(0) Obligation:

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

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

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

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

(2) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(3) DependencyGraphProof (EQUIVALENT transformation)

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

(4) Complex Obligation (AND)

(5) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


A__ISNATILISTKIND(cons(V1, V2)) → A__U51(a__isNatKind(V1), V2)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNATILISTKIND(take(V1, V2)) → A__U61(a__isNatKind(V1), V2)
A__ISNATILISTKIND(take(V1, V2)) → A__ISNATKIND(V1)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__U51(x1, x2)  =  x2
tt  =  tt
A__ISNATILISTKIND(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
a__isNatKind(x1)  =  a__isNatKind
A__ISNATKIND(x1)  =  x1
length(x1)  =  x1
take(x1, x2)  =  take(x1, x2)
A__U61(x1, x2)  =  x2
s(x1)  =  x1
a__U61(x1, x2)  =  a__U61(x1, x2)
U61(x1, x2)  =  U61
a__U62(x1)  =  a__U62
U62(x1)  =  U62(x1)
a__U51(x1, x2)  =  a__U51
a__U52(x1)  =  x1
a__isNatIListKind(x1)  =  x1
a__U71(x1)  =  a__U71(x1)
U71(x1)  =  x1
a__U81(x1)  =  a__U81(x1)
U81(x1)  =  U81(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
isNatIListKind(x1)  =  isNatIListKind(x1)
isNatKind(x1)  =  isNatKind(x1)
0  =  0
nil  =  nil
zeros  =  zeros

Lexicographic Path Order [LPO].
Precedence:
[aisNatKind, isNatKind1] > aU811
aU612 > [tt, aU62] > U621


The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(8) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

(9) Obligation:

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

A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)

The TRS R consists of the following rules:

a__zeroscons(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__zeroszeros
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.

(10) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__ISNATKIND(x1)  =  x1
s(x1)  =  s(x1)

Lexicographic Path Order [LPO].
Precedence:
trivial


The following usable rules [FROCOS05] were oriented: none

(11) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

a__zeroscons(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__zeroszeros
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.

(12) PisEmptyProof (EQUIVALENT transformation)

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

(13) TRUE

(14) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


A__U104(tt, V1, V2) → A__U105(a__isNat(V1), V2)
A__U31(tt, V) → A__U32(a__isNatIListKind(V), V)
A__U32(tt, V) → A__ISNATLIST(V)
A__ISNATLIST(cons(V1, V2)) → A__U91(a__isNatKind(V1), V1, V2)
A__U93(tt, V1, V2) → A__U94(a__isNatIListKind(V2), V1, V2)
A__U94(tt, V1, V2) → A__U95(a__isNat(V1), V2)
A__U95(tt, V2) → A__ISNATLIST(V2)
A__ISNATLIST(take(V1, V2)) → A__U101(a__isNatKind(V1), V1, V2)
A__U94(tt, V1, V2) → A__ISNAT(V1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNatKind(V1), V1, V2)
A__U44(tt, V1, V2) → A__U45(a__isNat(V1), V2)
A__U44(tt, V1, V2) → A__ISNAT(V1)
A__U104(tt, V1, V2) → A__ISNAT(V1)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__U102(x1, x2, x3)  =  A__U102(x2, x3)
tt  =  tt
A__U103(x1, x2, x3)  =  A__U103(x2, x3)
a__isNatIListKind(x1)  =  a__isNatIListKind(x1)
A__U104(x1, x2, x3)  =  A__U104(x2, x3)
A__U105(x1, x2)  =  A__U105(x2)
a__isNat(x1)  =  a__isNat(x1)
A__ISNATILIST(x1)  =  A__ISNATILIST(x1)
A__U31(x1, x2)  =  A__U31(x2)
A__U32(x1, x2)  =  A__U32(x1, x2)
A__ISNATLIST(x1)  =  A__ISNATLIST(x1)
cons(x1, x2)  =  cons(x1, x2)
A__U91(x1, x2, x3)  =  A__U91(x2, x3)
a__isNatKind(x1)  =  a__isNatKind
A__U92(x1, x2, x3)  =  A__U92(x2, x3)
A__U93(x1, x2, x3)  =  A__U93(x2, x3)
A__U94(x1, x2, x3)  =  A__U94(x1, x2, x3)
A__U95(x1, x2)  =  A__U95(x1, x2)
take(x1, x2)  =  take(x1, x2)
A__U101(x1, x2, x3)  =  A__U101(x2, x3)
A__ISNAT(x1)  =  x1
length(x1)  =  length(x1)
A__U11(x1, x2)  =  A__U11(x2)
A__U12(x1, x2)  =  A__U12(x2)
s(x1)  =  s(x1)
A__U21(x1, x2)  =  x2
A__U22(x1, x2)  =  x2
A__U41(x1, x2, x3)  =  A__U41(x2, x3)
A__U42(x1, x2, x3)  =  A__U42(x2, x3)
A__U43(x1, x2, x3)  =  A__U43(x2, x3)
A__U44(x1, x2, x3)  =  A__U44(x2, x3)
A__U45(x1, x2)  =  A__U45(x2)
a__isNatList(x1)  =  a__isNatList
a__U101(x1, x2, x3)  =  x1
a__U91(x1, x2, x3)  =  a__U91
nil  =  nil
a__U61(x1, x2)  =  a__U61(x1, x2)
U61(x1, x2)  =  U61(x1, x2)
a__U45(x1, x2)  =  a__U45
a__U46(x1)  =  x1
a__isNatIList(x1)  =  a__isNatIList
a__U62(x1)  =  a__U62
U62(x1)  =  U62
a__U44(x1, x2, x3)  =  a__U44
a__U51(x1, x2)  =  a__U51(x1)
a__U52(x1)  =  a__U52
a__U71(x1)  =  a__U71
U71(x1)  =  U71
a__U81(x1)  =  x1
U81(x1)  =  x1
U45(x1, x2)  =  U45
a__U41(x1, x2, x3)  =  a__U41
a__U42(x1, x2, x3)  =  a__U42
U46(x1)  =  U46
a__U33(x1)  =  a__U33
U51(x1, x2)  =  U51
a__U43(x1, x2, x3)  =  a__U43
U52(x1)  =  U52
U41(x1, x2, x3)  =  U41
a__U92(x1, x2, x3)  =  a__U92
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
a__U93(x1, x2, x3)  =  a__U93
a__U94(x1, x2, x3)  =  a__U94
U44(x1, x2, x3)  =  U44
a__U23(x1)  =  a__U23
U23(x1)  =  U23
a__U31(x1, x2)  =  a__U31
U31(x1, x2)  =  U31
a__U32(x1, x2)  =  a__U32
U32(x1, x2)  =  U32
U33(x1)  =  U33
a__U11(x1, x2)  =  a__U11(x1)
a__U21(x1, x2)  =  a__U21(x2)
zeros  =  zeros
a__U95(x1, x2)  =  a__U95
a__U96(x1)  =  x1
0  =  0
U96(x1)  =  x1
U95(x1, x2)  =  U95
U92(x1, x2, x3)  =  U92
U91(x1, x2, x3)  =  U91
U94(x1, x2, x3)  =  U94
U93(x1, x2, x3)  =  U93
a__U106(x1)  =  a__U106
U106(x1)  =  U106
isNatIList(x1)  =  isNatIList
a__U105(x1, x2)  =  a__U105
U105(x1, x2)  =  U105
isNat(x1)  =  isNat
a__U102(x1, x2, x3)  =  x1
a__U103(x1, x2, x3)  =  a__U103
U11(x1, x2)  =  U11
a__U12(x1, x2)  =  a__U12
U12(x1, x2)  =  U12
U101(x1, x2, x3)  =  U101
U102(x1, x2, x3)  =  x1
a__U104(x1, x2, x3)  =  a__U104
isNatIListKind(x1)  =  isNatIListKind
U104(x1, x2, x3)  =  U104
isNatKind(x1)  =  isNatKind
U103(x1, x2, x3)  =  U103
a__U13(x1)  =  a__U13
a__U22(x1, x2)  =  a__U22
U22(x1, x2)  =  U22
U21(x1, x2)  =  U21(x2)
U13(x1)  =  U13
isNatList(x1)  =  isNatList

Lexicographic Path Order [LPO].
Precedence:
[AU1022, AU1032, AU1042, take2, AU1012] > [AU1051, AISNATILIST1, AU311, AU451] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > AU952 > [aisNatIListKind1, AISNATLIST1, length1, AU111, AU121] > isNatIListKind > [U46, U23, U92, U101]
[AU1022, AU1032, AU1042, take2, AU1012] > [AU1051, AISNATILIST1, AU311, AU451] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > U52 > [U46, U23, U92, U101]
[AU1022, AU1032, AU1042, take2, AU1012] > [AU1051, AISNATILIST1, AU311, AU451] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > U33 > [U46, U23, U92, U101]
[AU1022, AU1032, AU1042, take2, AU1012] > [AU1051, AISNATILIST1, AU311, AU451] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > U106 > [U46, U23, U92, U101]
[AU1022, AU1032, AU1042, take2, AU1012] > [AU1051, AISNATILIST1, AU311, AU451] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > isNatKind > [U46, U23, U92, U101]
[AU1022, AU1032, AU1042, take2, AU1012] > [AU1051, AISNATILIST1, AU311, AU451] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > U13 > [U46, U23, U92, U101]
[AU1022, AU1032, AU1042, take2, AU1012] > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > AU952 > [aisNatIListKind1, AISNATLIST1, length1, AU111, AU121] > isNatIListKind > [U46, U23, U92, U101]
[AU1022, AU1032, AU1042, take2, AU1012] > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > U52 > [U46, U23, U92, U101]
[AU1022, AU1032, AU1042, take2, AU1012] > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > U33 > [U46, U23, U92, U101]
[AU1022, AU1032, AU1042, take2, AU1012] > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > U106 > [U46, U23, U92, U101]
[AU1022, AU1032, AU1042, take2, AU1012] > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > isNatKind > [U46, U23, U92, U101]
[AU1022, AU1032, AU1042, take2, AU1012] > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > U13 > [U46, U23, U92, U101]
[AU1022, AU1032, AU1042, take2, AU1012] > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > U45 > [U46, U23, U92, U101]
[AU1022, AU1032, AU1042, take2, AU1012] > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > U41 > [U46, U23, U92, U101]
[AU1022, AU1032, AU1042, take2, AU1012] > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > U42 > [U46, U23, U92, U101]
[AU1022, AU1032, AU1042, take2, AU1012] > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > U43 > [U46, U23, U92, U101]
[AU1022, AU1032, AU1042, take2, AU1012] > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > U95 > [U46, U23, U92, U101]
[AU1022, AU1032, AU1042, take2, AU1012] > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > U91 > [U46, U23, U92, U101]
[AU1022, AU1032, AU1042, take2, AU1012] > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > U93 > [U46, U23, U92, U101]
[AU1022, AU1032, AU1042, take2, AU1012] > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > isNatIList > [U46, U23, U92, U101]
[AU1022, AU1032, AU1042, take2, AU1012] > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > isNat > [U46, U23, U92, U101]
[AU1022, AU1032, AU1042, take2, AU1012] > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > U11 > [U46, U23, U92, U101]
[AU1022, AU1032, AU1042, take2, AU1012] > [aU612, U612] > [aU62, U62] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > AU952 > [aisNatIListKind1, AISNATLIST1, length1, AU111, AU121] > isNatIListKind > [U46, U23, U92, U101]
[AU1022, AU1032, AU1042, take2, AU1012] > [aU612, U612] > [aU62, U62] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > U52 > [U46, U23, U92, U101]
[AU1022, AU1032, AU1042, take2, AU1012] > [aU612, U612] > [aU62, U62] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > U33 > [U46, U23, U92, U101]
[AU1022, AU1032, AU1042, take2, AU1012] > [aU612, U612] > [aU62, U62] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > U106 > [U46, U23, U92, U101]
[AU1022, AU1032, AU1042, take2, AU1012] > [aU612, U612] > [aU62, U62] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > isNatKind > [U46, U23, U92, U101]
[AU1022, AU1032, AU1042, take2, AU1012] > [aU612, U612] > [aU62, U62] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > U13 > [U46, U23, U92, U101]
[cons2, AU912, AU922, AU932, AU412, AU422, AU432, AU442] > [AU1051, AISNATILIST1, AU311, AU451] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > AU952 > [aisNatIListKind1, AISNATLIST1, length1, AU111, AU121] > isNatIListKind > [U46, U23, U92, U101]
[cons2, AU912, AU922, AU932, AU412, AU422, AU432, AU442] > [AU1051, AISNATILIST1, AU311, AU451] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > U52 > [U46, U23, U92, U101]
[cons2, AU912, AU922, AU932, AU412, AU422, AU432, AU442] > [AU1051, AISNATILIST1, AU311, AU451] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > U33 > [U46, U23, U92, U101]
[cons2, AU912, AU922, AU932, AU412, AU422, AU432, AU442] > [AU1051, AISNATILIST1, AU311, AU451] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > U106 > [U46, U23, U92, U101]
[cons2, AU912, AU922, AU932, AU412, AU422, AU432, AU442] > [AU1051, AISNATILIST1, AU311, AU451] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > isNatKind > [U46, U23, U92, U101]
[cons2, AU912, AU922, AU932, AU412, AU422, AU432, AU442] > [AU1051, AISNATILIST1, AU311, AU451] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > U13 > [U46, U23, U92, U101]
[cons2, AU912, AU922, AU932, AU412, AU422, AU432, AU442] > AU943 > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > AU952 > [aisNatIListKind1, AISNATLIST1, length1, AU111, AU121] > isNatIListKind > [U46, U23, U92, U101]
[cons2, AU912, AU922, AU932, AU412, AU422, AU432, AU442] > AU943 > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > U52 > [U46, U23, U92, U101]
[cons2, AU912, AU922, AU932, AU412, AU422, AU432, AU442] > AU943 > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > U33 > [U46, U23, U92, U101]
[cons2, AU912, AU922, AU932, AU412, AU422, AU432, AU442] > AU943 > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > U106 > [U46, U23, U92, U101]
[cons2, AU912, AU922, AU932, AU412, AU422, AU432, AU442] > AU943 > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > isNatKind > [U46, U23, U92, U101]
[cons2, AU912, AU922, AU932, AU412, AU422, AU432, AU442] > AU943 > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > U13 > [U46, U23, U92, U101]
[cons2, AU912, AU922, AU932, AU412, AU422, AU432, AU442] > AU943 > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > U45 > [U46, U23, U92, U101]
[cons2, AU912, AU922, AU932, AU412, AU422, AU432, AU442] > AU943 > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > U41 > [U46, U23, U92, U101]
[cons2, AU912, AU922, AU932, AU412, AU422, AU432, AU442] > AU943 > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > U42 > [U46, U23, U92, U101]
[cons2, AU912, AU922, AU932, AU412, AU422, AU432, AU442] > AU943 > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > U43 > [U46, U23, U92, U101]
[cons2, AU912, AU922, AU932, AU412, AU422, AU432, AU442] > AU943 > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > U95 > [U46, U23, U92, U101]
[cons2, AU912, AU922, AU932, AU412, AU422, AU432, AU442] > AU943 > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > U91 > [U46, U23, U92, U101]
[cons2, AU912, AU922, AU932, AU412, AU422, AU432, AU442] > AU943 > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > U93 > [U46, U23, U92, U101]
[cons2, AU912, AU922, AU932, AU412, AU422, AU432, AU442] > AU943 > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > isNatIList > [U46, U23, U92, U101]
[cons2, AU912, AU922, AU932, AU412, AU422, AU432, AU442] > AU943 > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > isNat > [U46, U23, U92, U101]
[cons2, AU912, AU922, AU932, AU412, AU422, AU432, AU442] > AU943 > [aisNat1, aisNatList, aU91, aU45, aisNatIList, aU44, aU41, aU42, aU43, aU92, aU93, aU94, U44, aU111, aU95, U94, aU12, U12, isNatList] > U11 > [U46, U23, U92, U101]
[cons2, AU912, AU922, AU932, AU412, AU422, AU432, AU442] > aU511 > [aisNatIListKind1, AISNATLIST1, length1, AU111, AU121] > isNatIListKind > [U46, U23, U92, U101]
[cons2, AU912, AU922, AU932, AU412, AU422, AU432, AU442] > aU511 > U51 > [U46, U23, U92, U101]
[s1, aU211] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > AU952 > [aisNatIListKind1, AISNATLIST1, length1, AU111, AU121] > isNatIListKind > [U46, U23, U92, U101]
[s1, aU211] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > U52 > [U46, U23, U92, U101]
[s1, aU211] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > U33 > [U46, U23, U92, U101]
[s1, aU211] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > U106 > [U46, U23, U92, U101]
[s1, aU211] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > isNatKind > [U46, U23, U92, U101]
[s1, aU211] > [tt, AU322, aisNatKind, nil, aU52, aU71, U71, aU33, aU23, aU31, U31, aU32, U32, zeros, aU106, aU105, U105, aU103, aU104, U104, U103, aU13, aU22, U22] > U13 > [U46, U23, U92, U101]
[s1, aU211] > U211 > [U46, U23, U92, U101]
0 > [U46, U23, U92, U101]


The following usable rules [FROCOS05] were oriented:

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__isNatList(nil) → tt
a__U61(X1, X2) → U61(X1, X2)
a__U45(tt, V2) → a__U46(a__isNatIList(V2))
a__U62(X) → U62(X)
a__U44(tt, V1, V2) → a__U45(a__isNat(V1), V2)
a__U51(tt, V2) → a__U52(a__isNatIListKind(V2))
a__U71(X) → U71(X)
a__U81(X) → U81(X)
a__U46(tt) → tt
a__U45(X1, X2) → U45(X1, X2)
a__U41(tt, V1, V2) → a__U42(a__isNatKind(V1), V1, V2)
a__U46(X) → U46(X)
a__U33(tt) → tt
a__U51(X1, X2) → U51(X1, X2)
a__U43(tt, V1, V2) → a__U44(a__isNatIListKind(V2), V1, V2)
a__U52(X) → U52(X)
a__U42(tt, V1, V2) → a__U43(a__isNatIListKind(V2), V1, V2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U91(tt, V1, V2) → a__U92(a__isNatKind(V1), V1, V2)
a__U81(tt) → tt
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U93(tt, V1, V2) → a__U94(a__isNatIListKind(V2), V1, V2)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U92(tt, V1, V2) → a__U93(a__isNatIListKind(V2), V1, V2)
a__U61(tt, V2) → a__U62(a__isNatIListKind(V2))
a__U23(X) → U23(X)
a__U52(tt) → tt
a__U31(X1, X2) → U31(X1, X2)
a__U71(tt) → tt
a__U32(X1, X2) → U32(X1, X2)
a__U62(tt) → tt
a__U33(X) → U33(X)
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__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__isNatIListKind(take(V1, V2)) → a__U61(a__isNatKind(V1), V2)
a__U96(X) → U96(X)
a__isNatKind(0) → tt
a__U95(X1, X2) → U95(X1, X2)
a__isNatKind(length(V1)) → a__U71(a__isNatIListKind(V1))
a__isNatKind(s(V1)) → a__U81(a__isNatKind(V1))
a__isNatIList(cons(V1, V2)) → a__U41(a__isNatKind(V1), V1, V2)
a__U92(X1, X2, X3) → U92(X1, X2, X3)
a__isNatIListKind(nil) → tt
a__U91(X1, X2, X3) → U91(X1, X2, X3)
a__isNatIListKind(zeros) → tt
a__U94(X1, X2, X3) → U94(X1, X2, X3)
a__isNatIListKind(cons(V1, V2)) → a__U51(a__isNatKind(V1), V2)
a__U93(X1, X2, X3) → U93(X1, X2, X3)
a__U106(X) → U106(X)
a__isNatIList(X) → isNatIList(X)
a__U105(X1, X2) → U105(X1, X2)
a__isNat(X) → isNat(X)
a__U102(tt, V1, V2) → a__U103(a__isNatIListKind(V2), V1, V2)
a__U101(tt, V1, V2) → a__U102(a__isNatKind(V1), V1, V2)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U106(tt) → tt
a__U101(X1, X2, X3) → U101(X1, X2, X3)
a__U105(tt, V2) → a__U106(a__isNatIList(V2))
a__U102(X1, X2, X3) → U102(X1, X2, X3)
a__U104(tt, V1, V2) → a__U105(a__isNat(V1), V2)
a__U103(tt, V1, V2) → a__U104(a__isNatIListKind(V2), V1, V2)
a__isNatIListKind(X) → isNatIListKind(X)
a__U104(X1, X2, X3) → U104(X1, X2, X3)
a__isNatKind(X) → isNatKind(X)
a__U103(X1, X2, X3) → U103(X1, X2, X3)
a__U11(tt, V1) → a__U12(a__isNatIListKind(V1), V1)
a__U12(tt, V1) → a__U13(a__isNatList(V1))
a__U22(X1, X2) → U22(X1, X2)
a__U21(X1, X2) → U21(X1, X2)
a__U13(tt) → tt
a__U13(X) → U13(X)
a__U21(tt, V1) → a__U22(a__isNatKind(V1), V1)
a__U31(tt, V) → a__U32(a__isNatIListKind(V), V)
a__U32(tt, V) → a__U33(a__isNatList(V))
a__U22(tt, V1) → a__U23(a__isNat(V1))
a__isNatList(X) → isNatList(X)
a__U23(tt) → tt

(16) Obligation:

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

A__U102(tt, V1, V2) → A__U103(a__isNatIListKind(V2), V1, V2)
A__U103(tt, V1, V2) → A__U104(a__isNatIListKind(V2), V1, V2)
A__U105(tt, V2) → A__ISNATILIST(V2)
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
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__U101(tt, V1, V2) → A__U102(a__isNatKind(V1), V1, V2)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__U11(tt, V1) → A__U12(a__isNatIListKind(V1), V1)
A__U12(tt, V1) → A__ISNATLIST(V1)
A__U21(tt, V1) → A__U22(a__isNatKind(V1), V1)
A__U22(tt, V1) → A__ISNAT(V1)
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__U45(tt, V2) → A__ISNATILIST(V2)

The TRS R consists of the following rules:

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

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

(17) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 16 less nodes.

(18) TRUE

(19) Obligation:

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

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

The TRS R consists of the following rules:

a__zeroscons(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__zeroszeros
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.