Termination w.r.t. Q proof of /tmp/tmpbJxyWN/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 PisEmptyProof (⇔)

    ↳9 TRUE

  ↳10 QDP

    ↳11 QDPOrderProof (⇔)

    ↳12 QDP

    ↳13 DependencyGraphProof (⇔)

    ↳14 TRUE

  ↳15 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__U61(tt, V2) → A__ISNATILISTKIND(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) = A__ISNATILISTKIND(x1)
cons(x1, x2) = cons(x1, x2)
a__isNatKind(x1) = a__isNatKind(x1)
A__ISNATKIND(x1) = A__ISNATKIND(x1)
length(x1) = x1
take(x1, x2) = take(x1, x2)
A__U61(x1, x2) = A__U61(x1, x2)
s(x1) = s(x1)
0 = 0
a__U71(x1) = x1
a__isNatIListKind(x1) = a__isNatIListKind(x1)
a__U81(x1) = x1
isNatKind(x1) = x1
a__U51(x1, x2) = a__U51(x1)
U51(x1, x2) = x1
nil = nil
zeros = zeros
isNatIListKind(x1) = isNatIListKind(x1)
U71(x1) = x1
a__U61(x1, x2) = a__U61(x1)
U61(x1, x2) = U61
U81(x1) = U81
a__U62(x1) = a__U62
U62(x1) = U62
a__U52(x1) = a__U52
U52(x1) = U52

Recursive path order with status [RPO].
Precedence:

cons₂ > A_{_U51₁} > A_{_{ISNATILISTKIND₁}} > U81
cons₂ > a_{_isNatKind₁} > a_{_{isNatIListKind₁}} > a_{_U51₁} > a_{_U52} > tt > A_{_{ISNATILISTKIND₁}} > U81
cons₂ > a_{_isNatKind₁} > a_{_{isNatIListKind₁}} > a_{_U51₁} > a_{_U52} > U52 > U81
cons₂ > a_{_isNatKind₁} > a_{_{isNatIListKind₁}} > isNatIListKind₁ > U81
cons₂ > A_{_ISNATKIND₁} > A_{_{ISNATILISTKIND₁}} > U81
take₂ > a_{_isNatKind₁} > a_{_{isNatIListKind₁}} > a_{_U51₁} > a_{_U52} > tt > A_{_{ISNATILISTKIND₁}} > U81
take₂ > a_{_isNatKind₁} > a_{_{isNatIListKind₁}} > a_{_U51₁} > a_{_U52} > U52 > U81
take₂ > a_{_isNatKind₁} > a_{_{isNatIListKind₁}} > isNatIListKind₁ > U81
take₂ > A_{_ISNATKIND₁} > A_{_{ISNATILISTKIND₁}} > U81
take₂ > A_{_U61₂} > A_{_{ISNATILISTKIND₁}} > U81
take₂ > a_{_U61₁} > a_{_{isNatIListKind₁}} > a_{_U51₁} > a_{_U52} > tt > A_{_{ISNATILISTKIND₁}} > U81
take₂ > a_{_U61₁} > a_{_{isNatIListKind₁}} > a_{_U51₁} > a_{_U52} > U52 > U81
take₂ > a_{_U61₁} > a_{_{isNatIListKind₁}} > isNatIListKind₁ > U81
take₂ > a_{_U61₁} > U61 > U81
take₂ > a_{_U61₁} > a_{_U62} > tt > A_{_{ISNATILISTKIND₁}} > U81
take₂ > a_{_U61₁} > a_{_U62} > U62 > U81
s₁ > U81
0 > U81
nil > U81
zeros > U81

Status:

A_{_U51₁}: multiset
tt: multiset
A_{_{ISNATILISTKIND₁}}: multiset
cons₂: multiset
a_{_isNatKind₁}: [1]
A_{_ISNATKIND₁}: multiset
take₂: multiset
A_{_U61₂}: multiset
s₁: multiset
0: multiset
a_{_{isNatIListKind₁}}: multiset
a_{_U51₁}: multiset
nil: multiset
zeros: multiset
isNatIListKind₁: multiset
a_{_U61₁}: multiset
U61: multiset
U81: multiset
a_{_U62}: multiset
U62: multiset
a_{_U52}: multiset
U52: multiset

The following usable rules [FROCOS05] were oriented:

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

(7) Obligation:

Q DP problem:
P is empty.
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) PisEmptyProof (EQUIVALENT transformation)

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

(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__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__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__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 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(x2, x3)
tt = tt
A__U103(x1, x2, x3) = A__U103(x2, x3)
a__isNatIListKind(x1) = a__isNatIListKind
A__U104(x1, x2, x3) = A__U104(x2, x3)
A__U105(x1, x2) = A__U105(x1, x2)
a__isNat(x1) = x1
A__ISNATILIST(x1) = A__ISNATILIST(x1)
A__U31(x1, x2) = A__U31(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
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) = A__U95(x2)
take(x1, x2) = take(x1, x2)
A__U101(x1, x2, x3) = A__U101(x2, x3)
A__ISNAT(x1) = A__ISNAT(x1)
length(x1) = length(x1)
A__U11(x1, x2) = A__U11(x2)
A__U12(x1, x2) = x2
s(x1) = s(x1)
A__U21(x1, x2) = A__U21(x2)
A__U22(x1, x2) = A__U22(x2)
A__U41(x1, x2, x3) = A__U41(x2, x3)
A__U42(x1, x2, x3) = A__U42(x2, x3)
A__U43(x1, x2, x3) = A__U43(x2, x3)
A__U44(x1, x2, x3) = A__U44(x2, x3)
A__U45(x1, x2) = A__U45(x2)
nil = nil
zeros = zeros
a__U51(x1, x2) = a__U51
a__U61(x1, x2) = x1
isNatIListKind(x1) = isNatIListKind
0 = 0
a__U11(x1, x2) = a__U11
a__U21(x1, x2) = a__U21
isNat(x1) = x1
a__U71(x1) = a__U71
a__U81(x1) = a__U81
isNatKind(x1) = x1
a__U102(x1, x2, x3) = a__U102(x2, x3)
a__U103(x1, x2, x3) = x3
U103(x1, x2, x3) = x3
a__U104(x1, x2, x3) = x3
U104(x1, x2, x3) = x3
a__U105(x1, x2) = x2
U105(x1, x2) = x2
U11(x1, x2) = U11
a__U12(x1, x2) = a__U12
U12(x1, x2) = U12
a__U13(x1) = a__U13
a__isNatList(x1) = x1
isNatList(x1) = x1
U13(x1) = U13
a__U91(x1, x2, x3) = x3
U91(x1, x2, x3) = x3
a__U92(x1, x2, x3) = x3
U92(x1, x2, x3) = x3
a__U93(x1, x2, x3) = x3
U93(x1, x2, x3) = x3
a__U94(x1, x2, x3) = x3
U94(x1, x2, x3) = x3
a__U95(x1, x2) = x2
U95(x1, x2) = x2
U21(x1, x2) = U21
a__U22(x1, x2) = a__U22
U22(x1, x2) = U22
a__U23(x1) = a__U23
U23(x1) = U23
a__U96(x1) = x1
U96(x1) = U96
a__U101(x1, x2, x3) = a__U101(x2, x3)
U101(x1, x2, x3) = x2
U102(x1, x2, x3) = U102
a__U106(x1) = x1
a__isNatIList(x1) = x1
isNatIList(x1) = x1
U106(x1) = x1
a__U31(x1, x2) = x2
U31(x1, x2) = x2
a__U32(x1, x2) = x2
U32(x1, x2) = x2
a__U33(x1) = x1
U33(x1) = x1
a__U41(x1, x2, x3) = a__U41(x3)
U41(x1, x2, x3) = U41
a__U42(x1, x2, x3) = a__U42(x3)
U42(x1, x2, x3) = x3
a__U43(x1, x2, x3) = a__U43(x3)
U43(x1, x2, x3) = U43
a__U44(x1, x2, x3) = a__U44(x3)
U44(x1, x2, x3) = U44
a__U45(x1, x2) = a__U45(x2)
U45(x1, x2) = U45
a__U46(x1) = x1
U46(x1) = x1
U51(x1, x2) = U51
U71(x1) = x1
U61(x1, x2) = x1
U81(x1) = U81
a__U62(x1) = a__U62
U62(x1) = U62
a__U52(x1) = a__U52
U52(x1) = U52

Recursive path order with status [RPO].
Precedence:

cons₂ > A_{_U91₂} > A_{_U92₂} > A_{_U93₂} > a_{_{isNatIListKind}} > a_{_isNatKind} > a_{_U71} > U96
cons₂ > A_{_U91₂} > A_{_U92₂} > A_{_U93₂} > a_{_{isNatIListKind}} > a_{_U51} > U96
cons₂ > A_{_U91₂} > A_{_U92₂} > A_{_U93₂} > a_{_{isNatIListKind}} > isNatIListKind > U96
cons₂ > A_{_U91₂} > A_{_U92₂} > A_{_U93₂} > A_{_U94₂} > A_{_U95₁} > U96
cons₂ > A_{_U91₂} > A_{_U92₂} > A_{_U93₂} > A_{_U94₂} > A_{_ISNAT₁} > A_{_U11₁} > U96
cons₂ > A_{_U41₂} > A_{_U42₂} > A_{_U43₂} > A_{_U44₂} > A_{_ISNAT₁} > A_{_U11₁} > U96
cons₂ > A_{_U41₂} > A_{_U42₂} > A_{_U43₂} > A_{_U44₂} > A_{_U45₁} > A_{_{ISNATILIST₁}} > a_{_{isNatIListKind}} > a_{_isNatKind} > a_{_U71} > U96
cons₂ > A_{_U41₂} > A_{_U42₂} > A_{_U43₂} > A_{_U44₂} > A_{_U45₁} > A_{_{ISNATILIST₁}} > a_{_{isNatIListKind}} > a_{_U51} > U96
cons₂ > A_{_U41₂} > A_{_U42₂} > A_{_U43₂} > A_{_U44₂} > A_{_U45₁} > A_{_{ISNATILIST₁}} > a_{_{isNatIListKind}} > isNatIListKind > U96
cons₂ > A_{_U41₂} > A_{_U42₂} > A_{_U43₂} > A_{_U44₂} > A_{_U45₁} > A_{_{ISNATILIST₁}} > A_{_U31₁} > U96
cons₂ > a_{_U41₁} > U41 > U96
cons₂ > a_{_U41₁} > a_{_U42₁} > a_{_U43₁} > a_{_{isNatIListKind}} > a_{_isNatKind} > a_{_U71} > U96
cons₂ > a_{_U41₁} > a_{_U42₁} > a_{_U43₁} > a_{_{isNatIListKind}} > a_{_U51} > U96
cons₂ > a_{_U41₁} > a_{_U42₁} > a_{_U43₁} > a_{_{isNatIListKind}} > isNatIListKind > U96
cons₂ > a_{_U41₁} > a_{_U42₁} > a_{_U43₁} > U43 > U96
cons₂ > a_{_U41₁} > a_{_U42₁} > a_{_U43₁} > a_{_U44₁} > U44 > U96
cons₂ > a_{_U41₁} > a_{_U42₁} > a_{_U43₁} > a_{_U44₁} > a_{_U45₁} > U45 > U96
take₂ > A_{_U101₂} > A_{_U102₂} > A_{_U103₂} > A_{_U104₂} > A_{_U105₂} > A_{_{ISNATILIST₁}} > a_{_{isNatIListKind}} > a_{_isNatKind} > a_{_U71} > U96
take₂ > A_{_U101₂} > A_{_U102₂} > A_{_U103₂} > A_{_U104₂} > A_{_U105₂} > A_{_{ISNATILIST₁}} > a_{_{isNatIListKind}} > a_{_U51} > U96
take₂ > A_{_U101₂} > A_{_U102₂} > A_{_U103₂} > A_{_U104₂} > A_{_U105₂} > A_{_{ISNATILIST₁}} > a_{_{isNatIListKind}} > isNatIListKind > U96
take₂ > A_{_U101₂} > A_{_U102₂} > A_{_U103₂} > A_{_U104₂} > A_{_U105₂} > A_{_{ISNATILIST₁}} > A_{_U31₁} > U96
take₂ > A_{_U101₂} > A_{_U102₂} > A_{_U103₂} > A_{_U104₂} > A_{_ISNAT₁} > A_{_U11₁} > U96
take₂ > a_{_U101₂} > a_{_U102₂} > a_{_{isNatIListKind}} > a_{_isNatKind} > a_{_U71} > U96
take₂ > a_{_U101₂} > a_{_U102₂} > a_{_{isNatIListKind}} > a_{_U51} > U96
take₂ > a_{_U101₂} > a_{_U102₂} > a_{_{isNatIListKind}} > isNatIListKind > U96
take₂ > a_{_U101₂} > a_{_U102₂} > U102 > U96
length₁ > a_{_U11} > U11 > U96
length₁ > a_{_U11} > a_{_U12} > U12 > U96
length₁ > a_{_U11} > a_{_U12} > a_{_U13} > tt > A_{_U105₂} > A_{_{ISNATILIST₁}} > a_{_{isNatIListKind}} > a_{_isNatKind} > a_{_U71} > U96
length₁ > a_{_U11} > a_{_U12} > a_{_U13} > tt > A_{_U105₂} > A_{_{ISNATILIST₁}} > a_{_{isNatIListKind}} > a_{_U51} > U96
length₁ > a_{_U11} > a_{_U12} > a_{_U13} > tt > A_{_U105₂} > A_{_{ISNATILIST₁}} > a_{_{isNatIListKind}} > isNatIListKind > U96
length₁ > a_{_U11} > a_{_U12} > a_{_U13} > tt > A_{_U105₂} > A_{_{ISNATILIST₁}} > A_{_U31₁} > U96
length₁ > a_{_U11} > a_{_U12} > a_{_U13} > tt > A_{_U22₁} > A_{_ISNAT₁} > A_{_U11₁} > U96
length₁ > a_{_U11} > a_{_U12} > a_{_U13} > U13 > U96
s₁ > A_{_U21₁} > a_{_isNatKind} > a_{_U71} > U96
s₁ > A_{_U21₁} > A_{_U22₁} > A_{_ISNAT₁} > A_{_U11₁} > U96
s₁ > a_{_U21} > U21 > U96
s₁ > a_{_U21} > a_{_U22} > U22 > U96
s₁ > a_{_U21} > a_{_U22} > a_{_U23} > tt > A_{_U105₂} > A_{_{ISNATILIST₁}} > a_{_{isNatIListKind}} > a_{_isNatKind} > a_{_U71} > U96
s₁ > a_{_U21} > a_{_U22} > a_{_U23} > tt > A_{_U105₂} > A_{_{ISNATILIST₁}} > a_{_{isNatIListKind}} > a_{_U51} > U96
s₁ > a_{_U21} > a_{_U22} > a_{_U23} > tt > A_{_U105₂} > A_{_{ISNATILIST₁}} > a_{_{isNatIListKind}} > isNatIListKind > U96
s₁ > a_{_U21} > a_{_U22} > a_{_U23} > tt > A_{_U105₂} > A_{_{ISNATILIST₁}} > A_{_U31₁} > U96
s₁ > a_{_U21} > a_{_U22} > a_{_U23} > tt > A_{_U22₁} > A_{_ISNAT₁} > A_{_U11₁} > U96
s₁ > a_{_U21} > a_{_U22} > a_{_U23} > U23 > U96
s₁ > a_{_U81} > U81 > U96
nil > tt > A_{_U105₂} > A_{_{ISNATILIST₁}} > a_{_{isNatIListKind}} > a_{_isNatKind} > a_{_U71} > U96
nil > tt > A_{_U105₂} > A_{_{ISNATILIST₁}} > a_{_{isNatIListKind}} > a_{_U51} > U96
nil > tt > A_{_U105₂} > A_{_{ISNATILIST₁}} > a_{_{isNatIListKind}} > isNatIListKind > U96
nil > tt > A_{_U105₂} > A_{_{ISNATILIST₁}} > A_{_U31₁} > U96
nil > tt > A_{_U22₁} > A_{_ISNAT₁} > A_{_U11₁} > U96
zeros > tt > A_{_U105₂} > A_{_{ISNATILIST₁}} > a_{_{isNatIListKind}} > a_{_isNatKind} > a_{_U71} > U96
zeros > tt > A_{_U105₂} > A_{_{ISNATILIST₁}} > a_{_{isNatIListKind}} > a_{_U51} > U96
zeros > tt > A_{_U105₂} > A_{_{ISNATILIST₁}} > a_{_{isNatIListKind}} > isNatIListKind > U96
zeros > tt > A_{_U105₂} > A_{_{ISNATILIST₁}} > A_{_U31₁} > U96
zeros > tt > A_{_U22₁} > A_{_ISNAT₁} > A_{_U11₁} > U96
0 > tt > A_{_U105₂} > A_{_{ISNATILIST₁}} > a_{_{isNatIListKind}} > a_{_isNatKind} > a_{_U71} > U96
0 > tt > A_{_U105₂} > A_{_{ISNATILIST₁}} > a_{_{isNatIListKind}} > a_{_U51} > U96
0 > tt > A_{_U105₂} > A_{_{ISNATILIST₁}} > a_{_{isNatIListKind}} > isNatIListKind > U96
0 > tt > A_{_U105₂} > A_{_{ISNATILIST₁}} > A_{_U31₁} > U96
0 > tt > A_{_U22₁} > A_{_ISNAT₁} > A_{_U11₁} > U96
U51 > U96
a_{_U62} > tt > A_{_U105₂} > A_{_{ISNATILIST₁}} > a_{_{isNatIListKind}} > a_{_isNatKind} > a_{_U71} > U96
a_{_U62} > tt > A_{_U105₂} > A_{_{ISNATILIST₁}} > a_{_{isNatIListKind}} > a_{_U51} > U96
a_{_U62} > tt > A_{_U105₂} > A_{_{ISNATILIST₁}} > a_{_{isNatIListKind}} > isNatIListKind > U96
a_{_U62} > tt > A_{_U105₂} > A_{_{ISNATILIST₁}} > A_{_U31₁} > U96
a_{_U62} > tt > A_{_U22₁} > A_{_ISNAT₁} > A_{_U11₁} > U96
a_{_U62} > U62 > U96
a_{_U52} > tt > A_{_U105₂} > A_{_{ISNATILIST₁}} > a_{_{isNatIListKind}} > a_{_isNatKind} > a_{_U71} > U96
a_{_U52} > tt > A_{_U105₂} > A_{_{ISNATILIST₁}} > a_{_{isNatIListKind}} > a_{_U51} > U96
a_{_U52} > tt > A_{_U105₂} > A_{_{ISNATILIST₁}} > a_{_{isNatIListKind}} > isNatIListKind > U96
a_{_U52} > tt > A_{_U105₂} > A_{_{ISNATILIST₁}} > A_{_U31₁} > U96
a_{_U52} > tt > A_{_U22₁} > A_{_ISNAT₁} > A_{_U11₁} > U96
a_{_U52} > U52 > U96

Status:

A_{_U102₂}: multiset
tt: multiset
A_{_U103₂}: multiset
a_{_{isNatIListKind}}: multiset
A_{_U104₂}: multiset
A_{_U105₂}: multiset
A_{_{ISNATILIST₁}}: multiset
A_{_U31₁}: multiset
cons₂: multiset
A_{_U91₂}: multiset
a_{_isNatKind}: multiset
A_{_U92₂}: multiset
A_{_U93₂}: multiset
A_{_U94₂}: multiset
A_{_U95₁}: multiset
take₂: multiset
A_{_U101₂}: multiset
A_{_ISNAT₁}: multiset
length₁: multiset
A_{_U11₁}: multiset
s₁: multiset
A_{_U21₁}: multiset
A_{_U22₁}: multiset
A_{_U41₂}: multiset
A_{_U42₂}: multiset
A_{_U43₂}: multiset
A_{_U44₂}: multiset
A_{_U45₁}: multiset
nil: multiset
zeros: multiset
a_{_U51}: multiset
isNatIListKind: multiset
0: multiset
a_{_U11}: multiset
a_{_U21}: multiset
a_{_U71}: multiset
a_{_U81}: multiset
a_{_U102₂}: multiset
U11: multiset
a_{_U12}: multiset
U12: multiset
a_{_U13}: multiset
U13: multiset
U21: multiset
a_{_U22}: multiset
U22: multiset
a_{_U23}: multiset
U23: multiset
U96: multiset
a_{_U101₂}: multiset
U102: multiset
a_{_U41₁}: multiset
U41: multiset
a_{_U42₁}: multiset
a_{_U43₁}: multiset
U43: multiset
a_{_U44₁}: multiset
U44: multiset
a_{_U45₁}: multiset
U45: multiset
U51: multiset
U81: multiset
a_{_U62}: multiset
U62: multiset
a_{_U52}: multiset
U52: multiset

The following usable rules [FROCOS05] were oriented:

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__isNat(X) → isNat(X)
a__U102(tt, V1, V2) → a__U103(a__isNatIListKind(V2), V1, V2)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__U103(tt, V1, V2) → a__U104(a__isNatIListKind(V2), V1, V2)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U104(tt, V1, V2) → a__U105(a__isNat(V1), V2)
a__U105(X1, X2) → U105(X1, X2)
a__U11(X1, X2) → U11(X1, X2)
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__U12(X1, X2) → U12(X1, X2)
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__isNatList(nil) → tt
a__isNatList(X) → isNatList(X)
a__U13(tt) → tt
a__U13(X) → U13(X)
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U91(tt, V1, V2) → a__U92(a__isNatKind(V1), V1, V2)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__U92(tt, V1, V2) → a__U93(a__isNatIListKind(V2), V1, V2)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U93(tt, V1, V2) → a__U94(a__isNatIListKind(V2), V1, V2)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U94(tt, V1, V2) → a__U95(a__isNat(V1), V2)
a__U95(X1, X2) → U95(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U22(X1, X2) → U22(X1, X2)
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U23(tt) → tt
a__U23(X) → U23(X)
a__U95(tt, V2) → a__U96(a__isNatList(V2))
a__U96(tt) → tt
a__U96(X) → U96(X)
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U101(tt, V1, V2) → a__U102(a__isNatKind(V1), V1, V2)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__isNatIList(zeros) → tt
a__isNatIList(X) → isNatIList(X)
a__U106(tt) → tt
a__U106(X) → U106(X)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__U31(X1, X2) → U31(X1, X2)
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(X1, X2) → U32(X1, X2)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U33(tt) → tt
a__U33(X) → U33(X)
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U45(X1, X2) → U45(X1, X2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U46(tt) → tt
a__U46(X) → U46(X)

(12) Obligation:

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

A__U32(tt, V) → A__ISNATLIST(V)
A__U12(tt, V1) → A__ISNATLIST(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 0 SCCs with 2 less nodes.

(14) TRUE

(15) 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.