X3, X4)) → a__U134(mark(X1), X2, X3, X4)
mark(U135(X1, X2, X3, X4)) → a__U135(mark(X1), X2, X3, X4)
mark(U136(X1, X2, X3, X4)) → a__U136(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X)) → a__U71(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U94(X1, X2, X3)) → a__U94(mark(X1), X2, X3)
mark(U95(X1, X2)) → a__U95(mark(X1), X2)
mark(U96(X)) → a__U96(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U114(X1, X2) → U114(X1, X2)
a__length(X) → length(X)
a__U13(X) → U13(X)
a__isNatList(X) → isNatList(X)
a__U121(X1, X2) → U121(X1, X2)
a__U122(X) → U122(X)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U95(X1, X2) → U95(X1, X2)
a__U96(X) → U96(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.