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

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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.

↳ QTRS

  ↳ DependencyPairsProof

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.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

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

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.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

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

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.
The approximation of the Dependency Graph [15,17,22] contains 3 SCCs with 83 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

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

A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__U61(tt, V2) → A__ISNATILISTKIND(V2)
A__ISNATILISTKIND(take(V1, V2)) → A__ISNATKIND(V1)
A__ISNATILISTKIND(take(V1, V2)) → A__U61(a__isNatKind(V1), V2)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__U51(a__isNatKind(V1), V2)
A__U51(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.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

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__U81(tt) → tt
a__U81(X) → U81(X)
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__isNatIListKind(X) → isNatIListKind(X)
a__U71(tt) → tt
a__U71(X) → U71(X)
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U61(X1, X2) → U61(X1, X2)
a__U62(tt) → tt
a__U62(X) → U62(X)
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U51(X1, X2) → U51(X1, X2)
a__U52(tt) → tt
a__U52(X) → U52(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
The graph contains the following edges 1 > 1
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
The graph contains the following edges 1 > 1
A__ISNATILISTKIND(take(V1, V2)) → A__U61(a__isNatKind(V1), V2)
The graph contains the following edges 1 > 2
A__ISNATILISTKIND(cons(V1, V2)) → A__U51(a__isNatKind(V1), V2)
The graph contains the following edges 1 > 2
A__U61(tt, V2) → A__ISNATILISTKIND(V2)
The graph contains the following edges 2 >= 1
A__U51(tt, V2) → A__ISNATILISTKIND(V2)
The graph contains the following edges 2 >= 1
A__ISNATILISTKIND(take(V1, V2)) → A__ISNATKIND(V1)
The graph contains the following edges 1 > 1
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPSizeChangeProof

          ↳ QDP

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

A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
A__U12(tt, V1) → A__ISNATLIST(V1)
A__U94(tt, V1, V2) → A__U95(a__isNat(V1), V2)
A__U94(tt, V1, V2) → A__ISNAT(V1)
A__U31(tt, V) → A__U32(a__isNatIListKind(V), V)
A__U44(tt, V1, V2) → A__U45(a__isNat(V1), V2)
A__ISNATLIST(cons(V1, V2)) → A__U91(a__isNatKind(V1), V1, V2)
A__U101(tt, V1, V2) → A__U102(a__isNatKind(V1), V1, V2)
A__U93(tt, V1, V2) → A__U94(a__isNatIListKind(V2), V1, V2)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__U103(tt, V1, V2) → A__U104(a__isNatIListKind(V2), V1, V2)
A__U42(tt, V1, V2) → A__U43(a__isNatIListKind(V2), V1, V2)
A__U105(tt, V2) → A__ISNATILIST(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__U21(tt, V1) → A__U22(a__isNatKind(V1), V1)
A__U45(tt, V2) → A__ISNATILIST(V2)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNatKind(V1), V1, V2)
A__U104(tt, V1, V2) → A__ISNAT(V1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U102(tt, V1, V2) → A__U103(a__isNatIListKind(V2), V1, V2)
A__U43(tt, V1, V2) → A__U44(a__isNatIListKind(V2), V1, V2)
A__U41(tt, V1, V2) → A__U42(a__isNatKind(V1), V1, V2)
A__U44(tt, V1, V2) → A__ISNAT(V1)
A__U91(tt, V1, V2) → A__U92(a__isNatKind(V1), V1, V2)
A__U22(tt, V1) → A__ISNAT(V1)
A__U32(tt, V) → A__ISNATLIST(V)
A__U11(tt, V1) → A__U12(a__isNatIListKind(V1), V1)
A__U95(tt, V2) → A__ISNATLIST(V2)
A__ISNATLIST(take(V1, V2)) → A__U101(a__isNatKind(V1), V1, 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)

From the DPs we obtained the following set of size-change graphs:

A__U103(tt, V1, V2) → A__U104(a__isNatIListKind(V2), V1, V2)
The graph contains the following edges 2 >= 2, 3 >= 3
A__U101(tt, V1, V2) → A__U102(a__isNatKind(V1), V1, V2)
The graph contains the following edges 2 >= 2, 3 >= 3
A__U102(tt, V1, V2) → A__U103(a__isNatIListKind(V2), V1, V2)
The graph contains the following edges 2 >= 2, 3 >= 3
A__U105(tt, V2) → A__ISNATILIST(V2)
The graph contains the following edges 2 >= 1
A__U104(tt, V1, V2) → A__U105(a__isNat(V1), V2)
The graph contains the following edges 3 >= 2
A__U104(tt, V1, V2) → A__ISNAT(V1)
The graph contains the following edges 2 >= 1
A__U31(tt, V) → A__U32(a__isNatIListKind(V), V)
The graph contains the following edges 2 >= 2
A__U45(tt, V2) → A__ISNATILIST(V2)
The graph contains the following edges 2 >= 1
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
The graph contains the following edges 1 >= 2
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNatKind(V1), V1, V2)
The graph contains the following edges 1 > 2, 1 > 3
A__U32(tt, V) → A__ISNATLIST(V)
The graph contains the following edges 2 >= 1
A__U91(tt, V1, V2) → A__U92(a__isNatKind(V1), V1, V2)
The graph contains the following edges 2 >= 2, 3 >= 3
A__ISNATLIST(cons(V1, V2)) → A__U91(a__isNatKind(V1), V1, V2)
The graph contains the following edges 1 > 2, 1 > 3
A__ISNATLIST(take(V1, V2)) → A__U101(a__isNatKind(V1), V1, V2)
The graph contains the following edges 1 > 2, 1 > 3
A__U92(tt, V1, V2) → A__U93(a__isNatIListKind(V2), V1, V2)
The graph contains the following edges 2 >= 2, 3 >= 3
A__U93(tt, V1, V2) → A__U94(a__isNatIListKind(V2), V1, V2)
The graph contains the following edges 2 >= 2, 3 >= 3
A__U95(tt, V2) → A__ISNATLIST(V2)
The graph contains the following edges 2 >= 1
A__U12(tt, V1) → A__ISNATLIST(V1)
The graph contains the following edges 2 >= 1
A__U94(tt, V1, V2) → A__U95(a__isNat(V1), V2)
The graph contains the following edges 3 >= 2
A__U94(tt, V1, V2) → A__ISNAT(V1)
The graph contains the following edges 2 >= 1
A__U11(tt, V1) → A__U12(a__isNatIListKind(V1), V1)
The graph contains the following edges 2 >= 2
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
The graph contains the following edges 1 > 2
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
The graph contains the following edges 1 > 2
A__U21(tt, V1) → A__U22(a__isNatKind(V1), V1)
The graph contains the following edges 2 >= 2
A__U22(tt, V1) → A__ISNAT(V1)
The graph contains the following edges 2 >= 1
A__U44(tt, V1, V2) → A__ISNAT(V1)
The graph contains the following edges 2 >= 1
A__U41(tt, V1, V2) → A__U42(a__isNatKind(V1), V1, V2)
The graph contains the following edges 2 >= 2, 3 >= 3
A__U42(tt, V1, V2) → A__U43(a__isNatIListKind(V2), V1, V2)
The graph contains the following edges 2 >= 2, 3 >= 3
A__U43(tt, V1, V2) → A__U44(a__isNatIListKind(V2), V1, V2)
The graph contains the following edges 2 >= 2, 3 >= 3
A__U44(tt, V1, V2) → A__U45(a__isNat(V1), V2)
The graph contains the following edges 3 >= 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

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

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

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.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U112(X1, X2, X3)) → MARK(X1)
MARK(length(X)) → MARK(X)
MARK(U114(X1, X2)) → MARK(X1)
MARK(U114(X1, X2)) → A__U114(mark(X1), X2)
MARK(U111(X1, X2, X3)) → MARK(X1)
MARK(U113(X1, X2, X3)) → A__U113(mark(X1), X2, X3)
MARK(U111(X1, X2, X3)) → A__U111(mark(X1), X2, X3)
MARK(U112(X1, X2, X3)) → A__U112(mark(X1), X2, X3)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U113(X1, X2, X3)) → MARK(X1)

The remaining pairs can at least be oriented weakly.

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

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__LENGTH(x₁)) = x₁
POL(A__TAKE(x₁, x₂)) = x₂
POL(A__U111(x₁, x₂, x₃)) = x₂
POL(A__U112(x₁, x₂, x₃)) = x₂
POL(A__U113(x₁, x₂, x₃)) = x₂
POL(A__U114(x₁, x₂)) = x₂
POL(A__U131(x₁, x₂, x₃, x₄)) = x₄
POL(A__U132(x₁, x₂, x₃, x₄)) = x₄
POL(A__U133(x₁, x₂, x₃, x₄)) = x₄
POL(A__U134(x₁, x₂, x₃, x₄)) = x₄
POL(A__U135(x₁, x₂, x₃, x₄)) = x₄
POL(A__U136(x₁, x₂, x₃, x₄)) = x₄
POL(MARK(x₁)) = x₁
POL(U101(x₁, x₂, x₃)) = x₁
POL(U102(x₁, x₂, x₃)) = x₁
POL(U103(x₁, x₂, x₃)) = x₁
POL(U104(x₁, x₂, x₃)) = x₁
POL(U105(x₁, x₂)) = x₁
POL(U106(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁
POL(U111(x₁, x₂, x₃)) = 1 + x₁ + x₂
POL(U112(x₁, x₂, x₃)) = 1 + x₁ + x₂
POL(U113(x₁, x₂, x₃)) = 1 + x₁ + x₂
POL(U114(x₁, x₂)) = 1 + x₁ + x₂
POL(U12(x₁, x₂)) = x₁
POL(U121(x₁, x₂)) = x₁
POL(U122(x₁)) = x₁
POL(U13(x₁)) = x₁
POL(U131(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(U132(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(U133(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(U134(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(U135(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(U136(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁, x₂)) = x₁
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁, x₂)) = x₁
POL(U33(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = x₁
POL(U42(x₁, x₂, x₃)) = x₁
POL(U43(x₁, x₂, x₃)) = x₁
POL(U44(x₁, x₂, x₃)) = x₁
POL(U45(x₁, x₂)) = x₁
POL(U46(x₁)) = x₁
POL(U51(x₁, x₂)) = x₁
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂)) = x₁
POL(U62(x₁)) = x₁
POL(U71(x₁)) = x₁
POL(U81(x₁)) = x₁
POL(U91(x₁, x₂, x₃)) = x₁
POL(U92(x₁, x₂, x₃)) = x₁
POL(U93(x₁, x₂, x₃)) = x₁
POL(U94(x₁, x₂, x₃)) = x₁
POL(U95(x₁, x₂)) = x₁
POL(U96(x₁)) = x₁
POL(a__U101(x₁, x₂, x₃)) = x₁
POL(a__U102(x₁, x₂, x₃)) = x₁
POL(a__U103(x₁, x₂, x₃)) = x₁
POL(a__U104(x₁, x₂, x₃)) = x₁
POL(a__U105(x₁, x₂)) = x₁
POL(a__U106(x₁)) = x₁
POL(a__U11(x₁, x₂)) = x₁
POL(a__U111(x₁, x₂, x₃)) = 1 + x₁ + x₂
POL(a__U112(x₁, x₂, x₃)) = 1 + x₁ + x₂
POL(a__U113(x₁, x₂, x₃)) = 1 + x₁ + x₂
POL(a__U114(x₁, x₂)) = 1 + x₁ + x₂
POL(a__U12(x₁, x₂)) = x₁
POL(a__U121(x₁, x₂)) = x₁
POL(a__U122(x₁)) = x₁
POL(a__U13(x₁)) = x₁
POL(a__U131(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(a__U132(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(a__U133(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(a__U134(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(a__U135(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(a__U136(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(a__U21(x₁, x₂)) = x₁
POL(a__U22(x₁, x₂)) = x₁
POL(a__U23(x₁)) = x₁
POL(a__U31(x₁, x₂)) = x₁
POL(a__U32(x₁, x₂)) = x₁
POL(a__U33(x₁)) = x₁
POL(a__U41(x₁, x₂, x₃)) = x₁
POL(a__U42(x₁, x₂, x₃)) = x₁
POL(a__U43(x₁, x₂, x₃)) = x₁
POL(a__U44(x₁, x₂, x₃)) = x₁
POL(a__U45(x₁, x₂)) = x₁
POL(a__U46(x₁)) = x₁
POL(a__U51(x₁, x₂)) = x₁
POL(a__U52(x₁)) = x₁
POL(a__U61(x₁, x₂)) = x₁
POL(a__U62(x₁)) = x₁
POL(a__U71(x₁)) = x₁
POL(a__U81(x₁)) = x₁
POL(a__U91(x₁, x₂, x₃)) = x₁
POL(a__U92(x₁, x₂, x₃)) = x₁
POL(a__U93(x₁, x₂, x₃)) = x₁
POL(a__U94(x₁, x₂, x₃)) = x₁
POL(a__U95(x₁, x₂)) = x₁
POL(a__U96(x₁)) = x₁
POL(a__isNat(x₁)) = 0
POL(a__isNatIList(x₁)) = 0
POL(a__isNatIListKind(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__isNatList(x₁)) = 0
POL(a__length(x₁)) = 1 + x₁
POL(a__take(x₁, x₂)) = x₁ + x₂
POL(a__zeros) = 0
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 1 + x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = x₁ + x₂
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U113(tt, L, N) → a__U114(a__isNatKind(N), L)
a__U114(tt, L) → s(a__length(mark(L)))
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__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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__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__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__U122(tt) → nil
a__U13(tt) → tt
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U33(tt) → tt
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U23(tt) → tt
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U62(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U46(tt) → tt
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U103(tt, V1, V2) → a__U104(a__isNatIListKind(V2), V1, V2)
a__U102(tt, V1, V2) → a__U103(a__isNatIListKind(V2), V1, V2)
a__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U104(tt, V1, V2) → a__U105(a__isNat(V1), V2)
a__U101(tt, V1, V2) → a__U102(a__isNatKind(V1), V1, V2)
a__zeros → cons(0, zeros)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(U112(X1, X2, X3)) → a__U112(mark(X1), X2, X3)
mark(U113(X1, X2, X3)) → a__U113(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U111(X1, X2, X3)) → a__U111(mark(X1), X2, X3)
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U114(X1, X2)) → a__U114(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U132(X1, X2, X3, X4)) → a__U132(mark(X1), X2, X3, X4)
mark(U131(X1, X2, X3, X4)) → a__U131(mark(X1), X2, X3, X4)
mark(U122(X)) → a__U122(mark(X))
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U136(X1, X2, X3, X4)) → a__U136(mark(X1), X2, X3, X4)
mark(U135(X1, X2, X3, X4)) → a__U135(mark(X1), X2, X3, X4)
mark(U134(X1, X2, X3, X4)) → a__U134(mark(X1), X2, X3, X4)
mark(U133(X1, X2, X3, X4)) → a__U133(mark(X1), X2, X3, X4)
mark(U23(X)) → a__U23(mark(X))
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U33(X)) → a__U33(mark(X))
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
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(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(0) → tt
a__isNatList(nil) → tt
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
a__length(nil) → 0
a__take(s(M), cons(N, IL)) → a__U131(a__isNatIList(IL), IL, M, N)
a__take(0, IL) → a__U121(a__isNatIList(IL), IL)
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(zeros) → a__zeros
mark(isNatKind(X)) → a__isNatKind(X)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U122(X) → U122(X)
a__U121(X1, X2) → U121(X1, X2)
a__isNatList(X) → isNatList(X)
a__U13(X) → U13(X)
a__length(X) → length(X)
a__U114(X1, X2) → U114(X1, X2)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__U11(X1, X2) → U11(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U95(X1, X2)) → a__U95(mark(X1), X2)
mark(U94(X1, X2, X3)) → a__U94(mark(X1), X2, X3)
mark(U71(X)) → a__U71(mark(X))
mark(U62(X)) → a__U62(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U81(X)) → a__U81(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
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__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)
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
mark(U96(X)) → a__U96(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
a__U95(X1, X2) → U95(X1, X2)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U96(X) → U96(X)
a__U61(X1, X2) → U61(X1, X2)
a__U52(X) → U52(X)
a__U71(X) → U71(X)
a__U62(X) → U62(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U81(X) → U81(X)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

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

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

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.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                      ↳ QDPOrderProof

                    ↳ QDP

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

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

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.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U135(X1, X2, X3, X4)) → A__U135(mark(X1), X2, X3, X4)
MARK(cons(X1, X2)) → MARK(X1)
MARK(U135(X1, X2, X3, X4)) → MARK(X1)
MARK(U136(X1, X2, X3, X4)) → A__U136(mark(X1), X2, X3, X4)
MARK(take(X1, X2)) → MARK(X1)
MARK(U121(X1, X2)) → MARK(X1)
MARK(U136(X1, X2, X3, X4)) → MARK(X1)
A__U132(tt, IL, M, N) → A__U133(a__isNat(M), IL, M, N)
MARK(U133(X1, X2, X3, X4)) → A__U133(mark(X1), X2, X3, X4)
MARK(U122(X)) → MARK(X)
MARK(U132(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(U134(X1, X2, X3, X4)) → A__U134(mark(X1), X2, X3, X4)
MARK(U133(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
MARK(U131(X1, X2, X3, X4)) → MARK(X1)
MARK(U134(X1, X2, X3, X4)) → MARK(X1)

The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → MARK(X1)
MARK(U132(X1, X2, X3, X4)) → A__U132(mark(X1), X2, X3, X4)
MARK(U45(X1, X2)) → MARK(X1)
A__TAKE(s(M), cons(N, IL)) → A__U131(a__isNatIList(IL), IL, M, N)
MARK(U104(X1, X2, X3)) → MARK(X1)
MARK(U131(X1, X2, X3, X4)) → A__U131(mark(X1), X2, X3, X4)
MARK(U71(X)) → MARK(X)
A__U131(tt, IL, M, N) → A__U132(a__isNatIListKind(IL), IL, M, N)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U94(X1, X2, X3)) → MARK(X1)
MARK(U96(X)) → MARK(X)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U41(X1, X2, X3)) → MARK(X1)
A__U136(tt, IL, M, N) → MARK(N)
A__U133(tt, IL, M, N) → A__U134(a__isNatKind(M), IL, M, N)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)
A__U134(tt, IL, M, N) → A__U135(a__isNat(N), IL, M, N)
MARK(U105(X1, X2)) → MARK(X1)
MARK(U106(X)) → MARK(X)
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U92(X1, X2, X3)) → MARK(X1)
MARK(U91(X1, X2, X3)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U81(X)) → MARK(X)
MARK(U44(X1, X2, X3)) → MARK(X1)
MARK(U103(X1, X2, X3)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U13(X)) → MARK(X)
MARK(U43(X1, X2, X3)) → MARK(X1)
MARK(U32(X1, X2)) → MARK(X1)
A__U135(tt, IL, M, N) → A__U136(a__isNatKind(N), IL, M, N)
MARK(U101(X1, X2, X3)) → MARK(X1)
MARK(U46(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U93(X1, X2, X3)) → MARK(X1)
MARK(U102(X1, X2, X3)) → MARK(X1)
MARK(U33(X)) → MARK(X)
MARK(U23(X)) → MARK(X)
MARK(U95(X1, X2)) → MARK(X1)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__TAKE(x₁, x₂)) = x₂
POL(A__U131(x₁, x₂, x₃, x₄)) = 1 + x₄
POL(A__U132(x₁, x₂, x₃, x₄)) = 1 + x₄
POL(A__U133(x₁, x₂, x₃, x₄)) = x₄
POL(A__U134(x₁, x₂, x₃, x₄)) = x₄
POL(A__U135(x₁, x₂, x₃, x₄)) = x₄
POL(A__U136(x₁, x₂, x₃, x₄)) = x₄
POL(MARK(x₁)) = x₁
POL(U101(x₁, x₂, x₃)) = x₁
POL(U102(x₁, x₂, x₃)) = x₁
POL(U103(x₁, x₂, x₃)) = x₁
POL(U104(x₁, x₂, x₃)) = x₁
POL(U105(x₁, x₂)) = x₁
POL(U106(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁
POL(U111(x₁, x₂, x₃)) = 0
POL(U112(x₁, x₂, x₃)) = 0
POL(U113(x₁, x₂, x₃)) = 0
POL(U114(x₁, x₂)) = 0
POL(U12(x₁, x₂)) = x₁
POL(U121(x₁, x₂)) = 1 + x₁
POL(U122(x₁)) = 1 + x₁
POL(U13(x₁)) = x₁
POL(U131(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₄
POL(U132(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₄
POL(U133(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₄
POL(U134(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₄
POL(U135(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₄
POL(U136(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₄
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁, x₂)) = x₁
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁, x₂)) = x₁
POL(U33(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = x₁
POL(U42(x₁, x₂, x₃)) = x₁
POL(U43(x₁, x₂, x₃)) = x₁
POL(U44(x₁, x₂, x₃)) = x₁
POL(U45(x₁, x₂)) = x₁
POL(U46(x₁)) = x₁
POL(U51(x₁, x₂)) = x₁
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂)) = x₁
POL(U62(x₁)) = x₁
POL(U71(x₁)) = x₁
POL(U81(x₁)) = x₁
POL(U91(x₁, x₂, x₃)) = x₁
POL(U92(x₁, x₂, x₃)) = x₁
POL(U93(x₁, x₂, x₃)) = x₁
POL(U94(x₁, x₂, x₃)) = x₁
POL(U95(x₁, x₂)) = x₁
POL(U96(x₁)) = x₁
POL(a__U101(x₁, x₂, x₃)) = x₁
POL(a__U102(x₁, x₂, x₃)) = x₁
POL(a__U103(x₁, x₂, x₃)) = x₁
POL(a__U104(x₁, x₂, x₃)) = x₁
POL(a__U105(x₁, x₂)) = x₁
POL(a__U106(x₁)) = x₁
POL(a__U11(x₁, x₂)) = x₁
POL(a__U111(x₁, x₂, x₃)) = 0
POL(a__U112(x₁, x₂, x₃)) = 0
POL(a__U113(x₁, x₂, x₃)) = 0
POL(a__U114(x₁, x₂)) = 0
POL(a__U12(x₁, x₂)) = x₁
POL(a__U121(x₁, x₂)) = 1 + x₁
POL(a__U122(x₁)) = 1 + x₁
POL(a__U13(x₁)) = x₁
POL(a__U131(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₄
POL(a__U132(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₄
POL(a__U133(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₄
POL(a__U134(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₄
POL(a__U135(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₄
POL(a__U136(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₄
POL(a__U21(x₁, x₂)) = x₁
POL(a__U22(x₁, x₂)) = x₁
POL(a__U23(x₁)) = x₁
POL(a__U31(x₁, x₂)) = x₁
POL(a__U32(x₁, x₂)) = x₁
POL(a__U33(x₁)) = x₁
POL(a__U41(x₁, x₂, x₃)) = x₁
POL(a__U42(x₁, x₂, x₃)) = x₁
POL(a__U43(x₁, x₂, x₃)) = x₁
POL(a__U44(x₁, x₂, x₃)) = x₁
POL(a__U45(x₁, x₂)) = x₁
POL(a__U46(x₁)) = x₁
POL(a__U51(x₁, x₂)) = x₁
POL(a__U52(x₁)) = x₁
POL(a__U61(x₁, x₂)) = x₁
POL(a__U62(x₁)) = x₁
POL(a__U71(x₁)) = x₁
POL(a__U81(x₁)) = x₁
POL(a__U91(x₁, x₂, x₃)) = x₁
POL(a__U92(x₁, x₂, x₃)) = x₁
POL(a__U93(x₁, x₂, x₃)) = x₁
POL(a__U94(x₁, x₂, x₃)) = x₁
POL(a__U95(x₁, x₂)) = x₁
POL(a__U96(x₁)) = x₁
POL(a__isNat(x₁)) = 0
POL(a__isNatIList(x₁)) = 0
POL(a__isNatIListKind(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__isNatList(x₁)) = 0
POL(a__length(x₁)) = 0
POL(a__take(x₁, x₂)) = 1 + x₁ + x₂
POL(a__zeros) = 1
POL(cons(x₁, x₂)) = 1 + x₁
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 1 + x₁ + x₂
POL(tt) = 0
POL(zeros) = 1

The following usable rules [17] were oriented:

a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U113(tt, L, N) → a__U114(a__isNatKind(N), L)
a__U114(tt, L) → s(a__length(mark(L)))
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__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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__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__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__U122(tt) → nil
a__U13(tt) → tt
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U33(tt) → tt
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U23(tt) → tt
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U62(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U46(tt) → tt
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U103(tt, V1, V2) → a__U104(a__isNatIListKind(V2), V1, V2)
a__U102(tt, V1, V2) → a__U103(a__isNatIListKind(V2), V1, V2)
a__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U104(tt, V1, V2) → a__U105(a__isNat(V1), V2)
a__U101(tt, V1, V2) → a__U102(a__isNatKind(V1), V1, V2)
a__zeros → cons(0, zeros)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(U112(X1, X2, X3)) → a__U112(mark(X1), X2, X3)
mark(U113(X1, X2, X3)) → a__U113(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U111(X1, X2, X3)) → a__U111(mark(X1), X2, X3)
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U114(X1, X2)) → a__U114(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U132(X1, X2, X3, X4)) → a__U132(mark(X1), X2, X3, X4)
mark(U131(X1, X2, X3, X4)) → a__U131(mark(X1), X2, X3, X4)
mark(U122(X)) → a__U122(mark(X))
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U136(X1, X2, X3, X4)) → a__U136(mark(X1), X2, X3, X4)
mark(U135(X1, X2, X3, X4)) → a__U135(mark(X1), X2, X3, X4)
mark(U134(X1, X2, X3, X4)) → a__U134(mark(X1), X2, X3, X4)
mark(U133(X1, X2, X3, X4)) → a__U133(mark(X1), X2, X3, X4)
mark(U23(X)) → a__U23(mark(X))
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U33(X)) → a__U33(mark(X))
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
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(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(0) → tt
a__isNatList(nil) → tt
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
a__length(nil) → 0
a__take(s(M), cons(N, IL)) → a__U131(a__isNatIList(IL), IL, M, N)
a__take(0, IL) → a__U121(a__isNatIList(IL), IL)
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(zeros) → a__zeros
mark(isNatKind(X)) → a__isNatKind(X)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U122(X) → U122(X)
a__U121(X1, X2) → U121(X1, X2)
a__isNatList(X) → isNatList(X)
a__U13(X) → U13(X)
a__length(X) → length(X)
a__U114(X1, X2) → U114(X1, X2)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__U11(X1, X2) → U11(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U95(X1, X2)) → a__U95(mark(X1), X2)
mark(U94(X1, X2, X3)) → a__U94(mark(X1), X2, X3)
mark(U71(X)) → a__U71(mark(X))
mark(U62(X)) → a__U62(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U81(X)) → a__U81(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
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__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)
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
mark(U96(X)) → a__U96(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
a__U95(X1, X2) → U95(X1, X2)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U96(X) → U96(X)
a__U61(X1, X2) → U61(X1, X2)
a__U52(X) → U52(X)
a__U71(X) → U71(X)
a__U62(X) → U62(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U81(X) → U81(X)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                    ↳ QDP

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

MARK(U11(X1, X2)) → MARK(X1)
MARK(U62(X)) → MARK(X)
MARK(U45(X1, X2)) → MARK(X1)
MARK(U132(X1, X2, X3, X4)) → A__U132(mark(X1), X2, X3, X4)
MARK(s(X)) → MARK(X)
MARK(U92(X1, X2, X3)) → MARK(X1)
MARK(U91(X1, X2, X3)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
A__TAKE(s(M), cons(N, IL)) → A__U131(a__isNatIList(IL), IL, M, N)
MARK(U81(X)) → MARK(X)
MARK(U44(X1, X2, X3)) → MARK(X1)
MARK(U104(X1, X2, X3)) → MARK(X1)
MARK(U131(X1, X2, X3, X4)) → A__U131(mark(X1), X2, X3, X4)
MARK(U52(X)) → MARK(X)
MARK(U103(X1, X2, X3)) → MARK(X1)
MARK(U13(X)) → MARK(X)
MARK(U71(X)) → MARK(X)
A__U131(tt, IL, M, N) → A__U132(a__isNatIListKind(IL), IL, M, N)
MARK(U43(X1, X2, X3)) → MARK(X1)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U94(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U96(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → MARK(X1)
A__U135(tt, IL, M, N) → A__U136(a__isNatKind(N), IL, M, N)
A__U136(tt, IL, M, N) → MARK(N)
MARK(U101(X1, X2, X3)) → MARK(X1)
MARK(U46(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
A__U133(tt, IL, M, N) → A__U134(a__isNatKind(M), IL, M, N)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(U102(X1, X2, X3)) → MARK(X1)
MARK(U93(X1, X2, X3)) → MARK(X1)
MARK(U33(X)) → MARK(X)
MARK(U23(X)) → MARK(X)
MARK(U95(X1, X2)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)
A__U134(tt, IL, M, N) → A__U135(a__isNat(N), IL, M, N)
MARK(U105(X1, X2)) → MARK(X1)
MARK(U106(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.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 8 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ UsableRulesProof

                    ↳ QDP

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

MARK(U11(X1, X2)) → MARK(X1)
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U45(X1, X2)) → MARK(X1)
MARK(U92(X1, X2, X3)) → MARK(X1)
MARK(U91(X1, X2, X3)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U81(X)) → MARK(X)
MARK(U44(X1, X2, X3)) → MARK(X1)
MARK(U104(X1, X2, X3)) → MARK(X1)
MARK(U103(X1, X2, X3)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U13(X)) → MARK(X)
MARK(U71(X)) → MARK(X)
MARK(U43(X1, X2, X3)) → MARK(X1)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U94(X1, X2, X3)) → MARK(X1)
MARK(U96(X)) → MARK(X)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U101(X1, X2, X3)) → MARK(X1)
MARK(U46(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(U93(X1, X2, X3)) → MARK(X1)
MARK(U102(X1, X2, X3)) → MARK(X1)
MARK(U33(X)) → MARK(X)
MARK(U23(X)) → MARK(X)
MARK(U95(X1, X2)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U106(X)) → MARK(X)
MARK(U105(X1, X2)) → MARK(X1)

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)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ UsableRulesProof

                                ↳ QDP

                                  ↳ QDPSizeChangeProof

                    ↳ QDP

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

MARK(U62(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U45(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(U92(X1, X2, X3)) → MARK(X1)
MARK(U91(X1, X2, X3)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U81(X)) → MARK(X)
MARK(U44(X1, X2, X3)) → MARK(X1)
MARK(U104(X1, X2, X3)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U103(X1, X2, X3)) → MARK(X1)
MARK(U13(X)) → MARK(X)
MARK(U71(X)) → MARK(X)
MARK(U43(X1, X2, X3)) → MARK(X1)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U94(X1, X2, X3)) → MARK(X1)
MARK(U96(X)) → MARK(X)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U101(X1, X2, X3)) → MARK(X1)
MARK(U46(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(U102(X1, X2, X3)) → MARK(X1)
MARK(U93(X1, X2, X3)) → MARK(X1)
MARK(U33(X)) → MARK(X)
MARK(U23(X)) → MARK(X)
MARK(U95(X1, X2)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U105(X1, X2)) → MARK(X1)
MARK(U106(X)) → MARK(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

MARK(U11(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U62(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(s(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(U45(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U92(X1, X2, X3)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U91(X1, X2, X3)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U31(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U81(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(U44(X1, X2, X3)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U104(X1, X2, X3)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U103(X1, X2, X3)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U52(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(U13(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(U71(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(U43(X1, X2, X3)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U51(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U32(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U22(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U94(X1, X2, X3)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U96(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(U12(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U41(X1, X2, X3)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U101(X1, X2, X3)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U46(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(U61(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U42(X1, X2, X3)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U93(X1, X2, X3)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U102(X1, X2, X3)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U33(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(U23(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(U95(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U21(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U106(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(U105(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ Narrowing

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

A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__U113(tt, L, N) → A__U114(a__isNatKind(N), L)
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
A__U114(tt, L) → A__LENGTH(mark(L))

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.
By narrowing [15] the rule A__U114(tt, L) → A__LENGTH(mark(L)) at position [0] we obtained the following new rules:

A__U114(tt, U46(x0)) → A__LENGTH(a__U46(mark(x0)))
A__U114(tt, U102(x0, x1, x2)) → A__LENGTH(a__U102(mark(x0), x1, x2))
A__U114(tt, tt) → A__LENGTH(tt)
A__U114(tt, U111(x0, x1, x2)) → A__LENGTH(a__U111(mark(x0), x1, x2))
A__U114(tt, U133(x0, x1, x2, x3)) → A__LENGTH(a__U133(mark(x0), x1, x2, x3))
A__U114(tt, nil) → A__LENGTH(nil)
A__U114(tt, U21(x0, x1)) → A__LENGTH(a__U21(mark(x0), x1))
A__U114(tt, U61(x0, x1)) → A__LENGTH(a__U61(mark(x0), x1))
A__U114(tt, U106(x0)) → A__LENGTH(a__U106(mark(x0)))
A__U114(tt, U134(x0, x1, x2, x3)) → A__LENGTH(a__U134(mark(x0), x1, x2, x3))
A__U114(tt, U101(x0, x1, x2)) → A__LENGTH(a__U101(mark(x0), x1, x2))
A__U114(tt, U105(x0, x1)) → A__LENGTH(a__U105(mark(x0), x1))
A__U114(tt, U96(x0)) → A__LENGTH(a__U96(mark(x0)))
A__U114(tt, s(x0)) → A__LENGTH(s(mark(x0)))
A__U114(tt, U13(x0)) → A__LENGTH(a__U13(mark(x0)))
A__U114(tt, U131(x0, x1, x2, x3)) → A__LENGTH(a__U131(mark(x0), x1, x2, x3))
A__U114(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U114(tt, U136(x0, x1, x2, x3)) → A__LENGTH(a__U136(mark(x0), x1, x2, x3))
A__U114(tt, U23(x0)) → A__LENGTH(a__U23(mark(x0)))
A__U114(tt, U22(x0, x1)) → A__LENGTH(a__U22(mark(x0), x1))
A__U114(tt, U103(x0, x1, x2)) → A__LENGTH(a__U103(mark(x0), x1, x2))
A__U114(tt, U132(x0, x1, x2, x3)) → A__LENGTH(a__U132(mark(x0), x1, x2, x3))
A__U114(tt, U95(x0, x1)) → A__LENGTH(a__U95(mark(x0), x1))
A__U114(tt, U104(x0, x1, x2)) → A__LENGTH(a__U104(mark(x0), x1, x2))
A__U114(tt, U112(x0, x1, x2)) → A__LENGTH(a__U112(mark(x0), x1, x2))
A__U114(tt, U32(x0, x1)) → A__LENGTH(a__U32(mark(x0), x1))
A__U114(tt, U44(x0, x1, x2)) → A__LENGTH(a__U44(mark(x0), x1, x2))
A__U114(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U114(tt, U93(x0, x1, x2)) → A__LENGTH(a__U93(mark(x0), x1, x2))
A__U114(tt, zeros) → A__LENGTH(a__zeros)
A__U114(tt, U114(x0, x1)) → A__LENGTH(a__U114(mark(x0), x1))
A__U114(tt, U42(x0, x1, x2)) → A__LENGTH(a__U42(mark(x0), x1, x2))
A__U114(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U114(tt, U62(x0)) → A__LENGTH(a__U62(mark(x0)))
A__U114(tt, U113(x0, x1, x2)) → A__LENGTH(a__U113(mark(x0), x1, x2))
A__U114(tt, U121(x0, x1)) → A__LENGTH(a__U121(mark(x0), x1))
A__U114(tt, U31(x0, x1)) → A__LENGTH(a__U31(mark(x0), x1))
A__U114(tt, U33(x0)) → A__LENGTH(a__U33(mark(x0)))
A__U114(tt, U12(x0, x1)) → A__LENGTH(a__U12(mark(x0), x1))
A__U114(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__U114(tt, U43(x0, x1, x2)) → A__LENGTH(a__U43(mark(x0), x1, x2))
A__U114(tt, U92(x0, x1, x2)) → A__LENGTH(a__U92(mark(x0), x1, x2))
A__U114(tt, U52(x0)) → A__LENGTH(a__U52(mark(x0)))
A__U114(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U114(tt, U94(x0, x1, x2)) → A__LENGTH(a__U94(mark(x0), x1, x2))
A__U114(tt, U71(x0)) → A__LENGTH(a__U71(mark(x0)))
A__U114(tt, isNatKind(x0)) → A__LENGTH(a__isNatKind(x0))
A__U114(tt, U122(x0)) → A__LENGTH(a__U122(mark(x0)))
A__U114(tt, U51(x0, x1)) → A__LENGTH(a__U51(mark(x0), x1))
A__U114(tt, 0) → A__LENGTH(0)
A__U114(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))
A__U114(tt, U45(x0, x1)) → A__LENGTH(a__U45(mark(x0), x1))
A__U114(tt, U81(x0)) → A__LENGTH(a__U81(mark(x0)))
A__U114(tt, U135(x0, x1, x2, x3)) → A__LENGTH(a__U135(mark(x0), x1, x2, x3))
A__U114(tt, U91(x0, x1, x2)) → A__LENGTH(a__U91(mark(x0), x1, x2))
A__U114(tt, isNatIListKind(x0)) → A__LENGTH(a__isNatIListKind(x0))
A__U114(tt, U41(x0, x1, x2)) → A__LENGTH(a__U41(mark(x0), x1, x2))
A__U114(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

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

A__U114(tt, U46(x0)) → A__LENGTH(a__U46(mark(x0)))
A__U114(tt, U102(x0, x1, x2)) → A__LENGTH(a__U102(mark(x0), x1, x2))
A__U114(tt, tt) → A__LENGTH(tt)
A__U114(tt, U111(x0, x1, x2)) → A__LENGTH(a__U111(mark(x0), x1, x2))
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__U114(tt, U133(x0, x1, x2, x3)) → A__LENGTH(a__U133(mark(x0), x1, x2, x3))
A__U114(tt, nil) → A__LENGTH(nil)
A__U114(tt, U21(x0, x1)) → A__LENGTH(a__U21(mark(x0), x1))
A__U114(tt, U61(x0, x1)) → A__LENGTH(a__U61(mark(x0), x1))
A__U114(tt, U106(x0)) → A__LENGTH(a__U106(mark(x0)))
A__U114(tt, U134(x0, x1, x2, x3)) → A__LENGTH(a__U134(mark(x0), x1, x2, x3))
A__U114(tt, U101(x0, x1, x2)) → A__LENGTH(a__U101(mark(x0), x1, x2))
A__U114(tt, U105(x0, x1)) → A__LENGTH(a__U105(mark(x0), x1))
A__U114(tt, U96(x0)) → A__LENGTH(a__U96(mark(x0)))
A__U114(tt, s(x0)) → A__LENGTH(s(mark(x0)))
A__U114(tt, U13(x0)) → A__LENGTH(a__U13(mark(x0)))
A__U114(tt, U131(x0, x1, x2, x3)) → A__LENGTH(a__U131(mark(x0), x1, x2, x3))
A__U114(tt, U136(x0, x1, x2, x3)) → A__LENGTH(a__U136(mark(x0), x1, x2, x3))
A__U114(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U114(tt, U23(x0)) → A__LENGTH(a__U23(mark(x0)))
A__U114(tt, U22(x0, x1)) → A__LENGTH(a__U22(mark(x0), x1))
A__U114(tt, U103(x0, x1, x2)) → A__LENGTH(a__U103(mark(x0), x1, x2))
A__U114(tt, U132(x0, x1, x2, x3)) → A__LENGTH(a__U132(mark(x0), x1, x2, x3))
A__U114(tt, U95(x0, x1)) → A__LENGTH(a__U95(mark(x0), x1))
A__U114(tt, U104(x0, x1, x2)) → A__LENGTH(a__U104(mark(x0), x1, x2))
A__U114(tt, U112(x0, x1, x2)) → A__LENGTH(a__U112(mark(x0), x1, x2))
A__U114(tt, U32(x0, x1)) → A__LENGTH(a__U32(mark(x0), x1))
A__U114(tt, U44(x0, x1, x2)) → A__LENGTH(a__U44(mark(x0), x1, x2))
A__U114(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U114(tt, U93(x0, x1, x2)) → A__LENGTH(a__U93(mark(x0), x1, x2))
A__U114(tt, zeros) → A__LENGTH(a__zeros)
A__U114(tt, U114(x0, x1)) → A__LENGTH(a__U114(mark(x0), x1))
A__U114(tt, U42(x0, x1, x2)) → A__LENGTH(a__U42(mark(x0), x1, x2))
A__U114(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U114(tt, U62(x0)) → A__LENGTH(a__U62(mark(x0)))
A__U114(tt, U113(x0, x1, x2)) → A__LENGTH(a__U113(mark(x0), x1, x2))
A__U113(tt, L, N) → A__U114(a__isNatKind(N), L)
A__U114(tt, U121(x0, x1)) → A__LENGTH(a__U121(mark(x0), x1))
A__U114(tt, U31(x0, x1)) → A__LENGTH(a__U31(mark(x0), x1))
A__U114(tt, U33(x0)) → A__LENGTH(a__U33(mark(x0)))
A__U114(tt, U12(x0, x1)) → A__LENGTH(a__U12(mark(x0), x1))
A__U114(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__U114(tt, U43(x0, x1, x2)) → A__LENGTH(a__U43(mark(x0), x1, x2))
A__U114(tt, U92(x0, x1, x2)) → A__LENGTH(a__U92(mark(x0), x1, x2))
A__U114(tt, U52(x0)) → A__LENGTH(a__U52(mark(x0)))
A__U114(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U114(tt, U94(x0, x1, x2)) → A__LENGTH(a__U94(mark(x0), x1, x2))
A__U114(tt, U71(x0)) → A__LENGTH(a__U71(mark(x0)))
A__U114(tt, isNatKind(x0)) → A__LENGTH(a__isNatKind(x0))
A__U114(tt, U122(x0)) → A__LENGTH(a__U122(mark(x0)))
A__U114(tt, U51(x0, x1)) → A__LENGTH(a__U51(mark(x0), x1))
A__U114(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))
A__U114(tt, 0) → A__LENGTH(0)
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
A__U114(tt, U81(x0)) → A__LENGTH(a__U81(mark(x0)))
A__U114(tt, U45(x0, x1)) → A__LENGTH(a__U45(mark(x0), x1))
A__U114(tt, U135(x0, x1, x2, x3)) → A__LENGTH(a__U135(mark(x0), x1, x2, x3))
A__U114(tt, U91(x0, x1, x2)) → A__LENGTH(a__U91(mark(x0), x1, x2))
A__U114(tt, isNatIListKind(x0)) → A__LENGTH(a__isNatIListKind(x0))
A__U114(tt, U41(x0, x1, x2)) → A__LENGTH(a__U41(mark(x0), x1, x2))
A__U114(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)

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.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 4 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

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

A__U114(tt, U46(x0)) → A__LENGTH(a__U46(mark(x0)))
A__U114(tt, U102(x0, x1, x2)) → A__LENGTH(a__U102(mark(x0), x1, x2))
A__U114(tt, U111(x0, x1, x2)) → A__LENGTH(a__U111(mark(x0), x1, x2))
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__U114(tt, U133(x0, x1, x2, x3)) → A__LENGTH(a__U133(mark(x0), x1, x2, x3))
A__U114(tt, U21(x0, x1)) → A__LENGTH(a__U21(mark(x0), x1))
A__U114(tt, U61(x0, x1)) → A__LENGTH(a__U61(mark(x0), x1))
A__U114(tt, U106(x0)) → A__LENGTH(a__U106(mark(x0)))
A__U114(tt, U134(x0, x1, x2, x3)) → A__LENGTH(a__U134(mark(x0), x1, x2, x3))
A__U114(tt, U101(x0, x1, x2)) → A__LENGTH(a__U101(mark(x0), x1, x2))
A__U114(tt, U105(x0, x1)) → A__LENGTH(a__U105(mark(x0), x1))
A__U114(tt, U96(x0)) → A__LENGTH(a__U96(mark(x0)))
A__U114(tt, U13(x0)) → A__LENGTH(a__U13(mark(x0)))
A__U114(tt, U131(x0, x1, x2, x3)) → A__LENGTH(a__U131(mark(x0), x1, x2, x3))
A__U114(tt, U136(x0, x1, x2, x3)) → A__LENGTH(a__U136(mark(x0), x1, x2, x3))
A__U114(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U114(tt, U23(x0)) → A__LENGTH(a__U23(mark(x0)))
A__U114(tt, U22(x0, x1)) → A__LENGTH(a__U22(mark(x0), x1))
A__U114(tt, U103(x0, x1, x2)) → A__LENGTH(a__U103(mark(x0), x1, x2))
A__U114(tt, U132(x0, x1, x2, x3)) → A__LENGTH(a__U132(mark(x0), x1, x2, x3))
A__U114(tt, U104(x0, x1, x2)) → A__LENGTH(a__U104(mark(x0), x1, x2))
A__U114(tt, U95(x0, x1)) → A__LENGTH(a__U95(mark(x0), x1))
A__U114(tt, U112(x0, x1, x2)) → A__LENGTH(a__U112(mark(x0), x1, x2))
A__U114(tt, U32(x0, x1)) → A__LENGTH(a__U32(mark(x0), x1))
A__U114(tt, U44(x0, x1, x2)) → A__LENGTH(a__U44(mark(x0), x1, x2))
A__U114(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U114(tt, U93(x0, x1, x2)) → A__LENGTH(a__U93(mark(x0), x1, x2))
A__U114(tt, zeros) → A__LENGTH(a__zeros)
A__U114(tt, U114(x0, x1)) → A__LENGTH(a__U114(mark(x0), x1))
A__U114(tt, U42(x0, x1, x2)) → A__LENGTH(a__U42(mark(x0), x1, x2))
A__U114(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U114(tt, U62(x0)) → A__LENGTH(a__U62(mark(x0)))
A__U114(tt, U113(x0, x1, x2)) → A__LENGTH(a__U113(mark(x0), x1, x2))
A__U113(tt, L, N) → A__U114(a__isNatKind(N), L)
A__U114(tt, U121(x0, x1)) → A__LENGTH(a__U121(mark(x0), x1))
A__U114(tt, U31(x0, x1)) → A__LENGTH(a__U31(mark(x0), x1))
A__U114(tt, U33(x0)) → A__LENGTH(a__U33(mark(x0)))
A__U114(tt, U12(x0, x1)) → A__LENGTH(a__U12(mark(x0), x1))
A__U114(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__U114(tt, U43(x0, x1, x2)) → A__LENGTH(a__U43(mark(x0), x1, x2))
A__U114(tt, U92(x0, x1, x2)) → A__LENGTH(a__U92(mark(x0), x1, x2))
A__U114(tt, U52(x0)) → A__LENGTH(a__U52(mark(x0)))
A__U114(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U114(tt, U94(x0, x1, x2)) → A__LENGTH(a__U94(mark(x0), x1, x2))
A__U114(tt, U71(x0)) → A__LENGTH(a__U71(mark(x0)))
A__U114(tt, isNatKind(x0)) → A__LENGTH(a__isNatKind(x0))
A__U114(tt, U122(x0)) → A__LENGTH(a__U122(mark(x0)))
A__U114(tt, U51(x0, x1)) → A__LENGTH(a__U51(mark(x0), x1))
A__U114(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
A__U114(tt, U81(x0)) → A__LENGTH(a__U81(mark(x0)))
A__U114(tt, U45(x0, x1)) → A__LENGTH(a__U45(mark(x0), x1))
A__U114(tt, U135(x0, x1, x2, x3)) → A__LENGTH(a__U135(mark(x0), x1, x2, x3))
A__U114(tt, U91(x0, x1, x2)) → A__LENGTH(a__U91(mark(x0), x1, x2))
A__U114(tt, isNatIListKind(x0)) → A__LENGTH(a__isNatIListKind(x0))
A__U114(tt, U41(x0, x1, x2)) → A__LENGTH(a__U41(mark(x0), x1, x2))
A__U114(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)

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.
By narrowing [15] the rule A__U113(tt, L, N) → A__U114(a__isNatKind(N), L) at position [0] we obtained the following new rules:

A__U113(tt, y0, x0) → A__U114(isNatKind(x0), y0)
A__U113(tt, y0, length(x0)) → A__U114(a__U71(a__isNatIListKind(x0)), y0)
A__U113(tt, y0, 0) → A__U114(tt, y0)
A__U113(tt, y0, s(x0)) → A__U114(a__U81(a__isNatKind(x0)), y0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

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

A__U114(tt, U46(x0)) → A__LENGTH(a__U46(mark(x0)))
A__U114(tt, U102(x0, x1, x2)) → A__LENGTH(a__U102(mark(x0), x1, x2))
A__U114(tt, U111(x0, x1, x2)) → A__LENGTH(a__U111(mark(x0), x1, x2))
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__U114(tt, U133(x0, x1, x2, x3)) → A__LENGTH(a__U133(mark(x0), x1, x2, x3))
A__U114(tt, U21(x0, x1)) → A__LENGTH(a__U21(mark(x0), x1))
A__U114(tt, U61(x0, x1)) → A__LENGTH(a__U61(mark(x0), x1))
A__U114(tt, U106(x0)) → A__LENGTH(a__U106(mark(x0)))
A__U114(tt, U134(x0, x1, x2, x3)) → A__LENGTH(a__U134(mark(x0), x1, x2, x3))
A__U114(tt, U101(x0, x1, x2)) → A__LENGTH(a__U101(mark(x0), x1, x2))
A__U114(tt, U105(x0, x1)) → A__LENGTH(a__U105(mark(x0), x1))
A__U114(tt, U96(x0)) → A__LENGTH(a__U96(mark(x0)))
A__U114(tt, U13(x0)) → A__LENGTH(a__U13(mark(x0)))
A__U114(tt, U131(x0, x1, x2, x3)) → A__LENGTH(a__U131(mark(x0), x1, x2, x3))
A__U114(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U114(tt, U136(x0, x1, x2, x3)) → A__LENGTH(a__U136(mark(x0), x1, x2, x3))
A__U114(tt, U23(x0)) → A__LENGTH(a__U23(mark(x0)))
A__U114(tt, U22(x0, x1)) → A__LENGTH(a__U22(mark(x0), x1))
A__U114(tt, U103(x0, x1, x2)) → A__LENGTH(a__U103(mark(x0), x1, x2))
A__U113(tt, y0, s(x0)) → A__U114(a__U81(a__isNatKind(x0)), y0)
A__U113(tt, y0, 0) → A__U114(tt, y0)
A__U114(tt, U132(x0, x1, x2, x3)) → A__LENGTH(a__U132(mark(x0), x1, x2, x3))
A__U114(tt, U95(x0, x1)) → A__LENGTH(a__U95(mark(x0), x1))
A__U114(tt, U104(x0, x1, x2)) → A__LENGTH(a__U104(mark(x0), x1, x2))
A__U114(tt, U112(x0, x1, x2)) → A__LENGTH(a__U112(mark(x0), x1, x2))
A__U114(tt, U32(x0, x1)) → A__LENGTH(a__U32(mark(x0), x1))
A__U114(tt, U44(x0, x1, x2)) → A__LENGTH(a__U44(mark(x0), x1, x2))
A__U114(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U114(tt, zeros) → A__LENGTH(a__zeros)
A__U114(tt, U93(x0, x1, x2)) → A__LENGTH(a__U93(mark(x0), x1, x2))
A__U114(tt, U42(x0, x1, x2)) → A__LENGTH(a__U42(mark(x0), x1, x2))
A__U114(tt, U114(x0, x1)) → A__LENGTH(a__U114(mark(x0), x1))
A__U114(tt, U62(x0)) → A__LENGTH(a__U62(mark(x0)))
A__U114(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U114(tt, U113(x0, x1, x2)) → A__LENGTH(a__U113(mark(x0), x1, x2))
A__U113(tt, y0, length(x0)) → A__U114(a__U71(a__isNatIListKind(x0)), y0)
A__U114(tt, U121(x0, x1)) → A__LENGTH(a__U121(mark(x0), x1))
A__U113(tt, y0, x0) → A__U114(isNatKind(x0), y0)
A__U114(tt, U33(x0)) → A__LENGTH(a__U33(mark(x0)))
A__U114(tt, U31(x0, x1)) → A__LENGTH(a__U31(mark(x0), x1))
A__U114(tt, U12(x0, x1)) → A__LENGTH(a__U12(mark(x0), x1))
A__U114(tt, U92(x0, x1, x2)) → A__LENGTH(a__U92(mark(x0), x1, x2))
A__U114(tt, U43(x0, x1, x2)) → A__LENGTH(a__U43(mark(x0), x1, x2))
A__U114(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__U114(tt, U52(x0)) → A__LENGTH(a__U52(mark(x0)))
A__U114(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U114(tt, U94(x0, x1, x2)) → A__LENGTH(a__U94(mark(x0), x1, x2))
A__U114(tt, U71(x0)) → A__LENGTH(a__U71(mark(x0)))
A__U114(tt, isNatKind(x0)) → A__LENGTH(a__isNatKind(x0))
A__U114(tt, U122(x0)) → A__LENGTH(a__U122(mark(x0)))
A__U114(tt, U51(x0, x1)) → A__LENGTH(a__U51(mark(x0), x1))
A__U114(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
A__U114(tt, U45(x0, x1)) → A__LENGTH(a__U45(mark(x0), x1))
A__U114(tt, U81(x0)) → A__LENGTH(a__U81(mark(x0)))
A__U114(tt, U135(x0, x1, x2, x3)) → A__LENGTH(a__U135(mark(x0), x1, x2, x3))
A__U114(tt, U91(x0, x1, x2)) → A__LENGTH(a__U91(mark(x0), x1, x2))
A__U114(tt, isNatIListKind(x0)) → A__LENGTH(a__isNatIListKind(x0))
A__U114(tt, U41(x0, x1, x2)) → A__LENGTH(a__U41(mark(x0), x1, x2))
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)
A__U114(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))

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.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

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

A__U114(tt, U46(x0)) → A__LENGTH(a__U46(mark(x0)))
A__U114(tt, U102(x0, x1, x2)) → A__LENGTH(a__U102(mark(x0), x1, x2))
A__U114(tt, U111(x0, x1, x2)) → A__LENGTH(a__U111(mark(x0), x1, x2))
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__U114(tt, U133(x0, x1, x2, x3)) → A__LENGTH(a__U133(mark(x0), x1, x2, x3))
A__U114(tt, U21(x0, x1)) → A__LENGTH(a__U21(mark(x0), x1))
A__U114(tt, U61(x0, x1)) → A__LENGTH(a__U61(mark(x0), x1))
A__U114(tt, U106(x0)) → A__LENGTH(a__U106(mark(x0)))
A__U114(tt, U134(x0, x1, x2, x3)) → A__LENGTH(a__U134(mark(x0), x1, x2, x3))
A__U114(tt, U101(x0, x1, x2)) → A__LENGTH(a__U101(mark(x0), x1, x2))
A__U114(tt, U105(x0, x1)) → A__LENGTH(a__U105(mark(x0), x1))
A__U114(tt, U96(x0)) → A__LENGTH(a__U96(mark(x0)))
A__U114(tt, U13(x0)) → A__LENGTH(a__U13(mark(x0)))
A__U114(tt, U131(x0, x1, x2, x3)) → A__LENGTH(a__U131(mark(x0), x1, x2, x3))
A__U114(tt, U136(x0, x1, x2, x3)) → A__LENGTH(a__U136(mark(x0), x1, x2, x3))
A__U114(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U114(tt, U23(x0)) → A__LENGTH(a__U23(mark(x0)))
A__U114(tt, U22(x0, x1)) → A__LENGTH(a__U22(mark(x0), x1))
A__U114(tt, U103(x0, x1, x2)) → A__LENGTH(a__U103(mark(x0), x1, x2))
A__U113(tt, y0, s(x0)) → A__U114(a__U81(a__isNatKind(x0)), y0)
A__U113(tt, y0, 0) → A__U114(tt, y0)
A__U114(tt, U132(x0, x1, x2, x3)) → A__LENGTH(a__U132(mark(x0), x1, x2, x3))
A__U114(tt, U104(x0, x1, x2)) → A__LENGTH(a__U104(mark(x0), x1, x2))
A__U114(tt, U95(x0, x1)) → A__LENGTH(a__U95(mark(x0), x1))
A__U114(tt, U112(x0, x1, x2)) → A__LENGTH(a__U112(mark(x0), x1, x2))
A__U114(tt, U32(x0, x1)) → A__LENGTH(a__U32(mark(x0), x1))
A__U114(tt, U44(x0, x1, x2)) → A__LENGTH(a__U44(mark(x0), x1, x2))
A__U114(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U114(tt, U93(x0, x1, x2)) → A__LENGTH(a__U93(mark(x0), x1, x2))
A__U114(tt, zeros) → A__LENGTH(a__zeros)
A__U114(tt, U114(x0, x1)) → A__LENGTH(a__U114(mark(x0), x1))
A__U114(tt, U42(x0, x1, x2)) → A__LENGTH(a__U42(mark(x0), x1, x2))
A__U114(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U114(tt, U62(x0)) → A__LENGTH(a__U62(mark(x0)))
A__U114(tt, U113(x0, x1, x2)) → A__LENGTH(a__U113(mark(x0), x1, x2))
A__U113(tt, y0, length(x0)) → A__U114(a__U71(a__isNatIListKind(x0)), y0)
A__U114(tt, U121(x0, x1)) → A__LENGTH(a__U121(mark(x0), x1))
A__U114(tt, U31(x0, x1)) → A__LENGTH(a__U31(mark(x0), x1))
A__U114(tt, U33(x0)) → A__LENGTH(a__U33(mark(x0)))
A__U114(tt, U12(x0, x1)) → A__LENGTH(a__U12(mark(x0), x1))
A__U114(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__U114(tt, U43(x0, x1, x2)) → A__LENGTH(a__U43(mark(x0), x1, x2))
A__U114(tt, U92(x0, x1, x2)) → A__LENGTH(a__U92(mark(x0), x1, x2))
A__U114(tt, U52(x0)) → A__LENGTH(a__U52(mark(x0)))
A__U114(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U114(tt, U94(x0, x1, x2)) → A__LENGTH(a__U94(mark(x0), x1, x2))
A__U114(tt, U71(x0)) → A__LENGTH(a__U71(mark(x0)))
A__U114(tt, isNatKind(x0)) → A__LENGTH(a__isNatKind(x0))
A__U114(tt, U122(x0)) → A__LENGTH(a__U122(mark(x0)))
A__U114(tt, U51(x0, x1)) → A__LENGTH(a__U51(mark(x0), x1))
A__U114(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
A__U114(tt, U81(x0)) → A__LENGTH(a__U81(mark(x0)))
A__U114(tt, U45(x0, x1)) → A__LENGTH(a__U45(mark(x0), x1))
A__U114(tt, U135(x0, x1, x2, x3)) → A__LENGTH(a__U135(mark(x0), x1, x2, x3))
A__U114(tt, U91(x0, x1, x2)) → A__LENGTH(a__U91(mark(x0), x1, x2))
A__U114(tt, isNatIListKind(x0)) → A__LENGTH(a__isNatIListKind(x0))
A__U114(tt, U41(x0, x1, x2)) → A__LENGTH(a__U41(mark(x0), x1, x2))
A__U114(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)

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.
By narrowing [15] the rule A__U114(tt, zeros) → A__LENGTH(a__zeros) at position [0] we obtained the following new rules:

A__U114(tt, zeros) → A__LENGTH(zeros)
A__U114(tt, zeros) → A__LENGTH(cons(0, zeros))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

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

A__U114(tt, U46(x0)) → A__LENGTH(a__U46(mark(x0)))
A__U114(tt, U102(x0, x1, x2)) → A__LENGTH(a__U102(mark(x0), x1, x2))
A__U114(tt, U111(x0, x1, x2)) → A__LENGTH(a__U111(mark(x0), x1, x2))
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__U114(tt, U133(x0, x1, x2, x3)) → A__LENGTH(a__U133(mark(x0), x1, x2, x3))
A__U114(tt, zeros) → A__LENGTH(zeros)
A__U114(tt, U21(x0, x1)) → A__LENGTH(a__U21(mark(x0), x1))
A__U114(tt, U61(x0, x1)) → A__LENGTH(a__U61(mark(x0), x1))
A__U114(tt, U106(x0)) → A__LENGTH(a__U106(mark(x0)))
A__U114(tt, U134(x0, x1, x2, x3)) → A__LENGTH(a__U134(mark(x0), x1, x2, x3))
A__U114(tt, U101(x0, x1, x2)) → A__LENGTH(a__U101(mark(x0), x1, x2))
A__U114(tt, U105(x0, x1)) → A__LENGTH(a__U105(mark(x0), x1))
A__U114(tt, U96(x0)) → A__LENGTH(a__U96(mark(x0)))
A__U114(tt, U13(x0)) → A__LENGTH(a__U13(mark(x0)))
A__U114(tt, U131(x0, x1, x2, x3)) → A__LENGTH(a__U131(mark(x0), x1, x2, x3))
A__U114(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U114(tt, U136(x0, x1, x2, x3)) → A__LENGTH(a__U136(mark(x0), x1, x2, x3))
A__U114(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U114(tt, U23(x0)) → A__LENGTH(a__U23(mark(x0)))
A__U114(tt, U22(x0, x1)) → A__LENGTH(a__U22(mark(x0), x1))
A__U114(tt, U103(x0, x1, x2)) → A__LENGTH(a__U103(mark(x0), x1, x2))
A__U113(tt, y0, s(x0)) → A__U114(a__U81(a__isNatKind(x0)), y0)
A__U113(tt, y0, 0) → A__U114(tt, y0)
A__U114(tt, U132(x0, x1, x2, x3)) → A__LENGTH(a__U132(mark(x0), x1, x2, x3))
A__U114(tt, U95(x0, x1)) → A__LENGTH(a__U95(mark(x0), x1))
A__U114(tt, U104(x0, x1, x2)) → A__LENGTH(a__U104(mark(x0), x1, x2))
A__U114(tt, U112(x0, x1, x2)) → A__LENGTH(a__U112(mark(x0), x1, x2))
A__U114(tt, U32(x0, x1)) → A__LENGTH(a__U32(mark(x0), x1))
A__U114(tt, U44(x0, x1, x2)) → A__LENGTH(a__U44(mark(x0), x1, x2))
A__U114(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U114(tt, U93(x0, x1, x2)) → A__LENGTH(a__U93(mark(x0), x1, x2))
A__U114(tt, U42(x0, x1, x2)) → A__LENGTH(a__U42(mark(x0), x1, x2))
A__U114(tt, U114(x0, x1)) → A__LENGTH(a__U114(mark(x0), x1))
A__U114(tt, U62(x0)) → A__LENGTH(a__U62(mark(x0)))
A__U114(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U114(tt, U113(x0, x1, x2)) → A__LENGTH(a__U113(mark(x0), x1, x2))
A__U114(tt, U121(x0, x1)) → A__LENGTH(a__U121(mark(x0), x1))
A__U113(tt, y0, length(x0)) → A__U114(a__U71(a__isNatIListKind(x0)), y0)
A__U114(tt, U33(x0)) → A__LENGTH(a__U33(mark(x0)))
A__U114(tt, U31(x0, x1)) → A__LENGTH(a__U31(mark(x0), x1))
A__U114(tt, U12(x0, x1)) → A__LENGTH(a__U12(mark(x0), x1))
A__U114(tt, U92(x0, x1, x2)) → A__LENGTH(a__U92(mark(x0), x1, x2))
A__U114(tt, U43(x0, x1, x2)) → A__LENGTH(a__U43(mark(x0), x1, x2))
A__U114(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__U114(tt, U52(x0)) → A__LENGTH(a__U52(mark(x0)))
A__U114(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U114(tt, U94(x0, x1, x2)) → A__LENGTH(a__U94(mark(x0), x1, x2))
A__U114(tt, U71(x0)) → A__LENGTH(a__U71(mark(x0)))
A__U114(tt, isNatKind(x0)) → A__LENGTH(a__isNatKind(x0))
A__U114(tt, U122(x0)) → A__LENGTH(a__U122(mark(x0)))
A__U114(tt, U51(x0, x1)) → A__LENGTH(a__U51(mark(x0), x1))
A__U114(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
A__U114(tt, U45(x0, x1)) → A__LENGTH(a__U45(mark(x0), x1))
A__U114(tt, U81(x0)) → A__LENGTH(a__U81(mark(x0)))
A__U114(tt, U135(x0, x1, x2, x3)) → A__LENGTH(a__U135(mark(x0), x1, x2, x3))
A__U114(tt, U91(x0, x1, x2)) → A__LENGTH(a__U91(mark(x0), x1, x2))
A__U114(tt, isNatIListKind(x0)) → A__LENGTH(a__isNatIListKind(x0))
A__U114(tt, U41(x0, x1, x2)) → A__LENGTH(a__U41(mark(x0), x1, x2))
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)
A__U114(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))

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.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ QDPOrderProof

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

A__U114(tt, U46(x0)) → A__LENGTH(a__U46(mark(x0)))
A__U114(tt, U102(x0, x1, x2)) → A__LENGTH(a__U102(mark(x0), x1, x2))
A__U114(tt, U111(x0, x1, x2)) → A__LENGTH(a__U111(mark(x0), x1, x2))
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__U114(tt, U133(x0, x1, x2, x3)) → A__LENGTH(a__U133(mark(x0), x1, x2, x3))
A__U114(tt, U21(x0, x1)) → A__LENGTH(a__U21(mark(x0), x1))
A__U114(tt, U61(x0, x1)) → A__LENGTH(a__U61(mark(x0), x1))
A__U114(tt, U106(x0)) → A__LENGTH(a__U106(mark(x0)))
A__U114(tt, U134(x0, x1, x2, x3)) → A__LENGTH(a__U134(mark(x0), x1, x2, x3))
A__U114(tt, U101(x0, x1, x2)) → A__LENGTH(a__U101(mark(x0), x1, x2))
A__U114(tt, U105(x0, x1)) → A__LENGTH(a__U105(mark(x0), x1))
A__U114(tt, U96(x0)) → A__LENGTH(a__U96(mark(x0)))
A__U114(tt, U13(x0)) → A__LENGTH(a__U13(mark(x0)))
A__U114(tt, U131(x0, x1, x2, x3)) → A__LENGTH(a__U131(mark(x0), x1, x2, x3))
A__U114(tt, U136(x0, x1, x2, x3)) → A__LENGTH(a__U136(mark(x0), x1, x2, x3))
A__U114(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U114(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U114(tt, U23(x0)) → A__LENGTH(a__U23(mark(x0)))
A__U114(tt, U22(x0, x1)) → A__LENGTH(a__U22(mark(x0), x1))
A__U114(tt, U103(x0, x1, x2)) → A__LENGTH(a__U103(mark(x0), x1, x2))
A__U113(tt, y0, s(x0)) → A__U114(a__U81(a__isNatKind(x0)), y0)
A__U113(tt, y0, 0) → A__U114(tt, y0)
A__U114(tt, U132(x0, x1, x2, x3)) → A__LENGTH(a__U132(mark(x0), x1, x2, x3))
A__U114(tt, U95(x0, x1)) → A__LENGTH(a__U95(mark(x0), x1))
A__U114(tt, U104(x0, x1, x2)) → A__LENGTH(a__U104(mark(x0), x1, x2))
A__U114(tt, U112(x0, x1, x2)) → A__LENGTH(a__U112(mark(x0), x1, x2))
A__U114(tt, U32(x0, x1)) → A__LENGTH(a__U32(mark(x0), x1))
A__U114(tt, U44(x0, x1, x2)) → A__LENGTH(a__U44(mark(x0), x1, x2))
A__U114(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U114(tt, U93(x0, x1, x2)) → A__LENGTH(a__U93(mark(x0), x1, x2))
A__U114(tt, U114(x0, x1)) → A__LENGTH(a__U114(mark(x0), x1))
A__U114(tt, U42(x0, x1, x2)) → A__LENGTH(a__U42(mark(x0), x1, x2))
A__U114(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U114(tt, U62(x0)) → A__LENGTH(a__U62(mark(x0)))
A__U114(tt, U113(x0, x1, x2)) → A__LENGTH(a__U113(mark(x0), x1, x2))
A__U113(tt, y0, length(x0)) → A__U114(a__U71(a__isNatIListKind(x0)), y0)
A__U114(tt, U121(x0, x1)) → A__LENGTH(a__U121(mark(x0), x1))
A__U114(tt, U31(x0, x1)) → A__LENGTH(a__U31(mark(x0), x1))
A__U114(tt, U33(x0)) → A__LENGTH(a__U33(mark(x0)))
A__U114(tt, U12(x0, x1)) → A__LENGTH(a__U12(mark(x0), x1))
A__U114(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__U114(tt, U43(x0, x1, x2)) → A__LENGTH(a__U43(mark(x0), x1, x2))
A__U114(tt, U92(x0, x1, x2)) → A__LENGTH(a__U92(mark(x0), x1, x2))
A__U114(tt, U52(x0)) → A__LENGTH(a__U52(mark(x0)))
A__U114(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U114(tt, U94(x0, x1, x2)) → A__LENGTH(a__U94(mark(x0), x1, x2))
A__U114(tt, U71(x0)) → A__LENGTH(a__U71(mark(x0)))
A__U114(tt, isNatKind(x0)) → A__LENGTH(a__isNatKind(x0))
A__U114(tt, U122(x0)) → A__LENGTH(a__U122(mark(x0)))
A__U114(tt, U51(x0, x1)) → A__LENGTH(a__U51(mark(x0), x1))
A__U114(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
A__U114(tt, U81(x0)) → A__LENGTH(a__U81(mark(x0)))
A__U114(tt, U45(x0, x1)) → A__LENGTH(a__U45(mark(x0), x1))
A__U114(tt, U135(x0, x1, x2, x3)) → A__LENGTH(a__U135(mark(x0), x1, x2, x3))
A__U114(tt, U91(x0, x1, x2)) → A__LENGTH(a__U91(mark(x0), x1, x2))
A__U114(tt, isNatIListKind(x0)) → A__LENGTH(a__isNatIListKind(x0))
A__U114(tt, U41(x0, x1, x2)) → A__LENGTH(a__U41(mark(x0), x1, x2))
A__U114(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)

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.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A__U113(tt, y0, length(x0)) → A__U114(a__U71(a__isNatIListKind(x0)), y0)

The remaining pairs can at least be oriented weakly.

A__U114(tt, U46(x0)) → A__LENGTH(a__U46(mark(x0)))
A__U114(tt, U102(x0, x1, x2)) → A__LENGTH(a__U102(mark(x0), x1, x2))
A__U114(tt, U111(x0, x1, x2)) → A__LENGTH(a__U111(mark(x0), x1, x2))
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__U114(tt, U133(x0, x1, x2, x3)) → A__LENGTH(a__U133(mark(x0), x1, x2, x3))
A__U114(tt, U21(x0, x1)) → A__LENGTH(a__U21(mark(x0), x1))
A__U114(tt, U61(x0, x1)) → A__LENGTH(a__U61(mark(x0), x1))
A__U114(tt, U106(x0)) → A__LENGTH(a__U106(mark(x0)))
A__U114(tt, U134(x0, x1, x2, x3)) → A__LENGTH(a__U134(mark(x0), x1, x2, x3))
A__U114(tt, U101(x0, x1, x2)) → A__LENGTH(a__U101(mark(x0), x1, x2))
A__U114(tt, U105(x0, x1)) → A__LENGTH(a__U105(mark(x0), x1))
A__U114(tt, U96(x0)) → A__LENGTH(a__U96(mark(x0)))
A__U114(tt, U13(x0)) → A__LENGTH(a__U13(mark(x0)))
A__U114(tt, U131(x0, x1, x2, x3)) → A__LENGTH(a__U131(mark(x0), x1, x2, x3))
A__U114(tt, U136(x0, x1, x2, x3)) → A__LENGTH(a__U136(mark(x0), x1, x2, x3))
A__U114(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U114(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U114(tt, U23(x0)) → A__LENGTH(a__U23(mark(x0)))
A__U114(tt, U22(x0, x1)) → A__LENGTH(a__U22(mark(x0), x1))
A__U114(tt, U103(x0, x1, x2)) → A__LENGTH(a__U103(mark(x0), x1, x2))
A__U113(tt, y0, s(x0)) → A__U114(a__U81(a__isNatKind(x0)), y0)
A__U113(tt, y0, 0) → A__U114(tt, y0)
A__U114(tt, U132(x0, x1, x2, x3)) → A__LENGTH(a__U132(mark(x0), x1, x2, x3))
A__U114(tt, U95(x0, x1)) → A__LENGTH(a__U95(mark(x0), x1))
A__U114(tt, U104(x0, x1, x2)) → A__LENGTH(a__U104(mark(x0), x1, x2))
A__U114(tt, U112(x0, x1, x2)) → A__LENGTH(a__U112(mark(x0), x1, x2))
A__U114(tt, U32(x0, x1)) → A__LENGTH(a__U32(mark(x0), x1))
A__U114(tt, U44(x0, x1, x2)) → A__LENGTH(a__U44(mark(x0), x1, x2))
A__U114(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U114(tt, U93(x0, x1, x2)) → A__LENGTH(a__U93(mark(x0), x1, x2))
A__U114(tt, U114(x0, x1)) → A__LENGTH(a__U114(mark(x0), x1))
A__U114(tt, U42(x0, x1, x2)) → A__LENGTH(a__U42(mark(x0), x1, x2))
A__U114(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U114(tt, U62(x0)) → A__LENGTH(a__U62(mark(x0)))
A__U114(tt, U113(x0, x1, x2)) → A__LENGTH(a__U113(mark(x0), x1, x2))
A__U114(tt, U121(x0, x1)) → A__LENGTH(a__U121(mark(x0), x1))
A__U114(tt, U31(x0, x1)) → A__LENGTH(a__U31(mark(x0), x1))
A__U114(tt, U33(x0)) → A__LENGTH(a__U33(mark(x0)))
A__U114(tt, U12(x0, x1)) → A__LENGTH(a__U12(mark(x0), x1))
A__U114(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__U114(tt, U43(x0, x1, x2)) → A__LENGTH(a__U43(mark(x0), x1, x2))
A__U114(tt, U92(x0, x1, x2)) → A__LENGTH(a__U92(mark(x0), x1, x2))
A__U114(tt, U52(x0)) → A__LENGTH(a__U52(mark(x0)))
A__U114(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U114(tt, U94(x0, x1, x2)) → A__LENGTH(a__U94(mark(x0), x1, x2))
A__U114(tt, U71(x0)) → A__LENGTH(a__U71(mark(x0)))
A__U114(tt, isNatKind(x0)) → A__LENGTH(a__isNatKind(x0))
A__U114(tt, U122(x0)) → A__LENGTH(a__U122(mark(x0)))
A__U114(tt, U51(x0, x1)) → A__LENGTH(a__U51(mark(x0), x1))
A__U114(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
A__U114(tt, U81(x0)) → A__LENGTH(a__U81(mark(x0)))
A__U114(tt, U45(x0, x1)) → A__LENGTH(a__U45(mark(x0), x1))
A__U114(tt, U135(x0, x1, x2, x3)) → A__LENGTH(a__U135(mark(x0), x1, x2, x3))
A__U114(tt, U91(x0, x1, x2)) → A__LENGTH(a__U91(mark(x0), x1, x2))
A__U114(tt, isNatIListKind(x0)) → A__LENGTH(a__isNatIListKind(x0))
A__U114(tt, U41(x0, x1, x2)) → A__LENGTH(a__U41(mark(x0), x1, x2))
A__U114(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0
POL(A__LENGTH(x₁)) = x₁
POL(A__U111(x₁, x₂, x₃)) = x₂ + x₃
POL(A__U112(x₁, x₂, x₃)) = x₂ + x₃
POL(A__U113(x₁, x₂, x₃)) = x₂ + x₃
POL(A__U114(x₁, x₂)) = x₂
POL(U101(x₁, x₂, x₃)) = 0
POL(U102(x₁, x₂, x₃)) = 0
POL(U103(x₁, x₂, x₃)) = 0
POL(U104(x₁, x₂, x₃)) = 0
POL(U105(x₁, x₂)) = 0
POL(U106(x₁)) = 0
POL(U11(x₁, x₂)) = 0
POL(U111(x₁, x₂, x₃)) = 0
POL(U112(x₁, x₂, x₃)) = 0
POL(U113(x₁, x₂, x₃)) = 0
POL(U114(x₁, x₂)) = 0
POL(U12(x₁, x₂)) = 0
POL(U121(x₁, x₂)) = 0
POL(U122(x₁)) = 0
POL(U13(x₁)) = 0
POL(U131(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(U132(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(U133(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(U134(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(U135(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(U136(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(U21(x₁, x₂)) = 0
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁, x₂)) = 0
POL(U33(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = x₂ + x₃
POL(U42(x₁, x₂, x₃)) = x₃
POL(U43(x₁, x₂, x₃)) = x₃
POL(U44(x₁, x₂, x₃)) = 0
POL(U45(x₁, x₂)) = 0
POL(U46(x₁)) = 0
POL(U51(x₁, x₂)) = 0
POL(U52(x₁)) = 0
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁)) = 0
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃)) = 0
POL(U92(x₁, x₂, x₃)) = 0
POL(U93(x₁, x₂, x₃)) = 0
POL(U94(x₁, x₂, x₃)) = 0
POL(U95(x₁, x₂)) = 0
POL(U96(x₁)) = 0
POL(a__U101(x₁, x₂, x₃)) = 0
POL(a__U102(x₁, x₂, x₃)) = 0
POL(a__U103(x₁, x₂, x₃)) = 0
POL(a__U104(x₁, x₂, x₃)) = 0
POL(a__U105(x₁, x₂)) = 0
POL(a__U106(x₁)) = 0
POL(a__U11(x₁, x₂)) = 0
POL(a__U111(x₁, x₂, x₃)) = 0
POL(a__U112(x₁, x₂, x₃)) = 0
POL(a__U113(x₁, x₂, x₃)) = 0
POL(a__U114(x₁, x₂)) = 0
POL(a__U12(x₁, x₂)) = 0
POL(a__U121(x₁, x₂)) = 0
POL(a__U122(x₁)) = 0
POL(a__U13(x₁)) = 0
POL(a__U131(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(a__U132(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(a__U133(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(a__U134(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(a__U135(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(a__U136(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(a__U21(x₁, x₂)) = 0
POL(a__U22(x₁, x₂)) = 0
POL(a__U23(x₁)) = 0
POL(a__U31(x₁, x₂)) = 0
POL(a__U32(x₁, x₂)) = 0
POL(a__U33(x₁)) = 0
POL(a__U41(x₁, x₂, x₃)) = x₂ + x₃
POL(a__U42(x₁, x₂, x₃)) = x₃
POL(a__U43(x₁, x₂, x₃)) = x₃
POL(a__U44(x₁, x₂, x₃)) = 0
POL(a__U45(x₁, x₂)) = 0
POL(a__U46(x₁)) = 0
POL(a__U51(x₁, x₂)) = 0
POL(a__U52(x₁)) = 0
POL(a__U61(x₁, x₂)) = 0
POL(a__U62(x₁)) = 0
POL(a__U71(x₁)) = 0
POL(a__U81(x₁)) = 0
POL(a__U91(x₁, x₂, x₃)) = 0
POL(a__U92(x₁, x₂, x₃)) = 0
POL(a__U93(x₁, x₂, x₃)) = 0
POL(a__U94(x₁, x₂, x₃)) = 0
POL(a__U95(x₁, x₂)) = 0
POL(a__U96(x₁)) = 0
POL(a__isNat(x₁)) = 0
POL(a__isNatIList(x₁)) = x₁
POL(a__isNatIListKind(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__isNatList(x₁)) = 0
POL(a__length(x₁)) = 1
POL(a__take(x₁, x₂)) = x₂
POL(a__zeros) = 0
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = x₁
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 1
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = x₂
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U113(tt, L, N) → a__U114(a__isNatKind(N), L)
a__U114(tt, L) → s(a__length(mark(L)))
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__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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__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__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__U122(tt) → nil
a__U13(tt) → tt
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U33(tt) → tt
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U23(tt) → tt
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U62(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U46(tt) → tt
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U103(tt, V1, V2) → a__U104(a__isNatIListKind(V2), V1, V2)
a__U102(tt, V1, V2) → a__U103(a__isNatIListKind(V2), V1, V2)
a__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U104(tt, V1, V2) → a__U105(a__isNat(V1), V2)
a__U101(tt, V1, V2) → a__U102(a__isNatKind(V1), V1, V2)
a__zeros → cons(0, zeros)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(U112(X1, X2, X3)) → a__U112(mark(X1), X2, X3)
mark(U113(X1, X2, X3)) → a__U113(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U111(X1, X2, X3)) → a__U111(mark(X1), X2, X3)
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U114(X1, X2)) → a__U114(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U132(X1, X2, X3, X4)) → a__U132(mark(X1), X2, X3, X4)
mark(U131(X1, X2, X3, X4)) → a__U131(mark(X1), X2, X3, X4)
mark(U122(X)) → a__U122(mark(X))
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U136(X1, X2, X3, X4)) → a__U136(mark(X1), X2, X3, X4)
mark(U135(X1, X2, X3, X4)) → a__U135(mark(X1), X2, X3, X4)
mark(U134(X1, X2, X3, X4)) → a__U134(mark(X1), X2, X3, X4)
mark(U133(X1, X2, X3, X4)) → a__U133(mark(X1), X2, X3, X4)
mark(U23(X)) → a__U23(mark(X))
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U33(X)) → a__U33(mark(X))
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
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(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(0) → tt
a__isNatList(nil) → tt
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
a__length(nil) → 0
a__take(s(M), cons(N, IL)) → a__U131(a__isNatIList(IL), IL, M, N)
a__take(0, IL) → a__U121(a__isNatIList(IL), IL)
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(zeros) → a__zeros
mark(isNatKind(X)) → a__isNatKind(X)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U122(X) → U122(X)
a__U121(X1, X2) → U121(X1, X2)
a__isNatList(X) → isNatList(X)
a__U13(X) → U13(X)
a__length(X) → length(X)
a__U114(X1, X2) → U114(X1, X2)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__U11(X1, X2) → U11(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U95(X1, X2)) → a__U95(mark(X1), X2)
mark(U94(X1, X2, X3)) → a__U94(mark(X1), X2, X3)
mark(U71(X)) → a__U71(mark(X))
mark(U62(X)) → a__U62(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U81(X)) → a__U81(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
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__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)
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
mark(U96(X)) → a__U96(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
a__U95(X1, X2) → U95(X1, X2)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U96(X) → U96(X)
a__U61(X1, X2) → U61(X1, X2)
a__U52(X) → U52(X)
a__U71(X) → U71(X)
a__U62(X) → U62(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U81(X) → U81(X)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

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

A__U114(tt, U46(x0)) → A__LENGTH(a__U46(mark(x0)))
A__U114(tt, U102(x0, x1, x2)) → A__LENGTH(a__U102(mark(x0), x1, x2))
A__U114(tt, U111(x0, x1, x2)) → A__LENGTH(a__U111(mark(x0), x1, x2))
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__U114(tt, U133(x0, x1, x2, x3)) → A__LENGTH(a__U133(mark(x0), x1, x2, x3))
A__U114(tt, U21(x0, x1)) → A__LENGTH(a__U21(mark(x0), x1))
A__U114(tt, U61(x0, x1)) → A__LENGTH(a__U61(mark(x0), x1))
A__U114(tt, U106(x0)) → A__LENGTH(a__U106(mark(x0)))
A__U114(tt, U134(x0, x1, x2, x3)) → A__LENGTH(a__U134(mark(x0), x1, x2, x3))
A__U114(tt, U101(x0, x1, x2)) → A__LENGTH(a__U101(mark(x0), x1, x2))
A__U114(tt, U105(x0, x1)) → A__LENGTH(a__U105(mark(x0), x1))
A__U114(tt, U96(x0)) → A__LENGTH(a__U96(mark(x0)))
A__U114(tt, U13(x0)) → A__LENGTH(a__U13(mark(x0)))
A__U114(tt, U131(x0, x1, x2, x3)) → A__LENGTH(a__U131(mark(x0), x1, x2, x3))
A__U114(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U114(tt, U136(x0, x1, x2, x3)) → A__LENGTH(a__U136(mark(x0), x1, x2, x3))
A__U114(tt, U23(x0)) → A__LENGTH(a__U23(mark(x0)))
A__U114(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U114(tt, U22(x0, x1)) → A__LENGTH(a__U22(mark(x0), x1))
A__U114(tt, U103(x0, x1, x2)) → A__LENGTH(a__U103(mark(x0), x1, x2))
A__U113(tt, y0, s(x0)) → A__U114(a__U81(a__isNatKind(x0)), y0)
A__U113(tt, y0, 0) → A__U114(tt, y0)
A__U114(tt, U132(x0, x1, x2, x3)) → A__LENGTH(a__U132(mark(x0), x1, x2, x3))
A__U114(tt, U104(x0, x1, x2)) → A__LENGTH(a__U104(mark(x0), x1, x2))
A__U114(tt, U95(x0, x1)) → A__LENGTH(a__U95(mark(x0), x1))
A__U114(tt, U112(x0, x1, x2)) → A__LENGTH(a__U112(mark(x0), x1, x2))
A__U114(tt, U32(x0, x1)) → A__LENGTH(a__U32(mark(x0), x1))
A__U114(tt, U44(x0, x1, x2)) → A__LENGTH(a__U44(mark(x0), x1, x2))
A__U114(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U114(tt, U93(x0, x1, x2)) → A__LENGTH(a__U93(mark(x0), x1, x2))
A__U114(tt, U114(x0, x1)) → A__LENGTH(a__U114(mark(x0), x1))
A__U114(tt, U42(x0, x1, x2)) → A__LENGTH(a__U42(mark(x0), x1, x2))
A__U114(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U114(tt, U62(x0)) → A__LENGTH(a__U62(mark(x0)))
A__U114(tt, U113(x0, x1, x2)) → A__LENGTH(a__U113(mark(x0), x1, x2))
A__U114(tt, U121(x0, x1)) → A__LENGTH(a__U121(mark(x0), x1))
A__U114(tt, U31(x0, x1)) → A__LENGTH(a__U31(mark(x0), x1))
A__U114(tt, U33(x0)) → A__LENGTH(a__U33(mark(x0)))
A__U114(tt, U12(x0, x1)) → A__LENGTH(a__U12(mark(x0), x1))
A__U114(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__U114(tt, U43(x0, x1, x2)) → A__LENGTH(a__U43(mark(x0), x1, x2))
A__U114(tt, U92(x0, x1, x2)) → A__LENGTH(a__U92(mark(x0), x1, x2))
A__U114(tt, U52(x0)) → A__LENGTH(a__U52(mark(x0)))
A__U114(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U114(tt, U94(x0, x1, x2)) → A__LENGTH(a__U94(mark(x0), x1, x2))
A__U114(tt, U71(x0)) → A__LENGTH(a__U71(mark(x0)))
A__U114(tt, isNatKind(x0)) → A__LENGTH(a__isNatKind(x0))
A__U114(tt, U122(x0)) → A__LENGTH(a__U122(mark(x0)))
A__U114(tt, U51(x0, x1)) → A__LENGTH(a__U51(mark(x0), x1))
A__U114(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
A__U114(tt, U45(x0, x1)) → A__LENGTH(a__U45(mark(x0), x1))
A__U114(tt, U81(x0)) → A__LENGTH(a__U81(mark(x0)))
A__U114(tt, U135(x0, x1, x2, x3)) → A__LENGTH(a__U135(mark(x0), x1, x2, x3))
A__U114(tt, U91(x0, x1, x2)) → A__LENGTH(a__U91(mark(x0), x1, x2))
A__U114(tt, isNatIListKind(x0)) → A__LENGTH(a__isNatIListKind(x0))
A__U114(tt, U41(x0, x1, x2)) → A__LENGTH(a__U41(mark(x0), x1, x2))
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)
A__U114(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))

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.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A__U113(tt, y0, s(x0)) → A__U114(a__U81(a__isNatKind(x0)), y0)

The remaining pairs can at least be oriented weakly.

A__U114(tt, U46(x0)) → A__LENGTH(a__U46(mark(x0)))
A__U114(tt, U102(x0, x1, x2)) → A__LENGTH(a__U102(mark(x0), x1, x2))
A__U114(tt, U111(x0, x1, x2)) → A__LENGTH(a__U111(mark(x0), x1, x2))
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__U114(tt, U133(x0, x1, x2, x3)) → A__LENGTH(a__U133(mark(x0), x1, x2, x3))
A__U114(tt, U21(x0, x1)) → A__LENGTH(a__U21(mark(x0), x1))
A__U114(tt, U61(x0, x1)) → A__LENGTH(a__U61(mark(x0), x1))
A__U114(tt, U106(x0)) → A__LENGTH(a__U106(mark(x0)))
A__U114(tt, U134(x0, x1, x2, x3)) → A__LENGTH(a__U134(mark(x0), x1, x2, x3))
A__U114(tt, U101(x0, x1, x2)) → A__LENGTH(a__U101(mark(x0), x1, x2))
A__U114(tt, U105(x0, x1)) → A__LENGTH(a__U105(mark(x0), x1))
A__U114(tt, U96(x0)) → A__LENGTH(a__U96(mark(x0)))
A__U114(tt, U13(x0)) → A__LENGTH(a__U13(mark(x0)))
A__U114(tt, U131(x0, x1, x2, x3)) → A__LENGTH(a__U131(mark(x0), x1, x2, x3))
A__U114(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U114(tt, U136(x0, x1, x2, x3)) → A__LENGTH(a__U136(mark(x0), x1, x2, x3))
A__U114(tt, U23(x0)) → A__LENGTH(a__U23(mark(x0)))
A__U114(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U114(tt, U22(x0, x1)) → A__LENGTH(a__U22(mark(x0), x1))
A__U114(tt, U103(x0, x1, x2)) → A__LENGTH(a__U103(mark(x0), x1, x2))
A__U113(tt, y0, 0) → A__U114(tt, y0)
A__U114(tt, U132(x0, x1, x2, x3)) → A__LENGTH(a__U132(mark(x0), x1, x2, x3))
A__U114(tt, U104(x0, x1, x2)) → A__LENGTH(a__U104(mark(x0), x1, x2))
A__U114(tt, U95(x0, x1)) → A__LENGTH(a__U95(mark(x0), x1))
A__U114(tt, U112(x0, x1, x2)) → A__LENGTH(a__U112(mark(x0), x1, x2))
A__U114(tt, U32(x0, x1)) → A__LENGTH(a__U32(mark(x0), x1))
A__U114(tt, U44(x0, x1, x2)) → A__LENGTH(a__U44(mark(x0), x1, x2))
A__U114(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U114(tt, U93(x0, x1, x2)) → A__LENGTH(a__U93(mark(x0), x1, x2))
A__U114(tt, U114(x0, x1)) → A__LENGTH(a__U114(mark(x0), x1))
A__U114(tt, U42(x0, x1, x2)) → A__LENGTH(a__U42(mark(x0), x1, x2))
A__U114(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U114(tt, U62(x0)) → A__LENGTH(a__U62(mark(x0)))
A__U114(tt, U113(x0, x1, x2)) → A__LENGTH(a__U113(mark(x0), x1, x2))
A__U114(tt, U121(x0, x1)) → A__LENGTH(a__U121(mark(x0), x1))
A__U114(tt, U31(x0, x1)) → A__LENGTH(a__U31(mark(x0), x1))
A__U114(tt, U33(x0)) → A__LENGTH(a__U33(mark(x0)))
A__U114(tt, U12(x0, x1)) → A__LENGTH(a__U12(mark(x0), x1))
A__U114(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__U114(tt, U43(x0, x1, x2)) → A__LENGTH(a__U43(mark(x0), x1, x2))
A__U114(tt, U92(x0, x1, x2)) → A__LENGTH(a__U92(mark(x0), x1, x2))
A__U114(tt, U52(x0)) → A__LENGTH(a__U52(mark(x0)))
A__U114(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U114(tt, U94(x0, x1, x2)) → A__LENGTH(a__U94(mark(x0), x1, x2))
A__U114(tt, U71(x0)) → A__LENGTH(a__U71(mark(x0)))
A__U114(tt, isNatKind(x0)) → A__LENGTH(a__isNatKind(x0))
A__U114(tt, U122(x0)) → A__LENGTH(a__U122(mark(x0)))
A__U114(tt, U51(x0, x1)) → A__LENGTH(a__U51(mark(x0), x1))
A__U114(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
A__U114(tt, U45(x0, x1)) → A__LENGTH(a__U45(mark(x0), x1))
A__U114(tt, U81(x0)) → A__LENGTH(a__U81(mark(x0)))
A__U114(tt, U135(x0, x1, x2, x3)) → A__LENGTH(a__U135(mark(x0), x1, x2, x3))
A__U114(tt, U91(x0, x1, x2)) → A__LENGTH(a__U91(mark(x0), x1, x2))
A__U114(tt, isNatIListKind(x0)) → A__LENGTH(a__isNatIListKind(x0))
A__U114(tt, U41(x0, x1, x2)) → A__LENGTH(a__U41(mark(x0), x1, x2))
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)
A__U114(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0
POL(A__LENGTH(x₁)) = x₁
POL(A__U111(x₁, x₂, x₃)) = x₂ + x₃
POL(A__U112(x₁, x₂, x₃)) = x₂ + x₃
POL(A__U113(x₁, x₂, x₃)) = x₂ + x₃
POL(A__U114(x₁, x₂)) = x₂
POL(U101(x₁, x₂, x₃)) = 0
POL(U102(x₁, x₂, x₃)) = 0
POL(U103(x₁, x₂, x₃)) = 0
POL(U104(x₁, x₂, x₃)) = 0
POL(U105(x₁, x₂)) = 0
POL(U106(x₁)) = 0
POL(U11(x₁, x₂)) = 0
POL(U111(x₁, x₂, x₃)) = 1
POL(U112(x₁, x₂, x₃)) = 1
POL(U113(x₁, x₂, x₃)) = 1
POL(U114(x₁, x₂)) = 1
POL(U12(x₁, x₂)) = 0
POL(U121(x₁, x₂)) = 0
POL(U122(x₁)) = 0
POL(U13(x₁)) = 0
POL(U131(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(U132(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(U133(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(U134(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(U135(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(U136(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(U21(x₁, x₂)) = 0
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁, x₂)) = 0
POL(U33(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂, x₃)) = 0
POL(U43(x₁, x₂, x₃)) = 0
POL(U44(x₁, x₂, x₃)) = 0
POL(U45(x₁, x₂)) = 0
POL(U46(x₁)) = 0
POL(U51(x₁, x₂)) = 0
POL(U52(x₁)) = 0
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁)) = 0
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃)) = 0
POL(U92(x₁, x₂, x₃)) = 0
POL(U93(x₁, x₂, x₃)) = 0
POL(U94(x₁, x₂, x₃)) = 0
POL(U95(x₁, x₂)) = 0
POL(U96(x₁)) = 0
POL(a__U101(x₁, x₂, x₃)) = 0
POL(a__U102(x₁, x₂, x₃)) = 0
POL(a__U103(x₁, x₂, x₃)) = 0
POL(a__U104(x₁, x₂, x₃)) = 0
POL(a__U105(x₁, x₂)) = 0
POL(a__U106(x₁)) = 0
POL(a__U11(x₁, x₂)) = 0
POL(a__U111(x₁, x₂, x₃)) = 1
POL(a__U112(x₁, x₂, x₃)) = 1
POL(a__U113(x₁, x₂, x₃)) = 1
POL(a__U114(x₁, x₂)) = 1
POL(a__U12(x₁, x₂)) = 0
POL(a__U121(x₁, x₂)) = 0
POL(a__U122(x₁)) = 0
POL(a__U13(x₁)) = 0
POL(a__U131(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(a__U132(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(a__U133(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(a__U134(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(a__U135(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(a__U136(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(a__U21(x₁, x₂)) = 0
POL(a__U22(x₁, x₂)) = 0
POL(a__U23(x₁)) = 0
POL(a__U31(x₁, x₂)) = 0
POL(a__U32(x₁, x₂)) = 0
POL(a__U33(x₁)) = 0
POL(a__U41(x₁, x₂, x₃)) = 0
POL(a__U42(x₁, x₂, x₃)) = 0
POL(a__U43(x₁, x₂, x₃)) = 0
POL(a__U44(x₁, x₂, x₃)) = 0
POL(a__U45(x₁, x₂)) = 0
POL(a__U46(x₁)) = 0
POL(a__U51(x₁, x₂)) = 0
POL(a__U52(x₁)) = 0
POL(a__U61(x₁, x₂)) = 0
POL(a__U62(x₁)) = 0
POL(a__U71(x₁)) = 0
POL(a__U81(x₁)) = 0
POL(a__U91(x₁, x₂, x₃)) = 0
POL(a__U92(x₁, x₂, x₃)) = 0
POL(a__U93(x₁, x₂, x₃)) = 0
POL(a__U94(x₁, x₂, x₃)) = 0
POL(a__U95(x₁, x₂)) = 0
POL(a__U96(x₁)) = 0
POL(a__isNat(x₁)) = 0
POL(a__isNatIList(x₁)) = 0
POL(a__isNatIListKind(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__isNatList(x₁)) = 0
POL(a__length(x₁)) = 1
POL(a__take(x₁, x₂)) = x₂
POL(a__zeros) = 0
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 1
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = 1
POL(take(x₁, x₂)) = x₂
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U113(tt, L, N) → a__U114(a__isNatKind(N), L)
a__U114(tt, L) → s(a__length(mark(L)))
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__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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__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__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__U122(tt) → nil
a__U13(tt) → tt
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U33(tt) → tt
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U23(tt) → tt
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U62(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U46(tt) → tt
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U103(tt, V1, V2) → a__U104(a__isNatIListKind(V2), V1, V2)
a__U102(tt, V1, V2) → a__U103(a__isNatIListKind(V2), V1, V2)
a__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U104(tt, V1, V2) → a__U105(a__isNat(V1), V2)
a__U101(tt, V1, V2) → a__U102(a__isNatKind(V1), V1, V2)
a__zeros → cons(0, zeros)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(U112(X1, X2, X3)) → a__U112(mark(X1), X2, X3)
mark(U113(X1, X2, X3)) → a__U113(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U111(X1, X2, X3)) → a__U111(mark(X1), X2, X3)
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U114(X1, X2)) → a__U114(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U132(X1, X2, X3, X4)) → a__U132(mark(X1), X2, X3, X4)
mark(U131(X1, X2, X3, X4)) → a__U131(mark(X1), X2, X3, X4)
mark(U122(X)) → a__U122(mark(X))
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U136(X1, X2, X3, X4)) → a__U136(mark(X1), X2, X3, X4)
mark(U135(X1, X2, X3, X4)) → a__U135(mark(X1), X2, X3, X4)
mark(U134(X1, X2, X3, X4)) → a__U134(mark(X1), X2, X3, X4)
mark(U133(X1, X2, X3, X4)) → a__U133(mark(X1), X2, X3, X4)
mark(U23(X)) → a__U23(mark(X))
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U33(X)) → a__U33(mark(X))
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
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(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(0) → tt
a__isNatList(nil) → tt
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
a__length(nil) → 0
a__take(s(M), cons(N, IL)) → a__U131(a__isNatIList(IL), IL, M, N)
a__take(0, IL) → a__U121(a__isNatIList(IL), IL)
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(zeros) → a__zeros
mark(isNatKind(X)) → a__isNatKind(X)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U122(X) → U122(X)
a__U121(X1, X2) → U121(X1, X2)
a__isNatList(X) → isNatList(X)
a__U13(X) → U13(X)
a__length(X) → length(X)
a__U114(X1, X2) → U114(X1, X2)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__U11(X1, X2) → U11(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U95(X1, X2)) → a__U95(mark(X1), X2)
mark(U94(X1, X2, X3)) → a__U94(mark(X1), X2, X3)
mark(U71(X)) → a__U71(mark(X))
mark(U62(X)) → a__U62(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U81(X)) → a__U81(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
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__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)
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
mark(U96(X)) → a__U96(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
a__U95(X1, X2) → U95(X1, X2)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U96(X) → U96(X)
a__U61(X1, X2) → U61(X1, X2)
a__U52(X) → U52(X)
a__U71(X) → U71(X)
a__U62(X) → U62(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U81(X) → U81(X)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ QDPOrderProof

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

A__U114(tt, U46(x0)) → A__LENGTH(a__U46(mark(x0)))
A__U114(tt, U102(x0, x1, x2)) → A__LENGTH(a__U102(mark(x0), x1, x2))
A__U114(tt, U111(x0, x1, x2)) → A__LENGTH(a__U111(mark(x0), x1, x2))
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__U114(tt, U133(x0, x1, x2, x3)) → A__LENGTH(a__U133(mark(x0), x1, x2, x3))
A__U114(tt, U21(x0, x1)) → A__LENGTH(a__U21(mark(x0), x1))
A__U114(tt, U61(x0, x1)) → A__LENGTH(a__U61(mark(x0), x1))
A__U114(tt, U106(x0)) → A__LENGTH(a__U106(mark(x0)))
A__U114(tt, U134(x0, x1, x2, x3)) → A__LENGTH(a__U134(mark(x0), x1, x2, x3))
A__U114(tt, U101(x0, x1, x2)) → A__LENGTH(a__U101(mark(x0), x1, x2))
A__U114(tt, U105(x0, x1)) → A__LENGTH(a__U105(mark(x0), x1))
A__U114(tt, U96(x0)) → A__LENGTH(a__U96(mark(x0)))
A__U114(tt, U13(x0)) → A__LENGTH(a__U13(mark(x0)))
A__U114(tt, U131(x0, x1, x2, x3)) → A__LENGTH(a__U131(mark(x0), x1, x2, x3))
A__U114(tt, U136(x0, x1, x2, x3)) → A__LENGTH(a__U136(mark(x0), x1, x2, x3))
A__U114(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U114(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U114(tt, U23(x0)) → A__LENGTH(a__U23(mark(x0)))
A__U114(tt, U22(x0, x1)) → A__LENGTH(a__U22(mark(x0), x1))
A__U114(tt, U103(x0, x1, x2)) → A__LENGTH(a__U103(mark(x0), x1, x2))
A__U113(tt, y0, 0) → A__U114(tt, y0)
A__U114(tt, U132(x0, x1, x2, x3)) → A__LENGTH(a__U132(mark(x0), x1, x2, x3))
A__U114(tt, U95(x0, x1)) → A__LENGTH(a__U95(mark(x0), x1))
A__U114(tt, U104(x0, x1, x2)) → A__LENGTH(a__U104(mark(x0), x1, x2))
A__U114(tt, U112(x0, x1, x2)) → A__LENGTH(a__U112(mark(x0), x1, x2))
A__U114(tt, U32(x0, x1)) → A__LENGTH(a__U32(mark(x0), x1))
A__U114(tt, U44(x0, x1, x2)) → A__LENGTH(a__U44(mark(x0), x1, x2))
A__U114(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U114(tt, U93(x0, x1, x2)) → A__LENGTH(a__U93(mark(x0), x1, x2))
A__U114(tt, U114(x0, x1)) → A__LENGTH(a__U114(mark(x0), x1))
A__U114(tt, U42(x0, x1, x2)) → A__LENGTH(a__U42(mark(x0), x1, x2))
A__U114(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U114(tt, U62(x0)) → A__LENGTH(a__U62(mark(x0)))
A__U114(tt, U113(x0, x1, x2)) → A__LENGTH(a__U113(mark(x0), x1, x2))
A__U114(tt, U121(x0, x1)) → A__LENGTH(a__U121(mark(x0), x1))
A__U114(tt, U31(x0, x1)) → A__LENGTH(a__U31(mark(x0), x1))
A__U114(tt, U33(x0)) → A__LENGTH(a__U33(mark(x0)))
A__U114(tt, U12(x0, x1)) → A__LENGTH(a__U12(mark(x0), x1))
A__U114(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__U114(tt, U43(x0, x1, x2)) → A__LENGTH(a__U43(mark(x0), x1, x2))
A__U114(tt, U92(x0, x1, x2)) → A__LENGTH(a__U92(mark(x0), x1, x2))
A__U114(tt, U52(x0)) → A__LENGTH(a__U52(mark(x0)))
A__U114(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U114(tt, U94(x0, x1, x2)) → A__LENGTH(a__U94(mark(x0), x1, x2))
A__U114(tt, U71(x0)) → A__LENGTH(a__U71(mark(x0)))
A__U114(tt, isNatKind(x0)) → A__LENGTH(a__isNatKind(x0))
A__U114(tt, U122(x0)) → A__LENGTH(a__U122(mark(x0)))
A__U114(tt, U51(x0, x1)) → A__LENGTH(a__U51(mark(x0), x1))
A__U114(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
A__U114(tt, U81(x0)) → A__LENGTH(a__U81(mark(x0)))
A__U114(tt, U45(x0, x1)) → A__LENGTH(a__U45(mark(x0), x1))
A__U114(tt, U135(x0, x1, x2, x3)) → A__LENGTH(a__U135(mark(x0), x1, x2, x3))
A__U114(tt, U91(x0, x1, x2)) → A__LENGTH(a__U91(mark(x0), x1, x2))
A__U114(tt, isNatIListKind(x0)) → A__LENGTH(a__isNatIListKind(x0))
A__U114(tt, U41(x0, x1, x2)) → A__LENGTH(a__U41(mark(x0), x1, x2))
A__U114(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)

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.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A__U114(tt, U46(x0)) → A__LENGTH(a__U46(mark(x0)))
A__U114(tt, U21(x0, x1)) → A__LENGTH(a__U21(mark(x0), x1))
A__U114(tt, U61(x0, x1)) → A__LENGTH(a__U61(mark(x0), x1))
A__U114(tt, U106(x0)) → A__LENGTH(a__U106(mark(x0)))
A__U114(tt, U96(x0)) → A__LENGTH(a__U96(mark(x0)))
A__U114(tt, U13(x0)) → A__LENGTH(a__U13(mark(x0)))
A__U114(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U114(tt, U23(x0)) → A__LENGTH(a__U23(mark(x0)))
A__U114(tt, U22(x0, x1)) → A__LENGTH(a__U22(mark(x0), x1))
A__U114(tt, U95(x0, x1)) → A__LENGTH(a__U95(mark(x0), x1))
A__U114(tt, U32(x0, x1)) → A__LENGTH(a__U32(mark(x0), x1))
A__U114(tt, U44(x0, x1, x2)) → A__LENGTH(a__U44(mark(x0), x1, x2))
A__U114(tt, U93(x0, x1, x2)) → A__LENGTH(a__U93(mark(x0), x1, x2))
A__U114(tt, U42(x0, x1, x2)) → A__LENGTH(a__U42(mark(x0), x1, x2))
A__U114(tt, U62(x0)) → A__LENGTH(a__U62(mark(x0)))
A__U114(tt, U121(x0, x1)) → A__LENGTH(a__U121(mark(x0), x1))
A__U114(tt, U31(x0, x1)) → A__LENGTH(a__U31(mark(x0), x1))
A__U114(tt, U33(x0)) → A__LENGTH(a__U33(mark(x0)))
A__U114(tt, U12(x0, x1)) → A__LENGTH(a__U12(mark(x0), x1))
A__U114(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__U114(tt, U43(x0, x1, x2)) → A__LENGTH(a__U43(mark(x0), x1, x2))
A__U114(tt, U52(x0)) → A__LENGTH(a__U52(mark(x0)))
A__U114(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U114(tt, U94(x0, x1, x2)) → A__LENGTH(a__U94(mark(x0), x1, x2))
A__U114(tt, U71(x0)) → A__LENGTH(a__U71(mark(x0)))
A__U114(tt, isNatKind(x0)) → A__LENGTH(a__isNatKind(x0))
A__U114(tt, U122(x0)) → A__LENGTH(a__U122(mark(x0)))
A__U114(tt, U51(x0, x1)) → A__LENGTH(a__U51(mark(x0), x1))
A__U114(tt, U81(x0)) → A__LENGTH(a__U81(mark(x0)))
A__U114(tt, U45(x0, x1)) → A__LENGTH(a__U45(mark(x0), x1))
A__U114(tt, isNatIListKind(x0)) → A__LENGTH(a__isNatIListKind(x0))
A__U114(tt, U41(x0, x1, x2)) → A__LENGTH(a__U41(mark(x0), x1, x2))

The remaining pairs can at least be oriented weakly.

A__U114(tt, U102(x0, x1, x2)) → A__LENGTH(a__U102(mark(x0), x1, x2))
A__U114(tt, U111(x0, x1, x2)) → A__LENGTH(a__U111(mark(x0), x1, x2))
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__U114(tt, U133(x0, x1, x2, x3)) → A__LENGTH(a__U133(mark(x0), x1, x2, x3))
A__U114(tt, U134(x0, x1, x2, x3)) → A__LENGTH(a__U134(mark(x0), x1, x2, x3))
A__U114(tt, U101(x0, x1, x2)) → A__LENGTH(a__U101(mark(x0), x1, x2))
A__U114(tt, U105(x0, x1)) → A__LENGTH(a__U105(mark(x0), x1))
A__U114(tt, U131(x0, x1, x2, x3)) → A__LENGTH(a__U131(mark(x0), x1, x2, x3))
A__U114(tt, U136(x0, x1, x2, x3)) → A__LENGTH(a__U136(mark(x0), x1, x2, x3))
A__U114(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U114(tt, U103(x0, x1, x2)) → A__LENGTH(a__U103(mark(x0), x1, x2))
A__U113(tt, y0, 0) → A__U114(tt, y0)
A__U114(tt, U132(x0, x1, x2, x3)) → A__LENGTH(a__U132(mark(x0), x1, x2, x3))
A__U114(tt, U104(x0, x1, x2)) → A__LENGTH(a__U104(mark(x0), x1, x2))
A__U114(tt, U112(x0, x1, x2)) → A__LENGTH(a__U112(mark(x0), x1, x2))
A__U114(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U114(tt, U114(x0, x1)) → A__LENGTH(a__U114(mark(x0), x1))
A__U114(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U114(tt, U113(x0, x1, x2)) → A__LENGTH(a__U113(mark(x0), x1, x2))
A__U114(tt, U92(x0, x1, x2)) → A__LENGTH(a__U92(mark(x0), x1, x2))
A__U114(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
A__U114(tt, U135(x0, x1, x2, x3)) → A__LENGTH(a__U135(mark(x0), x1, x2, x3))
A__U114(tt, U91(x0, x1, x2)) → A__LENGTH(a__U91(mark(x0), x1, x2))
A__U114(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__LENGTH(x₁)) = x₁
POL(A__U111(x₁, x₂, x₃)) = 1
POL(A__U112(x₁, x₂, x₃)) = 1
POL(A__U113(x₁, x₂, x₃)) = 1
POL(A__U114(x₁, x₂)) = 1
POL(U101(x₁, x₂, x₃)) = 1
POL(U102(x₁, x₂, x₃)) = 1
POL(U103(x₁, x₂, x₃)) = 1
POL(U104(x₁, x₂, x₃)) = 1
POL(U105(x₁, x₂)) = 1
POL(U106(x₁)) = 0
POL(U11(x₁, x₂)) = 0
POL(U111(x₁, x₂, x₃)) = 1
POL(U112(x₁, x₂, x₃)) = 1
POL(U113(x₁, x₂, x₃)) = 1
POL(U114(x₁, x₂)) = 1
POL(U12(x₁, x₂)) = 0
POL(U121(x₁, x₂)) = 0
POL(U122(x₁)) = 0
POL(U13(x₁)) = 0
POL(U131(x₁, x₂, x₃, x₄)) = 1
POL(U132(x₁, x₂, x₃, x₄)) = 1
POL(U133(x₁, x₂, x₃, x₄)) = 1
POL(U134(x₁, x₂, x₃, x₄)) = 1
POL(U135(x₁, x₂, x₃, x₄)) = 1
POL(U136(x₁, x₂, x₃, x₄)) = 1
POL(U21(x₁, x₂)) = 0
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁, x₂)) = 0
POL(U33(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂, x₃)) = 0
POL(U43(x₁, x₂, x₃)) = 0
POL(U44(x₁, x₂, x₃)) = 0
POL(U45(x₁, x₂)) = 0
POL(U46(x₁)) = 0
POL(U51(x₁, x₂)) = 0
POL(U52(x₁)) = 0
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁)) = 0
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃)) = 1
POL(U92(x₁, x₂, x₃)) = 1
POL(U93(x₁, x₂, x₃)) = 0
POL(U94(x₁, x₂, x₃)) = 0
POL(U95(x₁, x₂)) = 0
POL(U96(x₁)) = 0
POL(a__U101(x₁, x₂, x₃)) = 1
POL(a__U102(x₁, x₂, x₃)) = 1
POL(a__U103(x₁, x₂, x₃)) = 1
POL(a__U104(x₁, x₂, x₃)) = 1
POL(a__U105(x₁, x₂)) = 1
POL(a__U106(x₁)) = 0
POL(a__U11(x₁, x₂)) = 0
POL(a__U111(x₁, x₂, x₃)) = 1
POL(a__U112(x₁, x₂, x₃)) = 1
POL(a__U113(x₁, x₂, x₃)) = 1
POL(a__U114(x₁, x₂)) = 1
POL(a__U12(x₁, x₂)) = 0
POL(a__U121(x₁, x₂)) = 0
POL(a__U122(x₁)) = 0
POL(a__U13(x₁)) = 0
POL(a__U131(x₁, x₂, x₃, x₄)) = 1
POL(a__U132(x₁, x₂, x₃, x₄)) = 1
POL(a__U133(x₁, x₂, x₃, x₄)) = 1
POL(a__U134(x₁, x₂, x₃, x₄)) = 1
POL(a__U135(x₁, x₂, x₃, x₄)) = 1
POL(a__U136(x₁, x₂, x₃, x₄)) = 1
POL(a__U21(x₁, x₂)) = 0
POL(a__U22(x₁, x₂)) = 0
POL(a__U23(x₁)) = 0
POL(a__U31(x₁, x₂)) = 0
POL(a__U32(x₁, x₂)) = 0
POL(a__U33(x₁)) = 0
POL(a__U41(x₁, x₂, x₃)) = 0
POL(a__U42(x₁, x₂, x₃)) = 0
POL(a__U43(x₁, x₂, x₃)) = 0
POL(a__U44(x₁, x₂, x₃)) = 0
POL(a__U45(x₁, x₂)) = 0
POL(a__U46(x₁)) = 0
POL(a__U51(x₁, x₂)) = 0
POL(a__U52(x₁)) = 0
POL(a__U61(x₁, x₂)) = 0
POL(a__U62(x₁)) = 0
POL(a__U71(x₁)) = 0
POL(a__U81(x₁)) = 0
POL(a__U91(x₁, x₂, x₃)) = 1
POL(a__U92(x₁, x₂, x₃)) = 1
POL(a__U93(x₁, x₂, x₃)) = 0
POL(a__U94(x₁, x₂, x₃)) = 0
POL(a__U95(x₁, x₂)) = 0
POL(a__U96(x₁)) = 0
POL(a__isNat(x₁)) = 0
POL(a__isNatIList(x₁)) = 0
POL(a__isNatIListKind(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__isNatList(x₁)) = 1
POL(a__length(x₁)) = 1
POL(a__take(x₁, x₂)) = 1
POL(a__zeros) = 0
POL(cons(x₁, x₂)) = 1
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 1
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 1
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U113(tt, L, N) → a__U114(a__isNatKind(N), L)
a__U114(tt, L) → s(a__length(mark(L)))
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__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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__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__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__U122(tt) → nil
a__U13(tt) → tt
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U33(tt) → tt
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U23(tt) → tt
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U62(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U46(tt) → tt
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U103(tt, V1, V2) → a__U104(a__isNatIListKind(V2), V1, V2)
a__U102(tt, V1, V2) → a__U103(a__isNatIListKind(V2), V1, V2)
a__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U104(tt, V1, V2) → a__U105(a__isNat(V1), V2)
a__U101(tt, V1, V2) → a__U102(a__isNatKind(V1), V1, V2)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U106(X)) → a__U106(mark(X))
mark(U112(X1, X2, X3)) → a__U112(mark(X1), X2, X3)
mark(U113(X1, X2, X3)) → a__U113(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U111(X1, X2, X3)) → a__U111(mark(X1), X2, X3)
mark(U13(X)) → a__U13(mark(X))
mark(U114(X1, X2)) → a__U114(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U132(X1, X2, X3, X4)) → a__U132(mark(X1), X2, X3, X4)
mark(U131(X1, X2, X3, X4)) → a__U131(mark(X1), X2, X3, X4)
mark(U122(X)) → a__U122(mark(X))
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U136(X1, X2, X3, X4)) → a__U136(mark(X1), X2, X3, X4)
mark(U135(X1, X2, X3, X4)) → a__U135(mark(X1), X2, X3, X4)
mark(U134(X1, X2, X3, X4)) → a__U134(mark(X1), X2, X3, X4)
mark(U133(X1, X2, X3, X4)) → a__U133(mark(X1), X2, X3, X4)
mark(U23(X)) → a__U23(mark(X))
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U33(X)) → a__U33(mark(X))
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
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(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(0) → tt
a__isNatList(nil) → tt
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
a__length(nil) → 0
a__take(s(M), cons(N, IL)) → a__U131(a__isNatIList(IL), IL, M, N)
a__take(0, IL) → a__U121(a__isNatIList(IL), IL)
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U122(X) → U122(X)
a__U121(X1, X2) → U121(X1, X2)
a__isNatList(X) → isNatList(X)
a__U13(X) → U13(X)
a__length(X) → length(X)
a__U114(X1, X2) → U114(X1, X2)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__U11(X1, X2) → U11(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U95(X1, X2)) → a__U95(mark(X1), X2)
mark(U94(X1, X2, X3)) → a__U94(mark(X1), X2, X3)
mark(U71(X)) → a__U71(mark(X))
mark(U62(X)) → a__U62(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U81(X)) → a__U81(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
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__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)
mark(s(X)) → s(mark(X))
a__U101(X1, X2, X3) → U101(X1, X2, X3)
mark(U96(X)) → a__U96(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__U95(X1, X2) → U95(X1, X2)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U96(X) → U96(X)
a__U61(X1, X2) → U61(X1, X2)
a__U52(X) → U52(X)
a__U71(X) → U71(X)
a__U62(X) → U62(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U81(X) → U81(X)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ QDPOrderProof

                                                        ↳ QDP

                                                          ↳ QDPOrderProof

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

A__U114(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U114(tt, U102(x0, x1, x2)) → A__LENGTH(a__U102(mark(x0), x1, x2))
A__U114(tt, U114(x0, x1)) → A__LENGTH(a__U114(mark(x0), x1))
A__U114(tt, U111(x0, x1, x2)) → A__LENGTH(a__U111(mark(x0), x1, x2))
A__U114(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__U114(tt, U113(x0, x1, x2)) → A__LENGTH(a__U113(mark(x0), x1, x2))
A__U114(tt, U133(x0, x1, x2, x3)) → A__LENGTH(a__U133(mark(x0), x1, x2, x3))
A__U114(tt, U92(x0, x1, x2)) → A__LENGTH(a__U92(mark(x0), x1, x2))
A__U114(tt, U134(x0, x1, x2, x3)) → A__LENGTH(a__U134(mark(x0), x1, x2, x3))
A__U114(tt, U101(x0, x1, x2)) → A__LENGTH(a__U101(mark(x0), x1, x2))
A__U114(tt, U105(x0, x1)) → A__LENGTH(a__U105(mark(x0), x1))
A__U114(tt, U131(x0, x1, x2, x3)) → A__LENGTH(a__U131(mark(x0), x1, x2, x3))
A__U114(tt, U136(x0, x1, x2, x3)) → A__LENGTH(a__U136(mark(x0), x1, x2, x3))
A__U114(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U114(tt, U103(x0, x1, x2)) → A__LENGTH(a__U103(mark(x0), x1, x2))
A__U114(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
A__U114(tt, U135(x0, x1, x2, x3)) → A__LENGTH(a__U135(mark(x0), x1, x2, x3))
A__U113(tt, y0, 0) → A__U114(tt, y0)
A__U114(tt, U132(x0, x1, x2, x3)) → A__LENGTH(a__U132(mark(x0), x1, x2, x3))
A__U114(tt, U104(x0, x1, x2)) → A__LENGTH(a__U104(mark(x0), x1, x2))
A__U114(tt, U112(x0, x1, x2)) → A__LENGTH(a__U112(mark(x0), x1, x2))
A__U114(tt, U91(x0, x1, x2)) → A__LENGTH(a__U91(mark(x0), x1, x2))
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)
A__U114(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))

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.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A__U114(tt, U102(x0, x1, x2)) → A__LENGTH(a__U102(mark(x0), x1, x2))
A__U114(tt, U114(x0, x1)) → A__LENGTH(a__U114(mark(x0), x1))
A__U114(tt, U111(x0, x1, x2)) → A__LENGTH(a__U111(mark(x0), x1, x2))
A__U114(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U114(tt, U113(x0, x1, x2)) → A__LENGTH(a__U113(mark(x0), x1, x2))
A__U114(tt, U92(x0, x1, x2)) → A__LENGTH(a__U92(mark(x0), x1, x2))
A__U114(tt, U101(x0, x1, x2)) → A__LENGTH(a__U101(mark(x0), x1, x2))
A__U114(tt, U105(x0, x1)) → A__LENGTH(a__U105(mark(x0), x1))
A__U114(tt, U103(x0, x1, x2)) → A__LENGTH(a__U103(mark(x0), x1, x2))
A__U114(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))
A__U114(tt, U104(x0, x1, x2)) → A__LENGTH(a__U104(mark(x0), x1, x2))
A__U114(tt, U112(x0, x1, x2)) → A__LENGTH(a__U112(mark(x0), x1, x2))
A__U114(tt, U91(x0, x1, x2)) → A__LENGTH(a__U91(mark(x0), x1, x2))

The remaining pairs can at least be oriented weakly.

A__U114(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__U114(tt, U133(x0, x1, x2, x3)) → A__LENGTH(a__U133(mark(x0), x1, x2, x3))
A__U114(tt, U134(x0, x1, x2, x3)) → A__LENGTH(a__U134(mark(x0), x1, x2, x3))
A__U114(tt, U131(x0, x1, x2, x3)) → A__LENGTH(a__U131(mark(x0), x1, x2, x3))
A__U114(tt, U136(x0, x1, x2, x3)) → A__LENGTH(a__U136(mark(x0), x1, x2, x3))
A__U114(tt, zeros) → A__LENGTH(cons(0, zeros))
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
A__U114(tt, U135(x0, x1, x2, x3)) → A__LENGTH(a__U135(mark(x0), x1, x2, x3))
A__U113(tt, y0, 0) → A__U114(tt, y0)
A__U114(tt, U132(x0, x1, x2, x3)) → A__LENGTH(a__U132(mark(x0), x1, x2, x3))
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)
A__U114(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0
POL(A__LENGTH(x₁)) = x₁
POL(A__U111(x₁, x₂, x₃)) = 1
POL(A__U112(x₁, x₂, x₃)) = 1
POL(A__U113(x₁, x₂, x₃)) = 1
POL(A__U114(x₁, x₂)) = 1
POL(U101(x₁, x₂, x₃)) = 0
POL(U102(x₁, x₂, x₃)) = 0
POL(U103(x₁, x₂, x₃)) = 0
POL(U104(x₁, x₂, x₃)) = 0
POL(U105(x₁, x₂)) = 0
POL(U106(x₁)) = 0
POL(U11(x₁, x₂)) = 0
POL(U111(x₁, x₂, x₃)) = 0
POL(U112(x₁, x₂, x₃)) = 0
POL(U113(x₁, x₂, x₃)) = 0
POL(U114(x₁, x₂)) = 0
POL(U12(x₁, x₂)) = 0
POL(U121(x₁, x₂)) = 0
POL(U122(x₁)) = 0
POL(U13(x₁)) = 0
POL(U131(x₁, x₂, x₃, x₄)) = 1
POL(U132(x₁, x₂, x₃, x₄)) = 1
POL(U133(x₁, x₂, x₃, x₄)) = 1
POL(U134(x₁, x₂, x₃, x₄)) = 1
POL(U135(x₁, x₂, x₃, x₄)) = 1
POL(U136(x₁, x₂, x₃, x₄)) = 1
POL(U21(x₁, x₂)) = 0
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁, x₂)) = 0
POL(U33(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 1
POL(U42(x₁, x₂, x₃)) = 1
POL(U43(x₁, x₂, x₃)) = 0
POL(U44(x₁, x₂, x₃)) = 0
POL(U45(x₁, x₂)) = 0
POL(U46(x₁)) = 0
POL(U51(x₁, x₂)) = 0
POL(U52(x₁)) = 0
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁)) = 0
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃)) = 0
POL(U92(x₁, x₂, x₃)) = 0
POL(U93(x₁, x₂, x₃)) = 0
POL(U94(x₁, x₂, x₃)) = 0
POL(U95(x₁, x₂)) = 0
POL(U96(x₁)) = 0
POL(a__U101(x₁, x₂, x₃)) = 0
POL(a__U102(x₁, x₂, x₃)) = 0
POL(a__U103(x₁, x₂, x₃)) = 0
POL(a__U104(x₁, x₂, x₃)) = 0
POL(a__U105(x₁, x₂)) = 0
POL(a__U106(x₁)) = 0
POL(a__U11(x₁, x₂)) = 0
POL(a__U111(x₁, x₂, x₃)) = 0
POL(a__U112(x₁, x₂, x₃)) = 0
POL(a__U113(x₁, x₂, x₃)) = 0
POL(a__U114(x₁, x₂)) = 0
POL(a__U12(x₁, x₂)) = 0
POL(a__U121(x₁, x₂)) = 0
POL(a__U122(x₁)) = 0
POL(a__U13(x₁)) = 0
POL(a__U131(x₁, x₂, x₃, x₄)) = 1
POL(a__U132(x₁, x₂, x₃, x₄)) = 1
POL(a__U133(x₁, x₂, x₃, x₄)) = 1
POL(a__U134(x₁, x₂, x₃, x₄)) = 1
POL(a__U135(x₁, x₂, x₃, x₄)) = 1
POL(a__U136(x₁, x₂, x₃, x₄)) = 1
POL(a__U21(x₁, x₂)) = 0
POL(a__U22(x₁, x₂)) = 0
POL(a__U23(x₁)) = 0
POL(a__U31(x₁, x₂)) = 0
POL(a__U32(x₁, x₂)) = 0
POL(a__U33(x₁)) = 0
POL(a__U41(x₁, x₂, x₃)) = 1
POL(a__U42(x₁, x₂, x₃)) = 1
POL(a__U43(x₁, x₂, x₃)) = 0
POL(a__U44(x₁, x₂, x₃)) = 0
POL(a__U45(x₁, x₂)) = 0
POL(a__U46(x₁)) = 0
POL(a__U51(x₁, x₂)) = 0
POL(a__U52(x₁)) = 0
POL(a__U61(x₁, x₂)) = 0
POL(a__U62(x₁)) = 0
POL(a__U71(x₁)) = 0
POL(a__U81(x₁)) = 0
POL(a__U91(x₁, x₂, x₃)) = 0
POL(a__U92(x₁, x₂, x₃)) = 0
POL(a__U93(x₁, x₂, x₃)) = 0
POL(a__U94(x₁, x₂, x₃)) = 0
POL(a__U95(x₁, x₂)) = 0
POL(a__U96(x₁)) = 0
POL(a__isNat(x₁)) = 0
POL(a__isNatIList(x₁)) = x₁
POL(a__isNatIListKind(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__isNatList(x₁)) = 0
POL(a__length(x₁)) = 0
POL(a__take(x₁, x₂)) = 1
POL(a__zeros) = 0
POL(cons(x₁, x₂)) = 1
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 1
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U113(tt, L, N) → a__U114(a__isNatKind(N), L)
a__U114(tt, L) → s(a__length(mark(L)))
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__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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__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__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__U122(tt) → nil
a__U13(tt) → tt
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U33(tt) → tt
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U23(tt) → tt
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U62(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U46(tt) → tt
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U103(tt, V1, V2) → a__U104(a__isNatIListKind(V2), V1, V2)
a__U102(tt, V1, V2) → a__U103(a__isNatIListKind(V2), V1, V2)
a__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U104(tt, V1, V2) → a__U105(a__isNat(V1), V2)
a__U101(tt, V1, V2) → a__U102(a__isNatKind(V1), V1, V2)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U106(X)) → a__U106(mark(X))
mark(U112(X1, X2, X3)) → a__U112(mark(X1), X2, X3)
mark(U113(X1, X2, X3)) → a__U113(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U111(X1, X2, X3)) → a__U111(mark(X1), X2, X3)
mark(U13(X)) → a__U13(mark(X))
mark(U114(X1, X2)) → a__U114(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U132(X1, X2, X3, X4)) → a__U132(mark(X1), X2, X3, X4)
mark(U131(X1, X2, X3, X4)) → a__U131(mark(X1), X2, X3, X4)
mark(U122(X)) → a__U122(mark(X))
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U136(X1, X2, X3, X4)) → a__U136(mark(X1), X2, X3, X4)
mark(U135(X1, X2, X3, X4)) → a__U135(mark(X1), X2, X3, X4)
mark(U134(X1, X2, X3, X4)) → a__U134(mark(X1), X2, X3, X4)
mark(U133(X1, X2, X3, X4)) → a__U133(mark(X1), X2, X3, X4)
mark(U23(X)) → a__U23(mark(X))
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U33(X)) → a__U33(mark(X))
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
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(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(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatList(nil) → tt
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
a__length(nil) → 0
a__take(s(M), cons(N, IL)) → a__U131(a__isNatIList(IL), IL, M, N)
a__take(0, IL) → a__U121(a__isNatIList(IL), IL)
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U122(X) → U122(X)
a__U121(X1, X2) → U121(X1, X2)
a__isNatList(X) → isNatList(X)
a__U13(X) → U13(X)
a__length(X) → length(X)
a__U114(X1, X2) → U114(X1, X2)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__U11(X1, X2) → U11(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U95(X1, X2)) → a__U95(mark(X1), X2)
mark(U94(X1, X2, X3)) → a__U94(mark(X1), X2, X3)
mark(U71(X)) → a__U71(mark(X))
mark(U62(X)) → a__U62(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U81(X)) → a__U81(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__U106(X) → U106(X)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
mark(s(X)) → s(mark(X))
a__U101(X1, X2, X3) → U101(X1, X2, X3)
mark(U96(X)) → a__U96(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__U95(X1, X2) → U95(X1, X2)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U96(X) → U96(X)
a__U61(X1, X2) → U61(X1, X2)
a__U52(X) → U52(X)
a__U71(X) → U71(X)
a__U62(X) → U62(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U81(X) → U81(X)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ QDPOrderProof

                                                        ↳ QDP

                                                          ↳ QDPOrderProof

                                                            ↳ QDP

                                                              ↳ QDPOrderProof

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

A__U114(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U114(tt, U131(x0, x1, x2, x3)) → A__LENGTH(a__U131(mark(x0), x1, x2, x3))
A__U114(tt, U136(x0, x1, x2, x3)) → A__LENGTH(a__U136(mark(x0), x1, x2, x3))
A__U114(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__U114(tt, U133(x0, x1, x2, x3)) → A__LENGTH(a__U133(mark(x0), x1, x2, x3))
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
A__U114(tt, U135(x0, x1, x2, x3)) → A__LENGTH(a__U135(mark(x0), x1, x2, x3))
A__U113(tt, y0, 0) → A__U114(tt, y0)
A__U114(tt, U132(x0, x1, x2, x3)) → A__LENGTH(a__U132(mark(x0), x1, x2, x3))
A__U114(tt, U134(x0, x1, x2, x3)) → A__LENGTH(a__U134(mark(x0), x1, x2, x3))
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)
A__U114(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))

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.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A__U114(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U114(tt, U132(x0, x1, x2, x3)) → A__LENGTH(a__U132(mark(x0), x1, x2, x3))

The remaining pairs can at least be oriented weakly.

A__U114(tt, U131(x0, x1, x2, x3)) → A__LENGTH(a__U131(mark(x0), x1, x2, x3))
A__U114(tt, U136(x0, x1, x2, x3)) → A__LENGTH(a__U136(mark(x0), x1, x2, x3))
A__U114(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__U114(tt, U133(x0, x1, x2, x3)) → A__LENGTH(a__U133(mark(x0), x1, x2, x3))
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
A__U114(tt, U135(x0, x1, x2, x3)) → A__LENGTH(a__U135(mark(x0), x1, x2, x3))
A__U113(tt, y0, 0) → A__U114(tt, y0)
A__U114(tt, U134(x0, x1, x2, x3)) → A__LENGTH(a__U134(mark(x0), x1, x2, x3))
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)
A__U114(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0
POL(A__LENGTH(x₁)) = x₁
POL(A__U111(x₁, x₂, x₃)) = 1 + x₂
POL(A__U112(x₁, x₂, x₃)) = 1 + x₂
POL(A__U113(x₁, x₂, x₃)) = 1 + x₂
POL(A__U114(x₁, x₂)) = 1 + x₂
POL(U101(x₁, x₂, x₃)) = 0
POL(U102(x₁, x₂, x₃)) = 0
POL(U103(x₁, x₂, x₃)) = 0
POL(U104(x₁, x₂, x₃)) = 0
POL(U105(x₁, x₂)) = 0
POL(U106(x₁)) = 0
POL(U11(x₁, x₂)) = 0
POL(U111(x₁, x₂, x₃)) = 1
POL(U112(x₁, x₂, x₃)) = 0
POL(U113(x₁, x₂, x₃)) = 0
POL(U114(x₁, x₂)) = 0
POL(U12(x₁, x₂)) = 0
POL(U121(x₁, x₂)) = 1
POL(U122(x₁)) = 0
POL(U13(x₁)) = 0
POL(U131(x₁, x₂, x₃, x₄)) = 0
POL(U132(x₁, x₂, x₃, x₄)) = 1
POL(U133(x₁, x₂, x₃, x₄)) = 0
POL(U134(x₁, x₂, x₃, x₄)) = 0
POL(U135(x₁, x₂, x₃, x₄)) = 0
POL(U136(x₁, x₂, x₃, x₄)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁, x₂)) = 0
POL(U33(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂, x₃)) = 0
POL(U43(x₁, x₂, x₃)) = 0
POL(U44(x₁, x₂, x₃)) = 0
POL(U45(x₁, x₂)) = 0
POL(U46(x₁)) = 0
POL(U51(x₁, x₂)) = 0
POL(U52(x₁)) = 0
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁)) = 0
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃)) = 0
POL(U92(x₁, x₂, x₃)) = 0
POL(U93(x₁, x₂, x₃)) = 0
POL(U94(x₁, x₂, x₃)) = 0
POL(U95(x₁, x₂)) = 0
POL(U96(x₁)) = 0
POL(a__U101(x₁, x₂, x₃)) = 0
POL(a__U102(x₁, x₂, x₃)) = 0
POL(a__U103(x₁, x₂, x₃)) = 0
POL(a__U104(x₁, x₂, x₃)) = 0
POL(a__U105(x₁, x₂)) = 0
POL(a__U106(x₁)) = 0
POL(a__U11(x₁, x₂)) = 0
POL(a__U111(x₁, x₂, x₃)) = 1
POL(a__U112(x₁, x₂, x₃)) = 0
POL(a__U113(x₁, x₂, x₃)) = 0
POL(a__U114(x₁, x₂)) = 0
POL(a__U12(x₁, x₂)) = 0
POL(a__U121(x₁, x₂)) = 1
POL(a__U122(x₁)) = 0
POL(a__U13(x₁)) = 0
POL(a__U131(x₁, x₂, x₃, x₄)) = 1
POL(a__U132(x₁, x₂, x₃, x₄)) = 1
POL(a__U133(x₁, x₂, x₃, x₄)) = 1
POL(a__U134(x₁, x₂, x₃, x₄)) = 1
POL(a__U135(x₁, x₂, x₃, x₄)) = 1
POL(a__U136(x₁, x₂, x₃, x₄)) = 1
POL(a__U21(x₁, x₂)) = 0
POL(a__U22(x₁, x₂)) = 0
POL(a__U23(x₁)) = 0
POL(a__U31(x₁, x₂)) = 0
POL(a__U32(x₁, x₂)) = 0
POL(a__U33(x₁)) = 0
POL(a__U41(x₁, x₂, x₃)) = 0
POL(a__U42(x₁, x₂, x₃)) = 0
POL(a__U43(x₁, x₂, x₃)) = 0
POL(a__U44(x₁, x₂, x₃)) = 0
POL(a__U45(x₁, x₂)) = 0
POL(a__U46(x₁)) = 0
POL(a__U51(x₁, x₂)) = 0
POL(a__U52(x₁)) = 0
POL(a__U61(x₁, x₂)) = 0
POL(a__U62(x₁)) = 0
POL(a__U71(x₁)) = 0
POL(a__U81(x₁)) = 0
POL(a__U91(x₁, x₂, x₃)) = 0
POL(a__U92(x₁, x₂, x₃)) = 0
POL(a__U93(x₁, x₂, x₃)) = 0
POL(a__U94(x₁, x₂, x₃)) = 0
POL(a__U95(x₁, x₂)) = 0
POL(a__U96(x₁)) = 0
POL(a__isNat(x₁)) = 0
POL(a__isNatIList(x₁)) = 0
POL(a__isNatIListKind(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__isNatList(x₁)) = 0
POL(a__length(x₁)) = 1
POL(a__take(x₁, x₂)) = 1
POL(a__zeros) = 0
POL(cons(x₁, x₂)) = 1 + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = 1 + x₁
POL(nil) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U113(tt, L, N) → a__U114(a__isNatKind(N), L)
a__U114(tt, L) → s(a__length(mark(L)))
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__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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__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__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__U122(tt) → nil
a__U13(tt) → tt
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U33(tt) → tt
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U23(tt) → tt
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U62(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U46(tt) → tt
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U103(tt, V1, V2) → a__U104(a__isNatIListKind(V2), V1, V2)
a__U102(tt, V1, V2) → a__U103(a__isNatIListKind(V2), V1, V2)
a__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U104(tt, V1, V2) → a__U105(a__isNat(V1), V2)
a__U101(tt, V1, V2) → a__U102(a__isNatKind(V1), V1, V2)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U106(X)) → a__U106(mark(X))
mark(U112(X1, X2, X3)) → a__U112(mark(X1), X2, X3)
mark(U113(X1, X2, X3)) → a__U113(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U111(X1, X2, X3)) → a__U111(mark(X1), X2, X3)
mark(U13(X)) → a__U13(mark(X))
mark(U114(X1, X2)) → a__U114(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U132(X1, X2, X3, X4)) → a__U132(mark(X1), X2, X3, X4)
mark(U131(X1, X2, X3, X4)) → a__U131(mark(X1), X2, X3, X4)
mark(U122(X)) → a__U122(mark(X))
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U136(X1, X2, X3, X4)) → a__U136(mark(X1), X2, X3, X4)
mark(U135(X1, X2, X3, X4)) → a__U135(mark(X1), X2, X3, X4)
mark(U134(X1, X2, X3, X4)) → a__U134(mark(X1), X2, X3, X4)
mark(U133(X1, X2, X3, X4)) → a__U133(mark(X1), X2, X3, X4)
mark(U23(X)) → a__U23(mark(X))
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U33(X)) → a__U33(mark(X))
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
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(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(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
a__length(nil) → 0
a__take(s(M), cons(N, IL)) → a__U131(a__isNatIList(IL), IL, M, N)
a__take(0, IL) → a__U121(a__isNatIList(IL), IL)
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U122(X) → U122(X)
a__U121(X1, X2) → U121(X1, X2)
a__U13(X) → U13(X)
a__length(X) → length(X)
a__U114(X1, X2) → U114(X1, X2)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__U11(X1, X2) → U11(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U95(X1, X2)) → a__U95(mark(X1), X2)
mark(U94(X1, X2, X3)) → a__U94(mark(X1), X2, X3)
mark(U71(X)) → a__U71(mark(X))
mark(U62(X)) → a__U62(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U81(X)) → a__U81(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__U106(X) → U106(X)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
mark(s(X)) → s(mark(X))
a__U101(X1, X2, X3) → U101(X1, X2, X3)
mark(U96(X)) → a__U96(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__U95(X1, X2) → U95(X1, X2)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U96(X) → U96(X)
a__U61(X1, X2) → U61(X1, X2)
a__U52(X) → U52(X)
a__U71(X) → U71(X)
a__U62(X) → U62(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U81(X) → U81(X)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ QDPOrderProof

                                                        ↳ QDP

                                                          ↳ QDPOrderProof

                                                            ↳ QDP

                                                              ↳ QDPOrderProof

                                                                ↳ QDP

                                                                  ↳ QDPOrderProof

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

A__U114(tt, U131(x0, x1, x2, x3)) → A__LENGTH(a__U131(mark(x0), x1, x2, x3))
A__U114(tt, U135(x0, x1, x2, x3)) → A__LENGTH(a__U135(mark(x0), x1, x2, x3))
A__U114(tt, U136(x0, x1, x2, x3)) → A__LENGTH(a__U136(mark(x0), x1, x2, x3))
A__U113(tt, y0, 0) → A__U114(tt, y0)
A__U114(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U114(tt, U134(x0, x1, x2, x3)) → A__LENGTH(a__U134(mark(x0), x1, x2, x3))
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__U114(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
A__U114(tt, U133(x0, x1, x2, x3)) → A__LENGTH(a__U133(mark(x0), x1, x2, x3))

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.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A__U114(tt, U134(x0, x1, x2, x3)) → A__LENGTH(a__U134(mark(x0), x1, x2, x3))

The remaining pairs can at least be oriented weakly.

A__U114(tt, U131(x0, x1, x2, x3)) → A__LENGTH(a__U131(mark(x0), x1, x2, x3))
A__U114(tt, U135(x0, x1, x2, x3)) → A__LENGTH(a__U135(mark(x0), x1, x2, x3))
A__U114(tt, U136(x0, x1, x2, x3)) → A__LENGTH(a__U136(mark(x0), x1, x2, x3))
A__U113(tt, y0, 0) → A__U114(tt, y0)
A__U114(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__U114(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
A__U114(tt, U133(x0, x1, x2, x3)) → A__LENGTH(a__U133(mark(x0), x1, x2, x3))

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0
POL(A__LENGTH(x₁)) = x₁
POL(A__U111(x₁, x₂, x₃)) = 1 + x₂
POL(A__U112(x₁, x₂, x₃)) = 1 + x₂
POL(A__U113(x₁, x₂, x₃)) = 1 + x₂
POL(A__U114(x₁, x₂)) = 1 + x₂
POL(U101(x₁, x₂, x₃)) = 0
POL(U102(x₁, x₂, x₃)) = 0
POL(U103(x₁, x₂, x₃)) = 0
POL(U104(x₁, x₂, x₃)) = 0
POL(U105(x₁, x₂)) = 0
POL(U106(x₁)) = 0
POL(U11(x₁, x₂)) = 0
POL(U111(x₁, x₂, x₃)) = 1
POL(U112(x₁, x₂, x₃)) = 0
POL(U113(x₁, x₂, x₃)) = 1
POL(U114(x₁, x₂)) = 0
POL(U12(x₁, x₂)) = 0
POL(U121(x₁, x₂)) = 0
POL(U122(x₁)) = 0
POL(U13(x₁)) = 0
POL(U131(x₁, x₂, x₃, x₄)) = 1 + x₁
POL(U132(x₁, x₂, x₃, x₄)) = 1
POL(U133(x₁, x₂, x₃, x₄)) = 0
POL(U134(x₁, x₂, x₃, x₄)) = 1
POL(U135(x₁, x₂, x₃, x₄)) = 0
POL(U136(x₁, x₂, x₃, x₄)) = 0
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁, x₂)) = 0
POL(U33(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂, x₃)) = 0
POL(U43(x₁, x₂, x₃)) = 0
POL(U44(x₁, x₂, x₃)) = 0
POL(U45(x₁, x₂)) = 0
POL(U46(x₁)) = 0
POL(U51(x₁, x₂)) = 0
POL(U52(x₁)) = 0
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁)) = 0
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃)) = 1 + x₃
POL(U92(x₁, x₂, x₃)) = 1 + x₃
POL(U93(x₁, x₂, x₃)) = x₃
POL(U94(x₁, x₂, x₃)) = x₃
POL(U95(x₁, x₂)) = 0
POL(U96(x₁)) = 0
POL(a__U101(x₁, x₂, x₃)) = 0
POL(a__U102(x₁, x₂, x₃)) = 0
POL(a__U103(x₁, x₂, x₃)) = 0
POL(a__U104(x₁, x₂, x₃)) = 0
POL(a__U105(x₁, x₂)) = 0
POL(a__U106(x₁)) = 0
POL(a__U11(x₁, x₂)) = 0
POL(a__U111(x₁, x₂, x₃)) = 1
POL(a__U112(x₁, x₂, x₃)) = 1
POL(a__U113(x₁, x₂, x₃)) = 1
POL(a__U114(x₁, x₂)) = 0
POL(a__U12(x₁, x₂)) = 0
POL(a__U121(x₁, x₂)) = 0
POL(a__U122(x₁)) = 0
POL(a__U13(x₁)) = 0
POL(a__U131(x₁, x₂, x₃, x₄)) = 1 + x₁
POL(a__U132(x₁, x₂, x₃, x₄)) = 1
POL(a__U133(x₁, x₂, x₃, x₄)) = 1
POL(a__U134(x₁, x₂, x₃, x₄)) = 1
POL(a__U135(x₁, x₂, x₃, x₄)) = 1
POL(a__U136(x₁, x₂, x₃, x₄)) = 1
POL(a__U21(x₁, x₂)) = x₁
POL(a__U22(x₁, x₂)) = 0
POL(a__U23(x₁)) = 0
POL(a__U31(x₁, x₂)) = 0
POL(a__U32(x₁, x₂)) = 0
POL(a__U33(x₁)) = 0
POL(a__U41(x₁, x₂, x₃)) = 0
POL(a__U42(x₁, x₂, x₃)) = 0
POL(a__U43(x₁, x₂, x₃)) = 0
POL(a__U44(x₁, x₂, x₃)) = 0
POL(a__U45(x₁, x₂)) = 0
POL(a__U46(x₁)) = 0
POL(a__U51(x₁, x₂)) = 0
POL(a__U52(x₁)) = 0
POL(a__U61(x₁, x₂)) = 0
POL(a__U62(x₁)) = 0
POL(a__U71(x₁)) = 0
POL(a__U81(x₁)) = 0
POL(a__U91(x₁, x₂, x₃)) = 1 + x₃
POL(a__U92(x₁, x₂, x₃)) = 1 + x₃
POL(a__U93(x₁, x₂, x₃)) = 1 + x₃
POL(a__U94(x₁, x₂, x₃)) = x₃
POL(a__U95(x₁, x₂)) = 0
POL(a__U96(x₁)) = 0
POL(a__isNat(x₁)) = 0
POL(a__isNatIList(x₁)) = 0
POL(a__isNatIListKind(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__isNatList(x₁)) = x₁
POL(a__length(x₁)) = 1
POL(a__take(x₁, x₂)) = 1
POL(a__zeros) = 1
POL(cons(x₁, x₂)) = 1 + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 1
POL(mark(x₁)) = 1 + x₁
POL(nil) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U113(tt, L, N) → a__U114(a__isNatKind(N), L)
a__U114(tt, L) → s(a__length(mark(L)))
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__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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__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__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__U122(tt) → nil
a__U13(tt) → tt
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U33(tt) → tt
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U23(tt) → tt
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U62(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U46(tt) → tt
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U103(tt, V1, V2) → a__U104(a__isNatIListKind(V2), V1, V2)
a__U102(tt, V1, V2) → a__U103(a__isNatIListKind(V2), V1, V2)
a__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U104(tt, V1, V2) → a__U105(a__isNat(V1), V2)
a__U101(tt, V1, V2) → a__U102(a__isNatKind(V1), V1, V2)
a__zeros → cons(0, zeros)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(U112(X1, X2, X3)) → a__U112(mark(X1), X2, X3)
mark(U113(X1, X2, X3)) → a__U113(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U111(X1, X2, X3)) → a__U111(mark(X1), X2, X3)
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U114(X1, X2)) → a__U114(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U132(X1, X2, X3, X4)) → a__U132(mark(X1), X2, X3, X4)
mark(U131(X1, X2, X3, X4)) → a__U131(mark(X1), X2, X3, X4)
mark(U122(X)) → a__U122(mark(X))
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U136(X1, X2, X3, X4)) → a__U136(mark(X1), X2, X3, X4)
mark(U135(X1, X2, X3, X4)) → a__U135(mark(X1), X2, X3, X4)
mark(U134(X1, X2, X3, X4)) → a__U134(mark(X1), X2, X3, X4)
mark(U133(X1, X2, X3, X4)) → a__U133(mark(X1), X2, X3, X4)
mark(U23(X)) → a__U23(mark(X))
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U33(X)) → a__U33(mark(X))
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
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(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(0) → tt
a__isNatList(nil) → tt
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
a__length(nil) → 0
a__take(s(M), cons(N, IL)) → a__U131(a__isNatIList(IL), IL, M, N)
a__take(0, IL) → a__U121(a__isNatIList(IL), IL)
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(zeros) → a__zeros
mark(isNatKind(X)) → a__isNatKind(X)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U122(X) → U122(X)
a__U121(X1, X2) → U121(X1, X2)
a__isNatList(X) → isNatList(X)
a__U13(X) → U13(X)
a__length(X) → length(X)
a__U114(X1, X2) → U114(X1, X2)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__U11(X1, X2) → U11(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U95(X1, X2)) → a__U95(mark(X1), X2)
mark(U94(X1, X2, X3)) → a__U94(mark(X1), X2, X3)
mark(U71(X)) → a__U71(mark(X))
mark(U62(X)) → a__U62(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U81(X)) → a__U81(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
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__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)
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
mark(U96(X)) → a__U96(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
a__U95(X1, X2) → U95(X1, X2)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U96(X) → U96(X)
a__U61(X1, X2) → U61(X1, X2)
a__U52(X) → U52(X)
a__U71(X) → U71(X)
a__U62(X) → U62(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U81(X) → U81(X)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ QDPOrderProof

                                                        ↳ QDP

                                                          ↳ QDPOrderProof

                                                            ↳ QDP

                                                              ↳ QDPOrderProof

                                                                ↳ QDP

                                                                  ↳ QDPOrderProof

                                                                    ↳ QDP

                                                                      ↳ QDPOrderProof

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

A__U114(tt, U131(x0, x1, x2, x3)) → A__LENGTH(a__U131(mark(x0), x1, x2, x3))
A__U114(tt, U136(x0, x1, x2, x3)) → A__LENGTH(a__U136(mark(x0), x1, x2, x3))
A__U114(tt, U135(x0, x1, x2, x3)) → A__LENGTH(a__U135(mark(x0), x1, x2, x3))
A__U114(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U113(tt, y0, 0) → A__U114(tt, y0)
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__U114(tt, U133(x0, x1, x2, x3)) → A__LENGTH(a__U133(mark(x0), x1, x2, x3))
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)
A__U114(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))

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.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A__U114(tt, U133(x0, x1, x2, x3)) → A__LENGTH(a__U133(mark(x0), x1, x2, x3))

The remaining pairs can at least be oriented weakly.

A__U114(tt, U131(x0, x1, x2, x3)) → A__LENGTH(a__U131(mark(x0), x1, x2, x3))
A__U114(tt, U136(x0, x1, x2, x3)) → A__LENGTH(a__U136(mark(x0), x1, x2, x3))
A__U114(tt, U135(x0, x1, x2, x3)) → A__LENGTH(a__U135(mark(x0), x1, x2, x3))
A__U114(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U113(tt, y0, 0) → A__U114(tt, y0)
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)
A__U114(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0
POL(A__LENGTH(x₁)) = x₁
POL(A__U111(x₁, x₂, x₃)) = 1 + x₂
POL(A__U112(x₁, x₂, x₃)) = 1 + x₂
POL(A__U113(x₁, x₂, x₃)) = 1 + x₂
POL(A__U114(x₁, x₂)) = 1 + x₂
POL(U101(x₁, x₂, x₃)) = 0
POL(U102(x₁, x₂, x₃)) = 0
POL(U103(x₁, x₂, x₃)) = 0
POL(U104(x₁, x₂, x₃)) = 0
POL(U105(x₁, x₂)) = 0
POL(U106(x₁)) = 0
POL(U11(x₁, x₂)) = 0
POL(U111(x₁, x₂, x₃)) = 0
POL(U112(x₁, x₂, x₃)) = 0
POL(U113(x₁, x₂, x₃)) = 0
POL(U114(x₁, x₂)) = 0
POL(U12(x₁, x₂)) = 0
POL(U121(x₁, x₂)) = 0
POL(U122(x₁)) = 0
POL(U13(x₁)) = 0
POL(U131(x₁, x₂, x₃, x₄)) = 0
POL(U132(x₁, x₂, x₃, x₄)) = 0
POL(U133(x₁, x₂, x₃, x₄)) = 1
POL(U134(x₁, x₂, x₃, x₄)) = 1
POL(U135(x₁, x₂, x₃, x₄)) = 0
POL(U136(x₁, x₂, x₃, x₄)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁, x₂)) = 0
POL(U33(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂, x₃)) = 0
POL(U43(x₁, x₂, x₃)) = 0
POL(U44(x₁, x₂, x₃)) = 0
POL(U45(x₁, x₂)) = 0
POL(U46(x₁)) = 0
POL(U51(x₁, x₂)) = 0
POL(U52(x₁)) = 0
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁)) = 0
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃)) = 0
POL(U92(x₁, x₂, x₃)) = 0
POL(U93(x₁, x₂, x₃)) = 0
POL(U94(x₁, x₂, x₃)) = 0
POL(U95(x₁, x₂)) = 0
POL(U96(x₁)) = 0
POL(a__U101(x₁, x₂, x₃)) = 0
POL(a__U102(x₁, x₂, x₃)) = 0
POL(a__U103(x₁, x₂, x₃)) = 0
POL(a__U104(x₁, x₂, x₃)) = 0
POL(a__U105(x₁, x₂)) = 0
POL(a__U106(x₁)) = 0
POL(a__U11(x₁, x₂)) = 0
POL(a__U111(x₁, x₂, x₃)) = 0
POL(a__U112(x₁, x₂, x₃)) = 0
POL(a__U113(x₁, x₂, x₃)) = 0
POL(a__U114(x₁, x₂)) = 0
POL(a__U12(x₁, x₂)) = 0
POL(a__U121(x₁, x₂)) = 0
POL(a__U122(x₁)) = 0
POL(a__U13(x₁)) = 0
POL(a__U131(x₁, x₂, x₃, x₄)) = 1
POL(a__U132(x₁, x₂, x₃, x₄)) = 1
POL(a__U133(x₁, x₂, x₃, x₄)) = 1
POL(a__U134(x₁, x₂, x₃, x₄)) = 1
POL(a__U135(x₁, x₂, x₃, x₄)) = 1
POL(a__U136(x₁, x₂, x₃, x₄)) = 1
POL(a__U21(x₁, x₂)) = 0
POL(a__U22(x₁, x₂)) = 0
POL(a__U23(x₁)) = 0
POL(a__U31(x₁, x₂)) = 0
POL(a__U32(x₁, x₂)) = 0
POL(a__U33(x₁)) = 0
POL(a__U41(x₁, x₂, x₃)) = 0
POL(a__U42(x₁, x₂, x₃)) = 0
POL(a__U43(x₁, x₂, x₃)) = 0
POL(a__U44(x₁, x₂, x₃)) = 0
POL(a__U45(x₁, x₂)) = 0
POL(a__U46(x₁)) = 0
POL(a__U51(x₁, x₂)) = 0
POL(a__U52(x₁)) = 0
POL(a__U61(x₁, x₂)) = 0
POL(a__U62(x₁)) = 0
POL(a__U71(x₁)) = 0
POL(a__U81(x₁)) = 0
POL(a__U91(x₁, x₂, x₃)) = 0
POL(a__U92(x₁, x₂, x₃)) = 0
POL(a__U93(x₁, x₂, x₃)) = 0
POL(a__U94(x₁, x₂, x₃)) = 0
POL(a__U95(x₁, x₂)) = 0
POL(a__U96(x₁)) = 0
POL(a__isNat(x₁)) = 0
POL(a__isNatIList(x₁)) = 0
POL(a__isNatIListKind(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__isNatList(x₁)) = 0
POL(a__length(x₁)) = 0
POL(a__take(x₁, x₂)) = 1
POL(a__zeros) = 0
POL(cons(x₁, x₂)) = 1 + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = 1 + x₁
POL(nil) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U113(tt, L, N) → a__U114(a__isNatKind(N), L)
a__U114(tt, L) → s(a__length(mark(L)))
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__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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__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__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__U122(tt) → nil
a__U13(tt) → tt
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U33(tt) → tt
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U23(tt) → tt
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U62(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U46(tt) → tt
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U103(tt, V1, V2) → a__U104(a__isNatIListKind(V2), V1, V2)
a__U102(tt, V1, V2) → a__U103(a__isNatIListKind(V2), V1, V2)
a__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U104(tt, V1, V2) → a__U105(a__isNat(V1), V2)
a__U101(tt, V1, V2) → a__U102(a__isNatKind(V1), V1, V2)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U106(X)) → a__U106(mark(X))
mark(U112(X1, X2, X3)) → a__U112(mark(X1), X2, X3)
mark(U113(X1, X2, X3)) → a__U113(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U111(X1, X2, X3)) → a__U111(mark(X1), X2, X3)
mark(U13(X)) → a__U13(mark(X))
mark(U114(X1, X2)) → a__U114(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U132(X1, X2, X3, X4)) → a__U132(mark(X1), X2, X3, X4)
mark(U131(X1, X2, X3, X4)) → a__U131(mark(X1), X2, X3, X4)
mark(U122(X)) → a__U122(mark(X))
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U136(X1, X2, X3, X4)) → a__U136(mark(X1), X2, X3, X4)
mark(U135(X1, X2, X3, X4)) → a__U135(mark(X1), X2, X3, X4)
mark(U134(X1, X2, X3, X4)) → a__U134(mark(X1), X2, X3, X4)
mark(U133(X1, X2, X3, X4)) → a__U133(mark(X1), X2, X3, X4)
mark(U23(X)) → a__U23(mark(X))
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U33(X)) → a__U33(mark(X))
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
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(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(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
a__length(nil) → 0
a__take(s(M), cons(N, IL)) → a__U131(a__isNatIList(IL), IL, M, N)
a__take(0, IL) → a__U121(a__isNatIList(IL), IL)
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U122(X) → U122(X)
a__U121(X1, X2) → U121(X1, X2)
a__U13(X) → U13(X)
a__length(X) → length(X)
a__U114(X1, X2) → U114(X1, X2)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__U11(X1, X2) → U11(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U95(X1, X2)) → a__U95(mark(X1), X2)
mark(U94(X1, X2, X3)) → a__U94(mark(X1), X2, X3)
mark(U71(X)) → a__U71(mark(X))
mark(U62(X)) → a__U62(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U81(X)) → a__U81(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__U106(X) → U106(X)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
mark(s(X)) → s(mark(X))
a__U101(X1, X2, X3) → U101(X1, X2, X3)
mark(U96(X)) → a__U96(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__U95(X1, X2) → U95(X1, X2)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U96(X) → U96(X)
a__U61(X1, X2) → U61(X1, X2)
a__U52(X) → U52(X)
a__U71(X) → U71(X)
a__U62(X) → U62(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U81(X) → U81(X)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ QDPOrderProof

                                                        ↳ QDP

                                                          ↳ QDPOrderProof

                                                            ↳ QDP

                                                              ↳ QDPOrderProof

                                                                ↳ QDP

                                                                  ↳ QDPOrderProof

                                                                    ↳ QDP

                                                                      ↳ QDPOrderProof

                                                                        ↳ QDP

                                                                          ↳ QDPOrderProof

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

A__U114(tt, U131(x0, x1, x2, x3)) → A__LENGTH(a__U131(mark(x0), x1, x2, x3))
A__U114(tt, U135(x0, x1, x2, x3)) → A__LENGTH(a__U135(mark(x0), x1, x2, x3))
A__U114(tt, U136(x0, x1, x2, x3)) → A__LENGTH(a__U136(mark(x0), x1, x2, x3))
A__U113(tt, y0, 0) → A__U114(tt, y0)
A__U114(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__U114(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)

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.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A__U114(tt, U131(x0, x1, x2, x3)) → A__LENGTH(a__U131(mark(x0), x1, x2, x3))
A__U114(tt, U135(x0, x1, x2, x3)) → A__LENGTH(a__U135(mark(x0), x1, x2, x3))
A__U114(tt, U136(x0, x1, x2, x3)) → A__LENGTH(a__U136(mark(x0), x1, x2, x3))

The remaining pairs can at least be oriented weakly.

A__U113(tt, y0, 0) → A__U114(tt, y0)
A__U114(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__U114(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__LENGTH(x₁)) = x₁
POL(A__U111(x₁, x₂, x₃)) = 1 + x₂
POL(A__U112(x₁, x₂, x₃)) = 1 + x₂
POL(A__U113(x₁, x₂, x₃)) = 1 + x₂
POL(A__U114(x₁, x₂)) = 1 + x₂
POL(U101(x₁, x₂, x₃)) = 0
POL(U102(x₁, x₂, x₃)) = 0
POL(U103(x₁, x₂, x₃)) = 0
POL(U104(x₁, x₂, x₃)) = 0
POL(U105(x₁, x₂)) = 0
POL(U106(x₁)) = 0
POL(U11(x₁, x₂)) = 0
POL(U111(x₁, x₂, x₃)) = 0
POL(U112(x₁, x₂, x₃)) = 0
POL(U113(x₁, x₂, x₃)) = 0
POL(U114(x₁, x₂)) = 0
POL(U12(x₁, x₂)) = 0
POL(U121(x₁, x₂)) = 0
POL(U122(x₁)) = 0
POL(U13(x₁)) = 0
POL(U131(x₁, x₂, x₃, x₄)) = 1
POL(U132(x₁, x₂, x₃, x₄)) = 0
POL(U133(x₁, x₂, x₃, x₄)) = 0
POL(U134(x₁, x₂, x₃, x₄)) = 0
POL(U135(x₁, x₂, x₃, x₄)) = 1
POL(U136(x₁, x₂, x₃, x₄)) = 1
POL(U21(x₁, x₂)) = 0
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = x₂
POL(U32(x₁, x₂)) = x₂
POL(U33(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂, x₃)) = 0
POL(U43(x₁, x₂, x₃)) = 0
POL(U44(x₁, x₂, x₃)) = 0
POL(U45(x₁, x₂)) = 0
POL(U46(x₁)) = 0
POL(U51(x₁, x₂)) = 0
POL(U52(x₁)) = 0
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁)) = 0
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃)) = 0
POL(U92(x₁, x₂, x₃)) = 0
POL(U93(x₁, x₂, x₃)) = 0
POL(U94(x₁, x₂, x₃)) = 0
POL(U95(x₁, x₂)) = 0
POL(U96(x₁)) = 0
POL(a__U101(x₁, x₂, x₃)) = 0
POL(a__U102(x₁, x₂, x₃)) = 0
POL(a__U103(x₁, x₂, x₃)) = 0
POL(a__U104(x₁, x₂, x₃)) = 0
POL(a__U105(x₁, x₂)) = 0
POL(a__U106(x₁)) = 0
POL(a__U11(x₁, x₂)) = 0
POL(a__U111(x₁, x₂, x₃)) = 1
POL(a__U112(x₁, x₂, x₃)) = 1
POL(a__U113(x₁, x₂, x₃)) = 1
POL(a__U114(x₁, x₂)) = 1
POL(a__U12(x₁, x₂)) = 0
POL(a__U121(x₁, x₂)) = 0
POL(a__U122(x₁)) = 0
POL(a__U13(x₁)) = 0
POL(a__U131(x₁, x₂, x₃, x₄)) = 1
POL(a__U132(x₁, x₂, x₃, x₄)) = 1
POL(a__U133(x₁, x₂, x₃, x₄)) = 1
POL(a__U134(x₁, x₂, x₃, x₄)) = 1
POL(a__U135(x₁, x₂, x₃, x₄)) = 1
POL(a__U136(x₁, x₂, x₃, x₄)) = 1
POL(a__U21(x₁, x₂)) = 0
POL(a__U22(x₁, x₂)) = 0
POL(a__U23(x₁)) = 0
POL(a__U31(x₁, x₂)) = x₂
POL(a__U32(x₁, x₂)) = x₂
POL(a__U33(x₁)) = 0
POL(a__U41(x₁, x₂, x₃)) = 1
POL(a__U42(x₁, x₂, x₃)) = 1
POL(a__U43(x₁, x₂, x₃)) = 1
POL(a__U44(x₁, x₂, x₃)) = 1
POL(a__U45(x₁, x₂)) = 1
POL(a__U46(x₁)) = 1
POL(a__U51(x₁, x₂)) = 0
POL(a__U52(x₁)) = 0
POL(a__U61(x₁, x₂)) = 0
POL(a__U62(x₁)) = 0
POL(a__U71(x₁)) = 0
POL(a__U81(x₁)) = 0
POL(a__U91(x₁, x₂, x₃)) = 0
POL(a__U92(x₁, x₂, x₃)) = 0
POL(a__U93(x₁, x₂, x₃)) = 0
POL(a__U94(x₁, x₂, x₃)) = 0
POL(a__U95(x₁, x₂)) = 0
POL(a__U96(x₁)) = 0
POL(a__isNat(x₁)) = 0
POL(a__isNatIList(x₁)) = x₁
POL(a__isNatIListKind(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__isNatList(x₁)) = 0
POL(a__length(x₁)) = 1
POL(a__take(x₁, x₂)) = 1
POL(a__zeros) = 0
POL(cons(x₁, x₂)) = 1 + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = 1 + x₁
POL(nil) = 0
POL(s(x₁)) = 1
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(zeros) = 1

The following usable rules [17] were oriented:

a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U113(tt, L, N) → a__U114(a__isNatKind(N), L)
a__U114(tt, L) → s(a__length(mark(L)))
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__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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__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__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__U122(tt) → nil
a__U13(tt) → tt
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U33(tt) → tt
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U23(tt) → tt
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U62(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U46(tt) → tt
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U103(tt, V1, V2) → a__U104(a__isNatIListKind(V2), V1, V2)
a__U102(tt, V1, V2) → a__U103(a__isNatIListKind(V2), V1, V2)
a__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U104(tt, V1, V2) → a__U105(a__isNat(V1), V2)
a__U101(tt, V1, V2) → a__U102(a__isNatKind(V1), V1, V2)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U106(X)) → a__U106(mark(X))
mark(U112(X1, X2, X3)) → a__U112(mark(X1), X2, X3)
mark(U113(X1, X2, X3)) → a__U113(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U111(X1, X2, X3)) → a__U111(mark(X1), X2, X3)
mark(U13(X)) → a__U13(mark(X))
mark(U114(X1, X2)) → a__U114(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U132(X1, X2, X3, X4)) → a__U132(mark(X1), X2, X3, X4)
mark(U131(X1, X2, X3, X4)) → a__U131(mark(X1), X2, X3, X4)
mark(U122(X)) → a__U122(mark(X))
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U136(X1, X2, X3, X4)) → a__U136(mark(X1), X2, X3, X4)
mark(U135(X1, X2, X3, X4)) → a__U135(mark(X1), X2, X3, X4)
mark(U134(X1, X2, X3, X4)) → a__U134(mark(X1), X2, X3, X4)
mark(U133(X1, X2, X3, X4)) → a__U133(mark(X1), X2, X3, X4)
mark(U23(X)) → a__U23(mark(X))
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U33(X)) → a__U33(mark(X))
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
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(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(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
a__length(nil) → 0
a__take(s(M), cons(N, IL)) → a__U131(a__isNatIList(IL), IL, M, N)
a__take(0, IL) → a__U121(a__isNatIList(IL), IL)
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U122(X) → U122(X)
a__U121(X1, X2) → U121(X1, X2)
a__U13(X) → U13(X)
a__length(X) → length(X)
a__U114(X1, X2) → U114(X1, X2)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__U11(X1, X2) → U11(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U95(X1, X2)) → a__U95(mark(X1), X2)
mark(U94(X1, X2, X3)) → a__U94(mark(X1), X2, X3)
mark(U71(X)) → a__U71(mark(X))
mark(U62(X)) → a__U62(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U81(X)) → a__U81(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__U105(X1, X2) → U105(X1, X2)
a__U106(X) → U106(X)
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
mark(s(X)) → s(mark(X))
a__U101(X1, X2, X3) → U101(X1, X2, X3)
mark(U96(X)) → a__U96(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__U95(X1, X2) → U95(X1, X2)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U96(X) → U96(X)
a__U61(X1, X2) → U61(X1, X2)
a__U52(X) → U52(X)
a__U71(X) → U71(X)
a__U62(X) → U62(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U81(X) → U81(X)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ QDPOrderProof

                                                        ↳ QDP

                                                          ↳ QDPOrderProof

                                                            ↳ QDP

                                                              ↳ QDPOrderProof

                                                                ↳ QDP

                                                                  ↳ QDPOrderProof

                                                                    ↳ QDP

                                                                      ↳ QDPOrderProof

                                                                        ↳ QDP

                                                                          ↳ QDPOrderProof

                                                                            ↳ QDP

                                                                              ↳ QDPOrderProof

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

A__U114(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U113(tt, y0, 0) → A__U114(tt, y0)
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)
A__U114(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))

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.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A__U114(tt, zeros) → A__LENGTH(cons(0, zeros))

The remaining pairs can at least be oriented weakly.

A__U113(tt, y0, 0) → A__U114(tt, y0)
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)
A__U114(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 1
POL(A__LENGTH(x₁)) = x₁
POL(A__U111(x₁, x₂, x₃)) = x₁
POL(A__U112(x₁, x₂, x₃)) = x₁
POL(A__U113(x₁, x₂, x₃)) = 1
POL(A__U114(x₁, x₂)) = 1
POL(U101(x₁, x₂, x₃)) = 1
POL(U102(x₁, x₂, x₃)) = 1
POL(U103(x₁, x₂, x₃)) = 1
POL(U104(x₁, x₂, x₃)) = 1
POL(U105(x₁, x₂)) = 1
POL(U106(x₁)) = 1
POL(U11(x₁, x₂)) = 1
POL(U111(x₁, x₂, x₃)) = 1
POL(U112(x₁, x₂, x₃)) = 1
POL(U113(x₁, x₂, x₃)) = 1
POL(U114(x₁, x₂)) = 1
POL(U12(x₁, x₂)) = x₁
POL(U121(x₁, x₂)) = 1
POL(U122(x₁)) = 1
POL(U13(x₁)) = 1
POL(U131(x₁, x₂, x₃, x₄)) = 1
POL(U132(x₁, x₂, x₃, x₄)) = x₁
POL(U133(x₁, x₂, x₃, x₄)) = 1
POL(U134(x₁, x₂, x₃, x₄)) = 1
POL(U135(x₁, x₂, x₃, x₄)) = 1
POL(U136(x₁, x₂, x₃, x₄)) = 1
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁, x₂)) = 1
POL(U23(x₁)) = 1
POL(U31(x₁, x₂)) = x₂
POL(U32(x₁, x₂)) = x₂
POL(U33(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = 1 + x₃
POL(U42(x₁, x₂, x₃)) = 1 + x₃
POL(U43(x₁, x₂, x₃)) = 1 + x₃
POL(U44(x₁, x₂, x₃)) = 1 + x₃
POL(U45(x₁, x₂)) = 1 + x₂
POL(U46(x₁)) = x₁
POL(U51(x₁, x₂)) = 1
POL(U52(x₁)) = 1
POL(U61(x₁, x₂)) = 1
POL(U62(x₁)) = 1
POL(U71(x₁)) = x₁
POL(U81(x₁)) = x₁
POL(U91(x₁, x₂, x₃)) = x₃
POL(U92(x₁, x₂, x₃)) = x₃
POL(U93(x₁, x₂, x₃)) = x₃
POL(U94(x₁, x₂, x₃)) = x₃
POL(U95(x₁, x₂)) = x₂
POL(U96(x₁)) = x₁
POL(a__U101(x₁, x₂, x₃)) = 1
POL(a__U102(x₁, x₂, x₃)) = 1
POL(a__U103(x₁, x₂, x₃)) = 1
POL(a__U104(x₁, x₂, x₃)) = 1
POL(a__U105(x₁, x₂)) = 1
POL(a__U106(x₁)) = 1
POL(a__U11(x₁, x₂)) = 1
POL(a__U111(x₁, x₂, x₃)) = 1
POL(a__U112(x₁, x₂, x₃)) = 1
POL(a__U113(x₁, x₂, x₃)) = 1
POL(a__U114(x₁, x₂)) = 1
POL(a__U12(x₁, x₂)) = x₁
POL(a__U121(x₁, x₂)) = 1
POL(a__U122(x₁)) = 1
POL(a__U13(x₁)) = 1
POL(a__U131(x₁, x₂, x₃, x₄)) = 1
POL(a__U132(x₁, x₂, x₃, x₄)) = x₁
POL(a__U133(x₁, x₂, x₃, x₄)) = 1
POL(a__U134(x₁, x₂, x₃, x₄)) = 1
POL(a__U135(x₁, x₂, x₃, x₄)) = 1
POL(a__U136(x₁, x₂, x₃, x₄)) = 1
POL(a__U21(x₁, x₂)) = x₁
POL(a__U22(x₁, x₂)) = 1
POL(a__U23(x₁)) = 1
POL(a__U31(x₁, x₂)) = x₂
POL(a__U32(x₁, x₂)) = x₂
POL(a__U33(x₁)) = x₁
POL(a__U41(x₁, x₂, x₃)) = 1 + x₃
POL(a__U42(x₁, x₂, x₃)) = 1 + x₃
POL(a__U43(x₁, x₂, x₃)) = 1 + x₃
POL(a__U44(x₁, x₂, x₃)) = 1 + x₃
POL(a__U45(x₁, x₂)) = 1 + x₂
POL(a__U46(x₁)) = x₁
POL(a__U51(x₁, x₂)) = 1
POL(a__U52(x₁)) = 1
POL(a__U61(x₁, x₂)) = 1
POL(a__U62(x₁)) = 1
POL(a__U71(x₁)) = x₁
POL(a__U81(x₁)) = x₁
POL(a__U91(x₁, x₂, x₃)) = x₃
POL(a__U92(x₁, x₂, x₃)) = x₃
POL(a__U93(x₁, x₂, x₃)) = x₃
POL(a__U94(x₁, x₂, x₃)) = x₃
POL(a__U95(x₁, x₂)) = x₂
POL(a__U96(x₁)) = x₁
POL(a__isNat(x₁)) = x₁
POL(a__isNatIList(x₁)) = 1 + x₁
POL(a__isNatIListKind(x₁)) = 1
POL(a__isNatKind(x₁)) = x₁
POL(a__isNatList(x₁)) = x₁
POL(a__length(x₁)) = 1
POL(a__take(x₁, x₂)) = 1
POL(a__zeros) = 0
POL(cons(x₁, x₂)) = x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 1 + x₁
POL(isNatIListKind(x₁)) = 1
POL(isNatKind(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 1
POL(mark(x₁)) = x₁
POL(nil) = 1
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 1
POL(tt) = 1
POL(zeros) = 0

The following usable rules [17] were oriented:

a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U121(tt, IL) → a__U122(a__isNatIListKind(IL))
a__U113(tt, L, N) → a__U114(a__isNatKind(N), L)
a__U114(tt, L) → s(a__length(mark(L)))
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__U106(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
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__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__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__U122(tt) → nil
a__U13(tt) → tt
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U33(tt) → tt
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U23(tt) → tt
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U62(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U52(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U46(tt) → tt
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U103(tt, V1, V2) → a__U104(a__isNatIListKind(V2), V1, V2)
a__U102(tt, V1, V2) → a__U103(a__isNatIListKind(V2), V1, V2)
a__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U104(tt, V1, V2) → a__U105(a__isNat(V1), V2)
a__U101(tt, V1, V2) → a__U102(a__isNatKind(V1), V1, V2)
a__zeros → cons(0, zeros)
mark(U104(X1, X2, X3)) → a__U104(mark(X1), X2, X3)
mark(U105(X1, X2)) → a__U105(mark(X1), X2)
mark(U103(X1, X2, X3)) → a__U103(mark(X1), X2, X3)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U106(X)) → a__U106(mark(X))
mark(U112(X1, X2, X3)) → a__U112(mark(X1), X2, X3)
mark(U113(X1, X2, X3)) → a__U113(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U111(X1, X2, X3)) → a__U111(mark(X1), X2, X3)
mark(U13(X)) → a__U13(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U114(X1, X2)) → a__U114(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U132(X1, X2, X3, X4)) → a__U132(mark(X1), X2, X3, X4)
mark(U131(X1, X2, X3, X4)) → a__U131(mark(X1), X2, X3, X4)
mark(U122(X)) → a__U122(mark(X))
mark(U121(X1, X2)) → a__U121(mark(X1), X2)
mark(U136(X1, X2, X3, X4)) → a__U136(mark(X1), X2, X3, X4)
mark(U135(X1, X2, X3, X4)) → a__U135(mark(X1), X2, X3, X4)
mark(U134(X1, X2, X3, X4)) → a__U134(mark(X1), X2, X3, X4)
mark(U133(X1, X2, X3, X4)) → a__U133(mark(X1), X2, X3, X4)
mark(U23(X)) → a__U23(mark(X))
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U33(X)) → a__U33(mark(X))
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
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(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(0) → tt
a__isNatList(nil) → tt
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatList(take(V1, V2)) → a__U101(a__isNatKind(V1), V1, V2)
a__isNatList(cons(V1, V2)) → a__U91(a__isNatKind(V1), V1, V2)
a__length(cons(N, L)) → a__U111(a__isNatList(L), L, N)
a__length(nil) → 0
a__take(s(M), cons(N, IL)) → a__U131(a__isNatIList(IL), IL, M, N)
a__take(0, IL) → a__U121(a__isNatIList(IL), IL)
mark(U101(X1, X2, X3)) → a__U101(mark(X1), X2, X3)
mark(zeros) → a__zeros
mark(isNatKind(X)) → a__isNatKind(X)
mark(U102(X1, X2, X3)) → a__U102(mark(X1), X2, X3)
a__U134(X1, X2, X3, X4) → U134(X1, X2, X3, X4)
a__U133(X1, X2, X3, X4) → U133(X1, X2, X3, X4)
a__U132(X1, X2, X3, X4) → U132(X1, X2, X3, X4)
a__U131(X1, X2, X3, X4) → U131(X1, X2, X3, X4)
a__U122(X) → U122(X)
a__U121(X1, X2) → U121(X1, X2)
a__isNatList(X) → isNatList(X)
a__U13(X) → U13(X)
a__length(X) → length(X)
a__U114(X1, X2) → U114(X1, X2)
a__U113(X1, X2, X3) → U113(X1, X2, X3)
a__U112(X1, X2, X3) → U112(X1, X2, X3)
a__U111(X1, X2, X3) → U111(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__U11(X1, X2) → U11(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__U46(X) → U46(X)
a__U51(X1, X2) → U51(X1, X2)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U33(X) → U33(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U22(X1, X2) → U22(X1, X2)
a__U23(X) → U23(X)
a__take(X1, X2) → take(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U135(X1, X2, X3, X4) → U135(X1, X2, X3, X4)
a__U136(X1, X2, X3, X4) → U136(X1, X2, X3, X4)
mark(U93(X1, X2, X3)) → a__U93(mark(X1), X2, X3)
mark(U92(X1, X2, X3)) → a__U92(mark(X1), X2, X3)
mark(U95(X1, X2)) → a__U95(mark(X1), X2)
mark(U94(X1, X2, X3)) → a__U94(mark(X1), X2, X3)
mark(U71(X)) → a__U71(mark(X))
mark(U62(X)) → a__U62(mark(X))
mark(U91(X1, X2, X3)) → a__U91(mark(X1), X2, X3)
mark(U81(X)) → a__U81(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
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__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)
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U101(X1, X2, X3) → U101(X1, X2, X3)
mark(U96(X)) → a__U96(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
a__U95(X1, X2) → U95(X1, X2)
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__U96(X) → U96(X)
a__U61(X1, X2) → U61(X1, X2)
a__U52(X) → U52(X)
a__U71(X) → U71(X)
a__U62(X) → U62(X)
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__U81(X) → U81(X)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U92(X1, X2, X3) → U92(X1, X2, X3)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ QDPOrderProof

                                                        ↳ QDP

                                                          ↳ QDPOrderProof

                                                            ↳ QDP

                                                              ↳ QDPOrderProof

                                                                ↳ QDP

                                                                  ↳ QDPOrderProof

                                                                    ↳ QDP

                                                                      ↳ QDPOrderProof

                                                                        ↳ QDP

                                                                          ↳ QDPOrderProof

                                                                            ↳ QDP

                                                                              ↳ QDPOrderProof

                                                                                ↳ QDP

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

A__U113(tt, y0, 0) → A__U114(tt, y0)
A__U111(tt, L, N) → A__U112(a__isNatIListKind(L), L, N)
A__U114(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U112(tt, L, N) → A__U113(a__isNat(N), L, N)
A__LENGTH(cons(N, L)) → A__U111(a__isNatList(L), L, N)

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.