Termination w.r.t. Q proof of /tmp/tmpv7LDMC/OvConsOS_complete

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

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

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, V1, V2) → a__U62(a__isNat(V1), V2)
a__U62(tt, V2) → a__U63(a__isNatIList(V2))
a__U63(tt) → tt
a__U71(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
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__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatIListKind(take(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatList(take(V1, V2)) → a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
a__take(0, IL) → a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL)))
a__take(s(M), cons(N, IL)) → a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2) → U71(X1, X2)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, V1, V2) → a__U62(a__isNat(V1), V2)
a__U62(tt, V2) → a__U63(a__isNatIList(V2))
a__U63(tt) → tt
a__U71(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
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__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatIListKind(take(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatList(take(V1, V2)) → a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
a__take(0, IL) → a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL)))
a__take(s(M), cons(N, IL)) → a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2) → U71(X1, X2)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

Q is empty.

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

A__ISNATILIST(V) → A__ISNATILISTKIND(V)
MARK(U62(X1, X2)) → A__U62(mark(X1), X2)
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
MARK(U11(X1, X2)) → MARK(X1)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))))
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
MARK(length(X)) → MARK(X)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U81(X)) → A__U81(mark(X))
MARK(U61(X1, X2, X3)) → A__U61(mark(X1), X2, X3)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__LENGTH(cons(N, L)) → A__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
A__U31(tt, V) → A__ISNATLIST(V)
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__TAKE(0, IL) → A__AND(a__isNatIList(IL), isNatIListKind(IL))
MARK(U63(X)) → MARK(X)
A__TAKE(0, IL) → A__ISNATILIST(IL)
A__U11(tt, V1) → A__U12(a__isNatList(V1))
MARK(U32(X)) → MARK(X)
A__ISNATLIST(take(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__U61(tt, V1, V2) → A__ISNAT(V1)
A__U41(tt, V1, V2) → A__ISNAT(V1)
A__ISNATLIST(take(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
A__U71(tt, L) → MARK(L)
MARK(U91(X1, X2, X3, X4)) → A__U91(mark(X1), X2, X3, X4)
MARK(take(X1, X2)) → MARK(X2)
MARK(s(X)) → MARK(X)
A__TAKE(s(M), cons(N, IL)) → A__ISNATILIST(IL)
MARK(U12(X)) → A__U12(mark(X))
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__U42(tt, V2) → A__U43(a__isNatIList(V2))
A__ISNATLIST(take(V1, V2)) → A__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNatIList(IL), isNatIListKind(IL))
MARK(and(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → A__U71(mark(X1), X2)
MARK(U22(X)) → A__U22(mark(X))
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__U31(tt, V) → A__U32(a__isNatList(V))
A__U41(tt, V1, V2) → A__U42(a__isNat(V1), V2)
A__TAKE(s(M), cons(N, IL)) → A__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
MARK(U62(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U32(X)) → A__U32(mark(X))
MARK(U43(X)) → MARK(X)
MARK(U53(X)) → A__U53(mark(X))
A__U42(tt, V2) → A__ISNATILIST(V2)
A__U71(tt, L) → A__LENGTH(mark(L))
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U91(tt, IL, M, N) → MARK(N)
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNatIListKind(L))
MARK(cons(X1, X2)) → MARK(X1)
A__U52(tt, V2) → A__U53(a__isNatList(V2))
A__AND(tt, X) → MARK(X)
A__ISNATILIST(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(take(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U43(X)) → A__U43(mark(X))
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(U53(X)) → MARK(X)
A__U62(tt, V2) → A__ISNATILIST(V2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__U51(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(X1, X2, X3)) → MARK(X1)
A__U61(tt, V1, V2) → A__U62(a__isNat(V1), V2)
A__LENGTH(cons(N, L)) → A__AND(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
MARK(U22(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
A__ISNATILISTKIND(take(V1, V2)) → A__ISNATKIND(V1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → MARK(X1)
MARK(zeros) → A__ZEROS
A__U21(tt, V1) → A__U22(a__isNat(V1))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U81(X)) → MARK(X)
MARK(U61(X1, X2, X3)) → MARK(X1)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
MARK(U63(X)) → A__U63(mark(X))
A__U11(tt, V1) → A__ISNATLIST(V1)
A__U62(tt, V2) → A__U63(a__isNatIList(V2))
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
A__ISNATILISTKIND(take(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
A__TAKE(0, IL) → A__U81(a__and(a__isNatIList(IL), isNatIListKind(IL)))
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, V1, V2) → a__U62(a__isNat(V1), V2)
a__U62(tt, V2) → a__U63(a__isNatIList(V2))
a__U63(tt) → tt
a__U71(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
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__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatIListKind(take(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatList(take(V1, V2)) → a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
a__take(0, IL) → a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL)))
a__take(s(M), cons(N, IL)) → a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2) → U71(X1, X2)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

A__ISNATILIST(V) → A__ISNATILISTKIND(V)
MARK(U62(X1, X2)) → A__U62(mark(X1), X2)
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
MARK(U11(X1, X2)) → MARK(X1)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))))
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
MARK(length(X)) → MARK(X)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U81(X)) → A__U81(mark(X))
MARK(U61(X1, X2, X3)) → A__U61(mark(X1), X2, X3)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__LENGTH(cons(N, L)) → A__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
A__U31(tt, V) → A__ISNATLIST(V)
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__TAKE(0, IL) → A__AND(a__isNatIList(IL), isNatIListKind(IL))
MARK(U63(X)) → MARK(X)
A__TAKE(0, IL) → A__ISNATILIST(IL)
A__U11(tt, V1) → A__U12(a__isNatList(V1))
MARK(U32(X)) → MARK(X)
A__ISNATLIST(take(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__U61(tt, V1, V2) → A__ISNAT(V1)
A__U41(tt, V1, V2) → A__ISNAT(V1)
A__ISNATLIST(take(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
A__U71(tt, L) → MARK(L)
MARK(U91(X1, X2, X3, X4)) → A__U91(mark(X1), X2, X3, X4)
MARK(take(X1, X2)) → MARK(X2)
MARK(s(X)) → MARK(X)
A__TAKE(s(M), cons(N, IL)) → A__ISNATILIST(IL)
MARK(U12(X)) → A__U12(mark(X))
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__U42(tt, V2) → A__U43(a__isNatIList(V2))
A__ISNATLIST(take(V1, V2)) → A__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNatIList(IL), isNatIListKind(IL))
MARK(and(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → A__U71(mark(X1), X2)
MARK(U22(X)) → A__U22(mark(X))
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__U31(tt, V) → A__U32(a__isNatList(V))
A__U41(tt, V1, V2) → A__U42(a__isNat(V1), V2)
A__TAKE(s(M), cons(N, IL)) → A__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
MARK(U62(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U32(X)) → A__U32(mark(X))
MARK(U43(X)) → MARK(X)
MARK(U53(X)) → A__U53(mark(X))
A__U42(tt, V2) → A__ISNATILIST(V2)
A__U71(tt, L) → A__LENGTH(mark(L))
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U91(tt, IL, M, N) → MARK(N)
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNatIListKind(L))
MARK(cons(X1, X2)) → MARK(X1)
A__U52(tt, V2) → A__U53(a__isNatList(V2))
A__AND(tt, X) → MARK(X)
A__ISNATILIST(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(take(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U43(X)) → A__U43(mark(X))
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(U53(X)) → MARK(X)
A__U62(tt, V2) → A__ISNATILIST(V2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__U51(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(X1, X2, X3)) → MARK(X1)
A__U61(tt, V1, V2) → A__U62(a__isNat(V1), V2)
A__LENGTH(cons(N, L)) → A__AND(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
MARK(U22(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
A__ISNATILISTKIND(take(V1, V2)) → A__ISNATKIND(V1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → MARK(X1)
MARK(zeros) → A__ZEROS
A__U21(tt, V1) → A__U22(a__isNat(V1))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U81(X)) → MARK(X)
MARK(U61(X1, X2, X3)) → MARK(X1)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
MARK(U63(X)) → A__U63(mark(X))
A__U11(tt, V1) → A__ISNATLIST(V1)
A__U62(tt, V2) → A__U63(a__isNatIList(V2))
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
A__ISNATILISTKIND(take(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
A__TAKE(0, IL) → A__U81(a__and(a__isNatIList(IL), isNatIListKind(IL)))
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, V1, V2) → a__U62(a__isNat(V1), V2)
a__U62(tt, V2) → a__U63(a__isNatIList(V2))
a__U63(tt) → tt
a__U71(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
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__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatIListKind(take(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatList(take(V1, V2)) → a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
a__take(0, IL) → a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL)))
a__take(s(M), cons(N, IL)) → a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2) → U71(X1, X2)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 15 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

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

MARK(U62(X1, X2)) → A__U62(mark(X1), X2)
A__ISNATILIST(V) → A__ISNATILISTKIND(V)
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNatIListKind(L))
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))))
A__AND(tt, X) → MARK(X)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
MARK(length(X)) → MARK(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNATILIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U61(X1, X2, X3)) → A__U61(mark(X1), X2, X3)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
MARK(take(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__LENGTH(cons(N, L)) → A__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U53(X)) → MARK(X)
A__U31(tt, V) → A__ISNATLIST(V)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__U62(tt, V2) → A__ISNATILIST(V2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__TAKE(0, IL) → A__AND(a__isNatIList(IL), isNatIListKind(IL))
MARK(U63(X)) → MARK(X)
A__U51(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(X1, X2, X3)) → MARK(X1)
A__U61(tt, V1, V2) → A__U62(a__isNat(V1), V2)
A__TAKE(0, IL) → A__ISNATILIST(IL)
MARK(U32(X)) → MARK(X)
A__LENGTH(cons(N, L)) → A__AND(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
A__ISNATLIST(take(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__U61(tt, V1, V2) → A__ISNAT(V1)
A__U41(tt, V1, V2) → A__ISNAT(V1)
MARK(U22(X)) → MARK(X)
A__ISNATLIST(take(V1, V2)) → A__ISNATKIND(V1)
MARK(U21(X1, X2)) → MARK(X1)
A__ISNATILISTKIND(take(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
A__U71(tt, L) → MARK(L)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U91(X1, X2, X3, X4)) → A__U91(mark(X1), X2, X3, X4)
MARK(take(X1, X2)) → MARK(X2)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(s(X)) → MARK(X)
A__TAKE(s(M), cons(N, IL)) → A__ISNATILIST(IL)
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U81(X)) → MARK(X)
MARK(U61(X1, X2, X3)) → MARK(X1)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATLIST(take(V1, V2)) → A__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNatIList(IL), isNatIListKind(IL))
MARK(and(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → A__U71(mark(X1), X2)
A__U11(tt, V1) → A__ISNATLIST(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__TAKE(s(M), cons(N, IL)) → A__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
A__U41(tt, V1, V2) → A__U42(a__isNat(V1), V2)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
MARK(U62(X1, X2)) → MARK(X1)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
MARK(U71(X1, X2)) → MARK(X1)
A__ISNATILISTKIND(take(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(U43(X)) → MARK(X)
A__U42(tt, V2) → A__ISNATILIST(V2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
A__U71(tt, L) → A__LENGTH(mark(L))
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U91(tt, IL, M, N) → MARK(N)
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, V1, V2) → a__U62(a__isNat(V1), V2)
a__U62(tt, V2) → a__U63(a__isNatIList(V2))
a__U63(tt) → tt
a__U71(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
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__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatIListKind(take(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatList(take(V1, V2)) → a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
a__take(0, IL) → a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL)))
a__take(s(M), cons(N, IL)) → a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2) → U71(X1, X2)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(length(X)) → MARK(X)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U71(X1, X2)) → A__U71(mark(X1), X2)
MARK(U71(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.

MARK(U62(X1, X2)) → A__U62(mark(X1), X2)
A__ISNATILIST(V) → A__ISNATILISTKIND(V)
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNatIListKind(L))
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))))
A__AND(tt, X) → MARK(X)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNATILIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U61(X1, X2, X3)) → A__U61(mark(X1), X2, X3)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
MARK(take(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__LENGTH(cons(N, L)) → A__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U53(X)) → MARK(X)
A__U31(tt, V) → A__ISNATLIST(V)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__U62(tt, V2) → A__ISNATILIST(V2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__TAKE(0, IL) → A__AND(a__isNatIList(IL), isNatIListKind(IL))
MARK(U63(X)) → MARK(X)
A__U51(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(X1, X2, X3)) → MARK(X1)
A__U61(tt, V1, V2) → A__U62(a__isNat(V1), V2)
A__TAKE(0, IL) → A__ISNATILIST(IL)
MARK(U32(X)) → MARK(X)
A__LENGTH(cons(N, L)) → A__AND(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
A__ISNATLIST(take(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__U61(tt, V1, V2) → A__ISNAT(V1)
A__U41(tt, V1, V2) → A__ISNAT(V1)
MARK(U22(X)) → MARK(X)
A__ISNATLIST(take(V1, V2)) → A__ISNATKIND(V1)
MARK(U21(X1, X2)) → MARK(X1)
A__ISNATILISTKIND(take(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
A__U71(tt, L) → MARK(L)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U91(X1, X2, X3, X4)) → A__U91(mark(X1), X2, X3, X4)
MARK(take(X1, X2)) → MARK(X2)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(s(X)) → MARK(X)
A__TAKE(s(M), cons(N, IL)) → A__ISNATILIST(IL)
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U81(X)) → MARK(X)
MARK(U61(X1, X2, X3)) → MARK(X1)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATLIST(take(V1, V2)) → A__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNatIList(IL), isNatIListKind(IL))
MARK(and(X1, X2)) → MARK(X1)
A__U11(tt, V1) → A__ISNATLIST(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__TAKE(s(M), cons(N, IL)) → A__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
A__U41(tt, V1, V2) → A__U42(a__isNat(V1), V2)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
MARK(U62(X1, X2)) → MARK(X1)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
A__ISNATILISTKIND(take(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(U43(X)) → MARK(X)
A__U42(tt, V2) → A__ISNATILIST(V2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
A__U71(tt, L) → A__LENGTH(mark(L))
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U91(tt, IL, M, N) → MARK(N)
A__U21(tt, V1) → A__ISNAT(V1)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__AND(x₁, x₂)) = x₂
POL(A__ISNAT(x₁)) = 0
POL(A__ISNATILIST(x₁)) = 0
POL(A__ISNATILISTKIND(x₁)) = 0
POL(A__ISNATKIND(x₁)) = 0
POL(A__ISNATLIST(x₁)) = 0
POL(A__LENGTH(x₁)) = x₁
POL(A__TAKE(x₁, x₂)) = x₂
POL(A__U11(x₁, x₂)) = 0
POL(A__U21(x₁, x₂)) = 0
POL(A__U31(x₁, x₂)) = 0
POL(A__U41(x₁, x₂, x₃)) = 0
POL(A__U42(x₁, x₂)) = 0
POL(A__U51(x₁, x₂, x₃)) = 0
POL(A__U52(x₁, x₂)) = 0
POL(A__U61(x₁, x₂, x₃)) = 0
POL(A__U62(x₁, x₂)) = 0
POL(A__U71(x₁, x₂)) = x₂
POL(A__U91(x₁, x₂, x₃, x₄)) = x₄
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁
POL(U12(x₁)) = x₁
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = x₁
POL(U42(x₁, x₂)) = x₁
POL(U43(x₁)) = x₁
POL(U51(x₁, x₂, x₃)) = x₁
POL(U52(x₁, x₂)) = x₁
POL(U53(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = x₁
POL(U62(x₁, x₂)) = x₁
POL(U63(x₁)) = x₁
POL(U71(x₁, x₂)) = 1 + x₁ + x₂
POL(U81(x₁)) = x₁
POL(U91(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(a__U11(x₁, x₂)) = x₁
POL(a__U12(x₁)) = x₁
POL(a__U21(x₁, x₂)) = x₁
POL(a__U22(x₁)) = x₁
POL(a__U31(x₁, x₂)) = x₁
POL(a__U32(x₁)) = x₁
POL(a__U41(x₁, x₂, x₃)) = x₁
POL(a__U42(x₁, x₂)) = x₁
POL(a__U43(x₁)) = x₁
POL(a__U51(x₁, x₂, x₃)) = x₁
POL(a__U52(x₁, x₂)) = x₁
POL(a__U53(x₁)) = x₁
POL(a__U61(x₁, x₂, x₃)) = x₁
POL(a__U62(x₁, x₂)) = x₁
POL(a__U63(x₁)) = x₁
POL(a__U71(x₁, x₂)) = 1 + x₁ + x₂
POL(a__U81(x₁)) = x₁
POL(a__U91(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(a__and(x₁, x₂)) = x₁ + x₂
POL(a__isNat(x₁)) = 0
POL(a__isNatIList(x₁)) = 0
POL(a__isNatIListKind(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__isNatList(x₁)) = 0
POL(a__length(x₁)) = 1 + x₁
POL(a__take(x₁, x₂)) = x₁ + x₂
POL(a__zeros) = 0
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 1 + x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = x₁ + x₂
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

a__isNat(0) → tt
a__U91(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__U81(tt) → nil
a__isNatIList(zeros) → tt
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNatIListKind(zeros) → tt
a__isNatIListKind(nil) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U63(tt) → tt
a__U71(tt, L) → s(a__length(mark(L)))
a__U61(tt, V1, V2) → a__U62(a__isNat(V1), V2)
a__U62(tt, V2) → a__U63(a__isNatIList(V2))
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(isNatIList(X)) → a__isNatIList(X)
mark(U43(X)) → a__U43(mark(X))
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U53(X)) → a__U53(mark(X))
mark(U63(X)) → a__U63(mark(X))
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(U81(X)) → a__U81(mark(X))
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__and(tt, X) → mark(X)
a__isNatIListKind(take(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatList(take(V1, V2)) → a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
a__take(0, IL) → a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL)))
a__take(s(M), cons(N, IL)) → a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U53(X) → U53(X)
a__U52(X1, X2) → U52(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__isNatIList(X) → isNatIList(X)
a__U43(X) → U43(X)
a__U42(X1, X2) → U42(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__and(X1, X2) → and(X1, X2)
a__take(X1, X2) → take(X1, X2)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__U81(X) → U81(X)
a__length(X) → length(X)
a__U71(X1, X2) → U71(X1, X2)
a__U63(X) → U63(X)
a__U62(X1, X2) → U62(X1, X2)
mark(nil) → nil
a__zeros → zeros
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__zeros → cons(0, zeros)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

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

MARK(U62(X1, X2)) → A__U62(mark(X1), X2)
A__ISNATILIST(V) → A__ISNATILISTKIND(V)
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNatIListKind(L))
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))))
A__AND(tt, X) → MARK(X)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNATILIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U61(X1, X2, X3)) → A__U61(mark(X1), X2, X3)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
MARK(take(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__LENGTH(cons(N, L)) → A__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__U31(tt, V) → A__ISNATLIST(V)
MARK(U53(X)) → MARK(X)
A__U62(tt, V2) → A__ISNATILIST(V2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__TAKE(0, IL) → A__AND(a__isNatIList(IL), isNatIListKind(IL))
MARK(U63(X)) → MARK(X)
A__U51(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(X1, X2, X3)) → MARK(X1)
A__U61(tt, V1, V2) → A__U62(a__isNat(V1), V2)
A__TAKE(0, IL) → A__ISNATILIST(IL)
MARK(U32(X)) → MARK(X)
A__LENGTH(cons(N, L)) → A__AND(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
A__ISNATLIST(take(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__U61(tt, V1, V2) → A__ISNAT(V1)
A__U41(tt, V1, V2) → A__ISNAT(V1)
MARK(U22(X)) → MARK(X)
A__ISNATLIST(take(V1, V2)) → A__ISNATKIND(V1)
MARK(U21(X1, X2)) → MARK(X1)
A__ISNATILISTKIND(take(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
A__U71(tt, L) → MARK(L)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U91(X1, X2, X3, X4)) → A__U91(mark(X1), X2, X3, X4)
MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
A__TAKE(s(M), cons(N, IL)) → A__ISNATILIST(IL)
MARK(s(X)) → MARK(X)
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U81(X)) → MARK(X)
MARK(U61(X1, X2, X3)) → MARK(X1)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATLIST(take(V1, V2)) → A__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNatIList(IL), isNatIListKind(IL))
MARK(and(X1, X2)) → MARK(X1)
A__U11(tt, V1) → A__ISNATLIST(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__U41(tt, V1, V2) → A__U42(a__isNat(V1), V2)
A__TAKE(s(M), cons(N, IL)) → A__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
MARK(U62(X1, X2)) → MARK(X1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
A__ISNATILISTKIND(take(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(U43(X)) → MARK(X)
A__U42(tt, V2) → A__ISNATILIST(V2)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
A__U71(tt, L) → A__LENGTH(mark(L))
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U91(tt, IL, M, N) → MARK(N)
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, V1, V2) → a__U62(a__isNat(V1), V2)
a__U62(tt, V2) → a__U63(a__isNatIList(V2))
a__U63(tt) → tt
a__U71(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
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__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatIListKind(take(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatList(take(V1, V2)) → a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
a__take(0, IL) → a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL)))
a__take(s(M), cons(N, IL)) → a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2) → U71(X1, X2)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 4 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                  ↳ QDP

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

MARK(U62(X1, X2)) → A__U62(mark(X1), X2)
A__ISNATILIST(V) → A__ISNATILISTKIND(V)
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))))
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__AND(tt, X) → MARK(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNATILIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U61(X1, X2, X3)) → A__U61(mark(X1), X2, X3)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
MARK(take(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U53(X)) → MARK(X)
A__U31(tt, V) → A__ISNATLIST(V)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__U62(tt, V2) → A__ISNATILIST(V2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__TAKE(0, IL) → A__AND(a__isNatIList(IL), isNatIListKind(IL))
MARK(U63(X)) → MARK(X)
A__U51(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(X1, X2, X3)) → MARK(X1)
A__U61(tt, V1, V2) → A__U62(a__isNat(V1), V2)
A__TAKE(0, IL) → A__ISNATILIST(IL)
MARK(U32(X)) → MARK(X)
A__U61(tt, V1, V2) → A__ISNAT(V1)
A__ISNATLIST(take(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(U22(X)) → MARK(X)
A__U41(tt, V1, V2) → A__ISNAT(V1)
A__ISNATLIST(take(V1, V2)) → A__ISNATKIND(V1)
MARK(U21(X1, X2)) → MARK(X1)
A__ISNATILISTKIND(take(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U91(X1, X2, X3, X4)) → A__U91(mark(X1), X2, X3, X4)
MARK(take(X1, X2)) → MARK(X2)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(s(X)) → MARK(X)
A__TAKE(s(M), cons(N, IL)) → A__ISNATILIST(IL)
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U81(X)) → MARK(X)
MARK(U61(X1, X2, X3)) → MARK(X1)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATLIST(take(V1, V2)) → A__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNatIList(IL), isNatIListKind(IL))
MARK(and(X1, X2)) → MARK(X1)
A__U11(tt, V1) → A__ISNATLIST(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__TAKE(s(M), cons(N, IL)) → A__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
A__U41(tt, V1, V2) → A__U42(a__isNat(V1), V2)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
MARK(U62(X1, X2)) → MARK(X1)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
A__ISNATILISTKIND(take(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(U43(X)) → MARK(X)
A__U42(tt, V2) → A__ISNATILIST(V2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U91(tt, IL, M, N) → MARK(N)
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, V1, V2) → a__U62(a__isNat(V1), V2)
a__U62(tt, V2) → a__U63(a__isNatIList(V2))
a__U63(tt) → tt
a__U71(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
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__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatIListKind(take(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatList(take(V1, V2)) → a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
a__take(0, IL) → a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL)))
a__take(s(M), cons(N, IL)) → a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2) → U71(X1, X2)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(U81(X)) → MARK(X)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))

The remaining pairs can at least be oriented weakly.

MARK(U62(X1, X2)) → A__U62(mark(X1), X2)
A__ISNATILIST(V) → A__ISNATILISTKIND(V)
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))))
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__AND(tt, X) → MARK(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNATILIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U61(X1, X2, X3)) → A__U61(mark(X1), X2, X3)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U53(X)) → MARK(X)
A__U31(tt, V) → A__ISNATLIST(V)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__U62(tt, V2) → A__ISNATILIST(V2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__TAKE(0, IL) → A__AND(a__isNatIList(IL), isNatIListKind(IL))
MARK(U63(X)) → MARK(X)
A__U51(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(X1, X2, X3)) → MARK(X1)
A__U61(tt, V1, V2) → A__U62(a__isNat(V1), V2)
A__TAKE(0, IL) → A__ISNATILIST(IL)
MARK(U32(X)) → MARK(X)
A__U61(tt, V1, V2) → A__ISNAT(V1)
A__ISNATLIST(take(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(U22(X)) → MARK(X)
A__U41(tt, V1, V2) → A__ISNAT(V1)
A__ISNATLIST(take(V1, V2)) → A__ISNATKIND(V1)
MARK(U21(X1, X2)) → MARK(X1)
A__ISNATILISTKIND(take(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U91(X1, X2, X3, X4)) → A__U91(mark(X1), X2, X3, X4)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(s(X)) → MARK(X)
A__TAKE(s(M), cons(N, IL)) → A__ISNATILIST(IL)
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U61(X1, X2, X3)) → MARK(X1)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATLIST(take(V1, V2)) → A__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNatIList(IL), isNatIListKind(IL))
MARK(and(X1, X2)) → MARK(X1)
A__U11(tt, V1) → A__ISNATLIST(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__TAKE(s(M), cons(N, IL)) → A__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
A__U41(tt, V1, V2) → A__U42(a__isNat(V1), V2)
MARK(U62(X1, X2)) → MARK(X1)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
A__ISNATILISTKIND(take(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(U43(X)) → MARK(X)
A__U42(tt, V2) → A__ISNATILIST(V2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U91(tt, IL, M, N) → MARK(N)
A__U21(tt, V1) → A__ISNAT(V1)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__AND(x₁, x₂)) = x₂
POL(A__ISNAT(x₁)) = 0
POL(A__ISNATILIST(x₁)) = 0
POL(A__ISNATILISTKIND(x₁)) = 0
POL(A__ISNATKIND(x₁)) = 0
POL(A__ISNATLIST(x₁)) = 0
POL(A__TAKE(x₁, x₂)) = x₂
POL(A__U11(x₁, x₂)) = 0
POL(A__U21(x₁, x₂)) = 0
POL(A__U31(x₁, x₂)) = 0
POL(A__U41(x₁, x₂, x₃)) = 0
POL(A__U42(x₁, x₂)) = 0
POL(A__U51(x₁, x₂, x₃)) = 0
POL(A__U52(x₁, x₂)) = 0
POL(A__U61(x₁, x₂, x₃)) = 0
POL(A__U62(x₁, x₂)) = 0
POL(A__U91(x₁, x₂, x₃, x₄)) = x₄
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁
POL(U12(x₁)) = x₁
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = x₁
POL(U42(x₁, x₂)) = x₁
POL(U43(x₁)) = x₁
POL(U51(x₁, x₂, x₃)) = x₁
POL(U52(x₁, x₂)) = x₁
POL(U53(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = x₁
POL(U62(x₁, x₂)) = x₁
POL(U63(x₁)) = x₁
POL(U71(x₁, x₂)) = 0
POL(U81(x₁)) = 1 + x₁
POL(U91(x₁, x₂, x₃, x₄)) = x₁ + x₄
POL(a__U11(x₁, x₂)) = x₁
POL(a__U12(x₁)) = x₁
POL(a__U21(x₁, x₂)) = x₁
POL(a__U22(x₁)) = x₁
POL(a__U31(x₁, x₂)) = x₁
POL(a__U32(x₁)) = x₁
POL(a__U41(x₁, x₂, x₃)) = x₁
POL(a__U42(x₁, x₂)) = x₁
POL(a__U43(x₁)) = x₁
POL(a__U51(x₁, x₂, x₃)) = x₁
POL(a__U52(x₁, x₂)) = x₁
POL(a__U53(x₁)) = x₁
POL(a__U61(x₁, x₂, x₃)) = x₁
POL(a__U62(x₁, x₂)) = x₁
POL(a__U63(x₁)) = x₁
POL(a__U71(x₁, x₂)) = 0
POL(a__U81(x₁)) = 1 + x₁
POL(a__U91(x₁, x₂, x₃, x₄)) = x₁ + x₄
POL(a__and(x₁, x₂)) = x₁ + x₂
POL(a__isNat(x₁)) = 0
POL(a__isNatIList(x₁)) = 0
POL(a__isNatIListKind(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__isNatList(x₁)) = 0
POL(a__length(x₁)) = 0
POL(a__take(x₁, x₂)) = 1 + x₁ + x₂
POL(a__zeros) = 0
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 1 + x₁ + x₂
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

a__isNat(0) → tt
a__U91(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__U81(tt) → nil
a__isNatIList(zeros) → tt
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNatIListKind(zeros) → tt
a__isNatIListKind(nil) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U63(tt) → tt
a__U71(tt, L) → s(a__length(mark(L)))
a__U61(tt, V1, V2) → a__U62(a__isNat(V1), V2)
a__U62(tt, V2) → a__U63(a__isNatIList(V2))
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(isNatIList(X)) → a__isNatIList(X)
mark(U43(X)) → a__U43(mark(X))
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U53(X)) → a__U53(mark(X))
mark(U63(X)) → a__U63(mark(X))
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(U81(X)) → a__U81(mark(X))
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__and(tt, X) → mark(X)
a__isNatIListKind(take(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatList(take(V1, V2)) → a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
a__take(0, IL) → a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL)))
a__take(s(M), cons(N, IL)) → a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U53(X) → U53(X)
a__U52(X1, X2) → U52(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__isNatIList(X) → isNatIList(X)
a__U43(X) → U43(X)
a__U42(X1, X2) → U42(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__and(X1, X2) → and(X1, X2)
a__take(X1, X2) → take(X1, X2)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__U81(X) → U81(X)
a__length(X) → length(X)
a__U71(X1, X2) → U71(X1, X2)
a__U63(X) → U63(X)
a__U62(X1, X2) → U62(X1, X2)
mark(nil) → nil
a__zeros → zeros
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__zeros → cons(0, zeros)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                  ↳ QDP

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

MARK(U62(X1, X2)) → A__U62(mark(X1), X2)
A__ISNATILIST(V) → A__ISNATILISTKIND(V)
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))))
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__AND(tt, X) → MARK(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNATILIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U61(X1, X2, X3)) → A__U61(mark(X1), X2, X3)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__U31(tt, V) → A__ISNATLIST(V)
MARK(U53(X)) → MARK(X)
A__U62(tt, V2) → A__ISNATILIST(V2)
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__TAKE(0, IL) → A__AND(a__isNatIList(IL), isNatIListKind(IL))
MARK(U63(X)) → MARK(X)
A__U51(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(X1, X2, X3)) → MARK(X1)
A__U61(tt, V1, V2) → A__U62(a__isNat(V1), V2)
A__TAKE(0, IL) → A__ISNATILIST(IL)
MARK(U32(X)) → MARK(X)
A__U61(tt, V1, V2) → A__ISNAT(V1)
A__ISNATLIST(take(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(U22(X)) → MARK(X)
A__U41(tt, V1, V2) → A__ISNAT(V1)
A__ISNATLIST(take(V1, V2)) → A__ISNATKIND(V1)
MARK(U21(X1, X2)) → MARK(X1)
A__ISNATILISTKIND(take(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U91(X1, X2, X3, X4)) → A__U91(mark(X1), X2, X3, X4)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(s(X)) → MARK(X)
A__TAKE(s(M), cons(N, IL)) → A__ISNATILIST(IL)
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U61(X1, X2, X3)) → MARK(X1)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATLIST(take(V1, V2)) → A__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__TAKE(s(M), cons(N, IL)) → A__AND(a__isNatIList(IL), isNatIListKind(IL))
MARK(and(X1, X2)) → MARK(X1)
A__U11(tt, V1) → A__ISNATLIST(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__U41(tt, V1, V2) → A__U42(a__isNat(V1), V2)
A__TAKE(s(M), cons(N, IL)) → A__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
MARK(U62(X1, X2)) → MARK(X1)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
A__ISNATILISTKIND(take(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(U43(X)) → MARK(X)
A__U42(tt, V2) → A__ISNATILIST(V2)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U91(tt, IL, M, N) → MARK(N)
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, V1, V2) → a__U62(a__isNat(V1), V2)
a__U62(tt, V2) → a__U63(a__isNatIList(V2))
a__U63(tt) → tt
a__U71(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
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__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatIListKind(take(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatList(take(V1, V2)) → a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
a__take(0, IL) → a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL)))
a__take(s(M), cons(N, IL)) → a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2) → U71(X1, X2)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 6 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                  ↳ QDP

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

MARK(U62(X1, X2)) → A__U62(mark(X1), X2)
A__ISNATILIST(V) → A__ISNATILISTKIND(V)
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__AND(tt, X) → MARK(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNATILIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U61(X1, X2, X3)) → A__U61(mark(X1), X2, X3)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U53(X)) → MARK(X)
A__U31(tt, V) → A__ISNATLIST(V)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__U62(tt, V2) → A__ISNATILIST(V2)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(U63(X)) → MARK(X)
A__U51(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(X1, X2, X3)) → MARK(X1)
A__U61(tt, V1, V2) → A__U62(a__isNat(V1), V2)
MARK(U32(X)) → MARK(X)
A__U61(tt, V1, V2) → A__ISNAT(V1)
A__ISNATLIST(take(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(U22(X)) → MARK(X)
A__U41(tt, V1, V2) → A__ISNAT(V1)
A__ISNATLIST(take(V1, V2)) → A__ISNATKIND(V1)
MARK(U21(X1, X2)) → MARK(X1)
A__ISNATILISTKIND(take(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U91(X1, X2, X3, X4)) → A__U91(mark(X1), X2, X3, X4)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U61(X1, X2, X3)) → MARK(X1)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATLIST(take(V1, V2)) → A__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(and(X1, X2)) → MARK(X1)
A__U11(tt, V1) → A__ISNATLIST(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__U41(tt, V1, V2) → A__U42(a__isNat(V1), V2)
MARK(U62(X1, X2)) → MARK(X1)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
A__ISNATILISTKIND(take(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(U43(X)) → MARK(X)
A__U42(tt, V2) → A__ISNATILIST(V2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U91(tt, IL, M, N) → MARK(N)
A__U21(tt, V1) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, V1, V2) → a__U62(a__isNat(V1), V2)
a__U62(tt, V2) → a__U63(a__isNatIList(V2))
a__U63(tt) → tt
a__U71(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
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__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatIListKind(take(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatList(take(V1, V2)) → a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
a__take(0, IL) → a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL)))
a__take(s(M), cons(N, IL)) → a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2) → U71(X1, X2)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U62(X1, X2)) → A__U62(mark(X1), X2)
MARK(U11(X1, X2)) → MARK(X1)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
A__ISNATILIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U61(X1, X2, X3)) → A__U61(mark(X1), X2, X3)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U53(X)) → MARK(X)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(U63(X)) → MARK(X)
A__U51(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
A__ISNATLIST(take(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(U22(X)) → MARK(X)
A__U41(tt, V1, V2) → A__ISNAT(V1)
A__ISNATLIST(take(V1, V2)) → A__ISNATKIND(V1)
MARK(U21(X1, X2)) → MARK(X1)
A__ISNATILISTKIND(take(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U61(X1, X2, X3)) → MARK(X1)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATLIST(take(V1, V2)) → A__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(and(X1, X2)) → MARK(X1)
A__U11(tt, V1) → A__ISNATLIST(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(U62(X1, X2)) → MARK(X1)
A__ISNATILISTKIND(take(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(U43(X)) → MARK(X)
A__U42(tt, V2) → A__ISNATILIST(V2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U91(tt, IL, M, N) → MARK(N)
A__U21(tt, V1) → A__ISNAT(V1)

The remaining pairs can at least be oriented weakly.

A__ISNATILIST(V) → A__ISNATILISTKIND(V)
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__AND(tt, X) → MARK(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__U31(tt, V) → A__ISNATLIST(V)
A__U62(tt, V2) → A__ISNATILIST(V2)
A__U61(tt, V1, V2) → A__U62(a__isNat(V1), V2)
A__U61(tt, V1, V2) → A__ISNAT(V1)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(U91(X1, X2, X3, X4)) → A__U91(mark(X1), X2, X3, X4)
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__U41(tt, V1, V2) → A__U42(a__isNat(V1), V2)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__AND(x₁, x₂)) = x₂
POL(A__ISNAT(x₁)) = x₁
POL(A__ISNATILIST(x₁)) = x₁
POL(A__ISNATILISTKIND(x₁)) = x₁
POL(A__ISNATKIND(x₁)) = x₁
POL(A__ISNATLIST(x₁)) = x₁
POL(A__U11(x₁, x₂)) = 1 + x₂
POL(A__U21(x₁, x₂)) = 1 + x₂
POL(A__U31(x₁, x₂)) = x₂
POL(A__U41(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(A__U42(x₁, x₂)) = 1 + x₂
POL(A__U51(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(A__U52(x₁, x₂)) = x₂
POL(A__U61(x₁, x₂, x₃)) = x₂ + x₃
POL(A__U62(x₁, x₂)) = x₂
POL(A__U91(x₁, x₂, x₃, x₄)) = 1 + x₄
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = 1 + x₁ + x₂
POL(U12(x₁)) = 1 + x₁
POL(U21(x₁, x₂)) = 1 + x₁ + x₂
POL(U22(x₁)) = 1 + x₁
POL(U31(x₁, x₂)) = 1 + x₁ + x₂
POL(U32(x₁)) = 1 + x₁
POL(U41(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U42(x₁, x₂)) = 1 + x₁ + x₂
POL(U43(x₁)) = 1 + x₁
POL(U51(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U52(x₁, x₂)) = 1 + x₁ + x₂
POL(U53(x₁)) = 1 + x₁
POL(U61(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U62(x₁, x₂)) = 1 + x₁ + x₂
POL(U63(x₁)) = 1 + x₁
POL(U71(x₁, x₂)) = 0
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₂ + x₃ + x₄
POL(a__U11(x₁, x₂)) = 0
POL(a__U12(x₁)) = 0
POL(a__U21(x₁, x₂)) = 0
POL(a__U22(x₁)) = 0
POL(a__U31(x₁, x₂)) = 0
POL(a__U32(x₁)) = 0
POL(a__U41(x₁, x₂, x₃)) = 0
POL(a__U42(x₁, x₂)) = 0
POL(a__U43(x₁)) = 0
POL(a__U51(x₁, x₂, x₃)) = 0
POL(a__U52(x₁, x₂)) = 0
POL(a__U53(x₁)) = 0
POL(a__U61(x₁, x₂, x₃)) = 0
POL(a__U62(x₁, x₂)) = 0
POL(a__U63(x₁)) = 0
POL(a__U71(x₁, x₂)) = 0
POL(a__U81(x₁)) = 0
POL(a__U91(x₁, x₂, x₃, x₄)) = 0
POL(a__and(x₁, x₂)) = 0
POL(a__isNat(x₁)) = 0
POL(a__isNatIList(x₁)) = 0
POL(a__isNatIListKind(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__isNatList(x₁)) = 0
POL(a__length(x₁)) = 0
POL(a__take(x₁, x₂)) = 0
POL(a__zeros) = 0
POL(and(x₁, x₂)) = 1 + x₁ + x₂
POL(cons(x₁, x₂)) = 1 + x₁ + x₂
POL(isNat(x₁)) = 1 + x₁
POL(isNatIList(x₁)) = 1 + x₁
POL(isNatIListKind(x₁)) = x₁
POL(isNatKind(x₁)) = 1 + x₁
POL(isNatList(x₁)) = 1 + x₁
POL(length(x₁)) = 1 + x₁
POL(mark(x₁)) = 0
POL(nil) = 0
POL(s(x₁)) = 1 + x₁
POL(take(x₁, x₂)) = 1 + x₁ + x₂
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                  ↳ QDP

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

A__ISNATILIST(V) → A__ISNATILISTKIND(V)
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
A__U61(tt, V1, V2) → A__U62(a__isNat(V1), V2)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__AND(tt, X) → MARK(X)
A__U61(tt, V1, V2) → A__ISNAT(V1)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
A__U41(tt, V1, V2) → A__U42(a__isNat(V1), V2)
A__U31(tt, V) → A__ISNATLIST(V)
MARK(U91(X1, X2, X3, X4)) → A__U91(mark(X1), X2, X3, X4)
A__U62(tt, V2) → A__ISNATILIST(V2)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, V1, V2) → a__U62(a__isNat(V1), V2)
a__U62(tt, V2) → a__U63(a__isNatIList(V2))
a__U63(tt) → tt
a__U71(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
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__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatIListKind(take(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatList(take(V1, V2)) → a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
a__take(0, IL) → a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL)))
a__take(s(M), cons(N, IL)) → a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2) → U71(X1, X2)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 20 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

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

A__U71(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, V1, V2) → a__U62(a__isNat(V1), V2)
a__U62(tt, V2) → a__U63(a__isNatIList(V2))
a__U63(tt) → tt
a__U71(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
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__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatIListKind(take(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatList(take(V1, V2)) → a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
a__take(0, IL) → a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL)))
a__take(s(M), cons(N, IL)) → a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2) → U71(X1, X2)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A__U71(tt, L) → A__LENGTH(mark(L)) at position [0] we obtained the following new rules:

A__U71(tt, and(x0, x1)) → A__LENGTH(a__and(mark(x0), x1))
A__U71(tt, U22(x0)) → A__LENGTH(a__U22(mark(x0)))
A__U71(tt, U52(x0, x1)) → A__LENGTH(a__U52(mark(x0), x1))
A__U71(tt, U91(x0, x1, x2, x3)) → A__LENGTH(a__U91(mark(x0), x1, x2, x3))
A__U71(tt, 0) → A__LENGTH(0)
A__U71(tt, U53(x0)) → A__LENGTH(a__U53(mark(x0)))
A__U71(tt, U43(x0)) → A__LENGTH(a__U43(mark(x0)))
A__U71(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__U71(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U71(tt, U51(x0, x1, x2)) → A__LENGTH(a__U51(mark(x0), x1, x2))
A__U71(tt, U71(x0, x1)) → A__LENGTH(a__U71(mark(x0), x1))
A__U71(tt, U62(x0, x1)) → A__LENGTH(a__U62(mark(x0), x1))
A__U71(tt, nil) → A__LENGTH(nil)
A__U71(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))
A__U71(tt, U42(x0, x1)) → A__LENGTH(a__U42(mark(x0), x1))
A__U71(tt, U63(x0)) → A__LENGTH(a__U63(mark(x0)))
A__U71(tt, U32(x0)) → A__LENGTH(a__U32(mark(x0)))
A__U71(tt, isNatKind(x0)) → A__LENGTH(a__isNatKind(x0))
A__U71(tt, s(x0)) → A__LENGTH(s(mark(x0)))
A__U71(tt, zeros) → A__LENGTH(a__zeros)
A__U71(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U71(tt, U41(x0, x1, x2)) → A__LENGTH(a__U41(mark(x0), x1, x2))
A__U71(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U71(tt, U21(x0, x1)) → A__LENGTH(a__U21(mark(x0), x1))
A__U71(tt, isNatIListKind(x0)) → A__LENGTH(a__isNatIListKind(x0))
A__U71(tt, U61(x0, x1, x2)) → A__LENGTH(a__U61(mark(x0), x1, x2))
A__U71(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U71(tt, U31(x0, x1)) → A__LENGTH(a__U31(mark(x0), x1))
A__U71(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U71(tt, U12(x0)) → A__LENGTH(a__U12(mark(x0)))
A__U71(tt, U81(x0)) → A__LENGTH(a__U81(mark(x0)))
A__U71(tt, tt) → A__LENGTH(tt)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

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

A__U71(tt, and(x0, x1)) → A__LENGTH(a__and(mark(x0), x1))
A__U71(tt, U22(x0)) → A__LENGTH(a__U22(mark(x0)))
A__U71(tt, U52(x0, x1)) → A__LENGTH(a__U52(mark(x0), x1))
A__U71(tt, U91(x0, x1, x2, x3)) → A__LENGTH(a__U91(mark(x0), x1, x2, x3))
A__U71(tt, 0) → A__LENGTH(0)
A__U71(tt, U53(x0)) → A__LENGTH(a__U53(mark(x0)))
A__U71(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__U71(tt, U43(x0)) → A__LENGTH(a__U43(mark(x0)))
A__U71(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U71(tt, U51(x0, x1, x2)) → A__LENGTH(a__U51(mark(x0), x1, x2))
A__U71(tt, U71(x0, x1)) → A__LENGTH(a__U71(mark(x0), x1))
A__U71(tt, U62(x0, x1)) → A__LENGTH(a__U62(mark(x0), x1))
A__U71(tt, nil) → A__LENGTH(nil)
A__U71(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))
A__LENGTH(cons(N, L)) → A__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
A__U71(tt, U42(x0, x1)) → A__LENGTH(a__U42(mark(x0), x1))
A__U71(tt, U63(x0)) → A__LENGTH(a__U63(mark(x0)))
A__U71(tt, U32(x0)) → A__LENGTH(a__U32(mark(x0)))
A__U71(tt, isNatKind(x0)) → A__LENGTH(a__isNatKind(x0))
A__U71(tt, s(x0)) → A__LENGTH(s(mark(x0)))
A__U71(tt, zeros) → A__LENGTH(a__zeros)
A__U71(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U71(tt, U41(x0, x1, x2)) → A__LENGTH(a__U41(mark(x0), x1, x2))
A__U71(tt, U21(x0, x1)) → A__LENGTH(a__U21(mark(x0), x1))
A__U71(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U71(tt, isNatIListKind(x0)) → A__LENGTH(a__isNatIListKind(x0))
A__U71(tt, U61(x0, x1, x2)) → A__LENGTH(a__U61(mark(x0), x1, x2))
A__U71(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U71(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U71(tt, U31(x0, x1)) → A__LENGTH(a__U31(mark(x0), x1))
A__U71(tt, U12(x0)) → A__LENGTH(a__U12(mark(x0)))
A__U71(tt, U81(x0)) → A__LENGTH(a__U81(mark(x0)))
A__U71(tt, tt) → A__LENGTH(tt)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, V1, V2) → a__U62(a__isNat(V1), V2)
a__U62(tt, V2) → a__U63(a__isNatIList(V2))
a__U63(tt) → tt
a__U71(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
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__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatIListKind(take(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatList(take(V1, V2)) → a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
a__take(0, IL) → a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL)))
a__take(s(M), cons(N, IL)) → a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2) → U71(X1, X2)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 4 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ Narrowing

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

A__U71(tt, and(x0, x1)) → A__LENGTH(a__and(mark(x0), x1))
A__U71(tt, U22(x0)) → A__LENGTH(a__U22(mark(x0)))
A__U71(tt, U52(x0, x1)) → A__LENGTH(a__U52(mark(x0), x1))
A__U71(tt, U91(x0, x1, x2, x3)) → A__LENGTH(a__U91(mark(x0), x1, x2, x3))
A__U71(tt, U53(x0)) → A__LENGTH(a__U53(mark(x0)))
A__U71(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__U71(tt, U43(x0)) → A__LENGTH(a__U43(mark(x0)))
A__U71(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U71(tt, U51(x0, x1, x2)) → A__LENGTH(a__U51(mark(x0), x1, x2))
A__U71(tt, U71(x0, x1)) → A__LENGTH(a__U71(mark(x0), x1))
A__U71(tt, U62(x0, x1)) → A__LENGTH(a__U62(mark(x0), x1))
A__U71(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))
A__LENGTH(cons(N, L)) → A__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
A__U71(tt, U42(x0, x1)) → A__LENGTH(a__U42(mark(x0), x1))
A__U71(tt, U63(x0)) → A__LENGTH(a__U63(mark(x0)))
A__U71(tt, U32(x0)) → A__LENGTH(a__U32(mark(x0)))
A__U71(tt, isNatKind(x0)) → A__LENGTH(a__isNatKind(x0))
A__U71(tt, zeros) → A__LENGTH(a__zeros)
A__U71(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U71(tt, U41(x0, x1, x2)) → A__LENGTH(a__U41(mark(x0), x1, x2))
A__U71(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U71(tt, U21(x0, x1)) → A__LENGTH(a__U21(mark(x0), x1))
A__U71(tt, isNatIListKind(x0)) → A__LENGTH(a__isNatIListKind(x0))
A__U71(tt, U61(x0, x1, x2)) → A__LENGTH(a__U61(mark(x0), x1, x2))
A__U71(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U71(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U71(tt, U31(x0, x1)) → A__LENGTH(a__U31(mark(x0), x1))
A__U71(tt, U12(x0)) → A__LENGTH(a__U12(mark(x0)))
A__U71(tt, U81(x0)) → A__LENGTH(a__U81(mark(x0)))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, V1, V2) → a__U62(a__isNat(V1), V2)
a__U62(tt, V2) → a__U63(a__isNatIList(V2))
a__U63(tt) → tt
a__U71(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
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__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatIListKind(take(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatList(take(V1, V2)) → a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
a__take(0, IL) → a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL)))
a__take(s(M), cons(N, IL)) → a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2) → U71(X1, X2)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A__U71(tt, zeros) → A__LENGTH(a__zeros) at position [0] we obtained the following new rules:

A__U71(tt, zeros) → A__LENGTH(zeros)
A__U71(tt, zeros) → A__LENGTH(cons(0, zeros))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

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

A__U71(tt, and(x0, x1)) → A__LENGTH(a__and(mark(x0), x1))
A__U71(tt, U22(x0)) → A__LENGTH(a__U22(mark(x0)))
A__U71(tt, zeros) → A__LENGTH(zeros)
A__U71(tt, U52(x0, x1)) → A__LENGTH(a__U52(mark(x0), x1))
A__U71(tt, U91(x0, x1, x2, x3)) → A__LENGTH(a__U91(mark(x0), x1, x2, x3))
A__U71(tt, U53(x0)) → A__LENGTH(a__U53(mark(x0)))
A__U71(tt, U43(x0)) → A__LENGTH(a__U43(mark(x0)))
A__U71(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__U71(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U71(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U71(tt, U51(x0, x1, x2)) → A__LENGTH(a__U51(mark(x0), x1, x2))
A__U71(tt, U71(x0, x1)) → A__LENGTH(a__U71(mark(x0), x1))
A__U71(tt, U62(x0, x1)) → A__LENGTH(a__U62(mark(x0), x1))
A__U71(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))
A__LENGTH(cons(N, L)) → A__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
A__U71(tt, U63(x0)) → A__LENGTH(a__U63(mark(x0)))
A__U71(tt, U42(x0, x1)) → A__LENGTH(a__U42(mark(x0), x1))
A__U71(tt, U32(x0)) → A__LENGTH(a__U32(mark(x0)))
A__U71(tt, isNatKind(x0)) → A__LENGTH(a__isNatKind(x0))
A__U71(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U71(tt, U41(x0, x1, x2)) → A__LENGTH(a__U41(mark(x0), x1, x2))
A__U71(tt, U21(x0, x1)) → A__LENGTH(a__U21(mark(x0), x1))
A__U71(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U71(tt, isNatIListKind(x0)) → A__LENGTH(a__isNatIListKind(x0))
A__U71(tt, U61(x0, x1, x2)) → A__LENGTH(a__U61(mark(x0), x1, x2))
A__U71(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U71(tt, U31(x0, x1)) → A__LENGTH(a__U31(mark(x0), x1))
A__U71(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U71(tt, U12(x0)) → A__LENGTH(a__U12(mark(x0)))
A__U71(tt, U81(x0)) → A__LENGTH(a__U81(mark(x0)))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, V1, V2) → a__U62(a__isNat(V1), V2)
a__U62(tt, V2) → a__U63(a__isNatIList(V2))
a__U63(tt) → tt
a__U71(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
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__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatIListKind(take(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatList(take(V1, V2)) → a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
a__take(0, IL) → a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL)))
a__take(s(M), cons(N, IL)) → a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2) → U71(X1, X2)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

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

A__U71(tt, and(x0, x1)) → A__LENGTH(a__and(mark(x0), x1))
A__U71(tt, U22(x0)) → A__LENGTH(a__U22(mark(x0)))
A__U71(tt, U52(x0, x1)) → A__LENGTH(a__U52(mark(x0), x1))
A__U71(tt, U91(x0, x1, x2, x3)) → A__LENGTH(a__U91(mark(x0), x1, x2, x3))
A__U71(tt, U53(x0)) → A__LENGTH(a__U53(mark(x0)))
A__U71(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__U71(tt, U43(x0)) → A__LENGTH(a__U43(mark(x0)))
A__U71(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U71(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U71(tt, U51(x0, x1, x2)) → A__LENGTH(a__U51(mark(x0), x1, x2))
A__U71(tt, U71(x0, x1)) → A__LENGTH(a__U71(mark(x0), x1))
A__U71(tt, U62(x0, x1)) → A__LENGTH(a__U62(mark(x0), x1))
A__U71(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))
A__LENGTH(cons(N, L)) → A__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
A__U71(tt, U42(x0, x1)) → A__LENGTH(a__U42(mark(x0), x1))
A__U71(tt, U63(x0)) → A__LENGTH(a__U63(mark(x0)))
A__U71(tt, U32(x0)) → A__LENGTH(a__U32(mark(x0)))
A__U71(tt, isNatKind(x0)) → A__LENGTH(a__isNatKind(x0))
A__U71(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U71(tt, U41(x0, x1, x2)) → A__LENGTH(a__U41(mark(x0), x1, x2))
A__U71(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U71(tt, U21(x0, x1)) → A__LENGTH(a__U21(mark(x0), x1))
A__U71(tt, isNatIListKind(x0)) → A__LENGTH(a__isNatIListKind(x0))
A__U71(tt, U61(x0, x1, x2)) → A__LENGTH(a__U61(mark(x0), x1, x2))
A__U71(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U71(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U71(tt, U31(x0, x1)) → A__LENGTH(a__U31(mark(x0), x1))
A__U71(tt, U12(x0)) → A__LENGTH(a__U12(mark(x0)))
A__U71(tt, U81(x0)) → A__LENGTH(a__U81(mark(x0)))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, V1, V2) → a__U62(a__isNat(V1), V2)
a__U62(tt, V2) → a__U63(a__isNatIList(V2))
a__U63(tt) → tt
a__U71(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
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__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatIListKind(take(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatList(take(V1, V2)) → a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
a__take(0, IL) → a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL)))
a__take(s(M), cons(N, IL)) → a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2) → U71(X1, X2)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A__U71(tt, U81(x0)) → A__LENGTH(a__U81(mark(x0)))

The remaining pairs can at least be oriented weakly.

A__U71(tt, and(x0, x1)) → A__LENGTH(a__and(mark(x0), x1))
A__U71(tt, U22(x0)) → A__LENGTH(a__U22(mark(x0)))
A__U71(tt, U52(x0, x1)) → A__LENGTH(a__U52(mark(x0), x1))
A__U71(tt, U91(x0, x1, x2, x3)) → A__LENGTH(a__U91(mark(x0), x1, x2, x3))
A__U71(tt, U53(x0)) → A__LENGTH(a__U53(mark(x0)))
A__U71(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__U71(tt, U43(x0)) → A__LENGTH(a__U43(mark(x0)))
A__U71(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U71(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U71(tt, U51(x0, x1, x2)) → A__LENGTH(a__U51(mark(x0), x1, x2))
A__U71(tt, U71(x0, x1)) → A__LENGTH(a__U71(mark(x0), x1))
A__U71(tt, U62(x0, x1)) → A__LENGTH(a__U62(mark(x0), x1))
A__U71(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))
A__LENGTH(cons(N, L)) → A__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
A__U71(tt, U42(x0, x1)) → A__LENGTH(a__U42(mark(x0), x1))
A__U71(tt, U63(x0)) → A__LENGTH(a__U63(mark(x0)))
A__U71(tt, U32(x0)) → A__LENGTH(a__U32(mark(x0)))
A__U71(tt, isNatKind(x0)) → A__LENGTH(a__isNatKind(x0))
A__U71(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U71(tt, U41(x0, x1, x2)) → A__LENGTH(a__U41(mark(x0), x1, x2))
A__U71(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U71(tt, U21(x0, x1)) → A__LENGTH(a__U21(mark(x0), x1))
A__U71(tt, isNatIListKind(x0)) → A__LENGTH(a__isNatIListKind(x0))
A__U71(tt, U61(x0, x1, x2)) → A__LENGTH(a__U61(mark(x0), x1, x2))
A__U71(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U71(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U71(tt, U31(x0, x1)) → A__LENGTH(a__U31(mark(x0), x1))
A__U71(tt, U12(x0)) → A__LENGTH(a__U12(mark(x0)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__LENGTH(x₁)) = x₁
POL(A__U71(x₁, x₂)) = x₁
POL(U11(x₁, x₂)) = 1
POL(U12(x₁)) = 1
POL(U21(x₁, x₂)) = 1
POL(U22(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 0
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 1
POL(U62(x₁, x₂)) = 0
POL(U63(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃, x₄)) = 0
POL(a__U11(x₁, x₂)) = 1
POL(a__U12(x₁)) = 1
POL(a__U21(x₁, x₂)) = 1
POL(a__U22(x₁)) = 1
POL(a__U31(x₁, x₂)) = 1
POL(a__U32(x₁)) = 1
POL(a__U41(x₁, x₂, x₃)) = 1
POL(a__U42(x₁, x₂)) = x₁
POL(a__U43(x₁)) = 1
POL(a__U51(x₁, x₂, x₃)) = x₁
POL(a__U52(x₁, x₂)) = 1
POL(a__U53(x₁)) = 1
POL(a__U61(x₁, x₂, x₃)) = 1
POL(a__U62(x₁, x₂)) = 1
POL(a__U63(x₁)) = x₁
POL(a__U71(x₁, x₂)) = x₁
POL(a__U81(x₁)) = 0
POL(a__U91(x₁, x₂, x₃, x₄)) = 1
POL(a__and(x₁, x₂)) = x₁
POL(a__isNat(x₁)) = 1
POL(a__isNatIList(x₁)) = 1
POL(a__isNatIListKind(x₁)) = 1
POL(a__isNatKind(x₁)) = 1
POL(a__isNatList(x₁)) = 1
POL(a__length(x₁)) = 1
POL(a__take(x₁, x₂)) = 1
POL(a__zeros) = 1
POL(and(x₁, x₂)) = 0
POL(cons(x₁, x₂)) = 1
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = 1
POL(nil) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 1
POL(zeros) = 0

The following usable rules [17] were oriented:

a__isNat(0) → tt
a__U91(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__U81(tt) → nil
a__isNatIList(zeros) → tt
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNatIListKind(zeros) → tt
a__isNatIListKind(nil) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U63(tt) → tt
a__U71(tt, L) → s(a__length(mark(L)))
a__U61(tt, V1, V2) → a__U62(a__isNat(V1), V2)
a__U62(tt, V2) → a__U63(a__isNatIList(V2))
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(isNatIList(X)) → a__isNatIList(X)
mark(U43(X)) → a__U43(mark(X))
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U53(X)) → a__U53(mark(X))
mark(U63(X)) → a__U63(mark(X))
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(U81(X)) → a__U81(mark(X))
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__and(tt, X) → mark(X)
a__isNatIListKind(take(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatList(take(V1, V2)) → a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
a__take(0, IL) → a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL)))
a__take(s(M), cons(N, IL)) → a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U53(X) → U53(X)
a__U52(X1, X2) → U52(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__isNatIList(X) → isNatIList(X)
a__U43(X) → U43(X)
a__U42(X1, X2) → U42(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__and(X1, X2) → and(X1, X2)
a__take(X1, X2) → take(X1, X2)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__U81(X) → U81(X)
a__length(X) → length(X)
a__U71(X1, X2) → U71(X1, X2)
a__U63(X) → U63(X)
a__U62(X1, X2) → U62(X1, X2)
mark(nil) → nil
a__zeros → zeros
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__zeros → cons(0, zeros)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

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

A__U71(tt, and(x0, x1)) → A__LENGTH(a__and(mark(x0), x1))
A__U71(tt, U22(x0)) → A__LENGTH(a__U22(mark(x0)))
A__U71(tt, U52(x0, x1)) → A__LENGTH(a__U52(mark(x0), x1))
A__U71(tt, U91(x0, x1, x2, x3)) → A__LENGTH(a__U91(mark(x0), x1, x2, x3))
A__U71(tt, U53(x0)) → A__LENGTH(a__U53(mark(x0)))
A__U71(tt, U43(x0)) → A__LENGTH(a__U43(mark(x0)))
A__U71(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__U71(tt, zeros) → A__LENGTH(cons(0, zeros))
A__U71(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U71(tt, U51(x0, x1, x2)) → A__LENGTH(a__U51(mark(x0), x1, x2))
A__U71(tt, U71(x0, x1)) → A__LENGTH(a__U71(mark(x0), x1))
A__U71(tt, U62(x0, x1)) → A__LENGTH(a__U62(mark(x0), x1))
A__U71(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))
A__LENGTH(cons(N, L)) → A__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
A__U71(tt, U42(x0, x1)) → A__LENGTH(a__U42(mark(x0), x1))
A__U71(tt, U63(x0)) → A__LENGTH(a__U63(mark(x0)))
A__U71(tt, U32(x0)) → A__LENGTH(a__U32(mark(x0)))
A__U71(tt, isNatKind(x0)) → A__LENGTH(a__isNatKind(x0))
A__U71(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U71(tt, U41(x0, x1, x2)) → A__LENGTH(a__U41(mark(x0), x1, x2))
A__U71(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U71(tt, U21(x0, x1)) → A__LENGTH(a__U21(mark(x0), x1))
A__U71(tt, isNatIListKind(x0)) → A__LENGTH(a__isNatIListKind(x0))
A__U71(tt, U61(x0, x1, x2)) → A__LENGTH(a__U61(mark(x0), x1, x2))
A__U71(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U71(tt, U31(x0, x1)) → A__LENGTH(a__U31(mark(x0), x1))
A__U71(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U71(tt, U12(x0)) → A__LENGTH(a__U12(mark(x0)))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, V1, V2) → a__U62(a__isNat(V1), V2)
a__U62(tt, V2) → a__U63(a__isNatIList(V2))
a__U63(tt) → tt
a__U71(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
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__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatIListKind(take(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatList(take(V1, V2)) → a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
a__take(0, IL) → a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL)))
a__take(s(M), cons(N, IL)) → a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2) → U71(X1, X2)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A__U71(tt, and(x0, x1)) → A__LENGTH(a__and(mark(x0), x1))
A__U71(tt, U22(x0)) → A__LENGTH(a__U22(mark(x0)))
A__U71(tt, U52(x0, x1)) → A__LENGTH(a__U52(mark(x0), x1))
A__U71(tt, U91(x0, x1, x2, x3)) → A__LENGTH(a__U91(mark(x0), x1, x2, x3))
A__U71(tt, U53(x0)) → A__LENGTH(a__U53(mark(x0)))
A__U71(tt, U43(x0)) → A__LENGTH(a__U43(mark(x0)))
A__U71(tt, isNat(x0)) → A__LENGTH(a__isNat(x0))
A__U71(tt, isNatIList(x0)) → A__LENGTH(a__isNatIList(x0))
A__U71(tt, U51(x0, x1, x2)) → A__LENGTH(a__U51(mark(x0), x1, x2))
A__U71(tt, U71(x0, x1)) → A__LENGTH(a__U71(mark(x0), x1))
A__U71(tt, U62(x0, x1)) → A__LENGTH(a__U62(mark(x0), x1))
A__U71(tt, isNatList(x0)) → A__LENGTH(a__isNatList(x0))
A__U71(tt, U42(x0, x1)) → A__LENGTH(a__U42(mark(x0), x1))
A__U71(tt, U63(x0)) → A__LENGTH(a__U63(mark(x0)))
A__U71(tt, U32(x0)) → A__LENGTH(a__U32(mark(x0)))
A__U71(tt, isNatKind(x0)) → A__LENGTH(a__isNatKind(x0))
A__U71(tt, cons(x0, x1)) → A__LENGTH(cons(mark(x0), x1))
A__U71(tt, U41(x0, x1, x2)) → A__LENGTH(a__U41(mark(x0), x1, x2))
A__U71(tt, U11(x0, x1)) → A__LENGTH(a__U11(mark(x0), x1))
A__U71(tt, U21(x0, x1)) → A__LENGTH(a__U21(mark(x0), x1))
A__U71(tt, U61(x0, x1, x2)) → A__LENGTH(a__U61(mark(x0), x1, x2))
A__U71(tt, U31(x0, x1)) → A__LENGTH(a__U31(mark(x0), x1))
A__U71(tt, U12(x0)) → A__LENGTH(a__U12(mark(x0)))

The remaining pairs can at least be oriented weakly.

A__U71(tt, zeros) → A__LENGTH(cons(0, zeros))
A__LENGTH(cons(N, L)) → A__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
A__U71(tt, isNatIListKind(x0)) → A__LENGTH(a__isNatIListKind(x0))
A__U71(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U71(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__LENGTH(x₁)) = x₁
POL(A__U71(x₁, x₂)) = 1 + x₂
POL(U11(x₁, x₂)) = 1
POL(U12(x₁)) = 1
POL(U21(x₁, x₂)) = 1
POL(U22(x₁)) = 1
POL(U31(x₁, x₂)) = 1
POL(U32(x₁)) = 1
POL(U41(x₁, x₂, x₃)) = 1 + x₃
POL(U42(x₁, x₂)) = 1 + x₂
POL(U43(x₁)) = 1
POL(U51(x₁, x₂, x₃)) = 1 + x₃
POL(U52(x₁, x₂)) = 1
POL(U53(x₁)) = 1
POL(U61(x₁, x₂, x₃)) = 1
POL(U62(x₁, x₂)) = 1
POL(U63(x₁)) = 1
POL(U71(x₁, x₂)) = 1
POL(U81(x₁)) = 1
POL(U91(x₁, x₂, x₃, x₄)) = 1
POL(a__U11(x₁, x₂)) = 1
POL(a__U12(x₁)) = 1
POL(a__U21(x₁, x₂)) = 1
POL(a__U22(x₁)) = 1
POL(a__U31(x₁, x₂)) = 1
POL(a__U32(x₁)) = 1
POL(a__U41(x₁, x₂, x₃)) = 1 + x₃
POL(a__U42(x₁, x₂)) = 1 + x₂
POL(a__U43(x₁)) = 1
POL(a__U51(x₁, x₂, x₃)) = 1 + x₃
POL(a__U52(x₁, x₂)) = 1
POL(a__U53(x₁)) = 1
POL(a__U61(x₁, x₂, x₃)) = 1
POL(a__U62(x₁, x₂)) = 1
POL(a__U63(x₁)) = 1
POL(a__U71(x₁, x₂)) = 1
POL(a__U81(x₁)) = 1
POL(a__U91(x₁, x₂, x₃, x₄)) = 1
POL(a__and(x₁, x₂)) = 1 + x₂
POL(a__isNat(x₁)) = 1
POL(a__isNatIList(x₁)) = 1 + x₁
POL(a__isNatIListKind(x₁)) = 1
POL(a__isNatKind(x₁)) = 1
POL(a__isNatList(x₁)) = 1 + x₁
POL(a__length(x₁)) = 1 + x₁
POL(a__take(x₁, x₂)) = 1
POL(a__zeros) = 1
POL(and(x₁, x₂)) = 1 + x₂
POL(cons(x₁, x₂)) = 1 + x₂
POL(isNat(x₁)) = 1
POL(isNatIList(x₁)) = 1 + x₁
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 1
POL(isNatList(x₁)) = 1 + x₁
POL(length(x₁)) = 1 + x₁
POL(mark(x₁)) = 1 + x₁
POL(nil) = 1
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

a__isNat(0) → tt
a__U91(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__U81(tt) → nil
a__isNatIList(zeros) → tt
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNatIListKind(zeros) → tt
a__isNatIListKind(nil) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U63(tt) → tt
a__U71(tt, L) → s(a__length(mark(L)))
a__U61(tt, V1, V2) → a__U62(a__isNat(V1), V2)
a__U62(tt, V2) → a__U63(a__isNatIList(V2))
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(isNatIList(X)) → a__isNatIList(X)
mark(U43(X)) → a__U43(mark(X))
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U53(X)) → a__U53(mark(X))
mark(U63(X)) → a__U63(mark(X))
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(U81(X)) → a__U81(mark(X))
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__and(tt, X) → mark(X)
a__isNatIListKind(take(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatList(take(V1, V2)) → a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
a__take(0, IL) → a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL)))
a__take(s(M), cons(N, IL)) → a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U53(X) → U53(X)
a__U52(X1, X2) → U52(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__isNatIList(X) → isNatIList(X)
a__U43(X) → U43(X)
a__U42(X1, X2) → U42(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__and(X1, X2) → and(X1, X2)
a__take(X1, X2) → take(X1, X2)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__U81(X) → U81(X)
a__length(X) → length(X)
a__U71(X1, X2) → U71(X1, X2)
a__U63(X) → U63(X)
a__U62(X1, X2) → U62(X1, X2)
mark(nil) → nil
a__zeros → zeros
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__zeros → cons(0, zeros)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

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

A__U71(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U71(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))
A__U71(tt, zeros) → A__LENGTH(cons(0, zeros))
A__LENGTH(cons(N, L)) → A__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
A__U71(tt, isNatIListKind(x0)) → A__LENGTH(a__isNatIListKind(x0))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, V1, V2) → a__U62(a__isNat(V1), V2)
a__U62(tt, V2) → a__U63(a__isNatIList(V2))
a__U63(tt) → tt
a__U71(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
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__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatIListKind(take(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatList(take(V1, V2)) → a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
a__take(0, IL) → a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL)))
a__take(s(M), cons(N, IL)) → a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2) → U71(X1, X2)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A__U71(tt, length(x0)) → A__LENGTH(a__length(mark(x0)))

The remaining pairs can at least be oriented weakly.

A__U71(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U71(tt, zeros) → A__LENGTH(cons(0, zeros))
A__LENGTH(cons(N, L)) → A__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
A__U71(tt, isNatIListKind(x0)) → A__LENGTH(a__isNatIListKind(x0))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__LENGTH(x₁)) = x₁
POL(A__U71(x₁, x₂)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = x₁
POL(U31(x₁, x₂)) = 1
POL(U32(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 0
POL(U43(x₁)) = 1
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 1
POL(U53(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 0
POL(U62(x₁, x₂)) = 0
POL(U63(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃, x₄)) = x₁
POL(a__U11(x₁, x₂)) = 1
POL(a__U12(x₁)) = 1
POL(a__U21(x₁, x₂)) = 1
POL(a__U22(x₁)) = x₁
POL(a__U31(x₁, x₂)) = 1
POL(a__U32(x₁)) = x₁
POL(a__U41(x₁, x₂, x₃)) = 1
POL(a__U42(x₁, x₂)) = 1
POL(a__U43(x₁)) = 1
POL(a__U51(x₁, x₂, x₃)) = 1
POL(a__U52(x₁, x₂)) = 1
POL(a__U53(x₁)) = 1
POL(a__U61(x₁, x₂, x₃)) = x₁
POL(a__U62(x₁, x₂)) = x₁
POL(a__U63(x₁)) = 1
POL(a__U71(x₁, x₂)) = 0
POL(a__U81(x₁)) = 1
POL(a__U91(x₁, x₂, x₃, x₄)) = x₁
POL(a__and(x₁, x₂)) = x₁
POL(a__isNat(x₁)) = 1
POL(a__isNatIList(x₁)) = 1
POL(a__isNatIListKind(x₁)) = 1
POL(a__isNatKind(x₁)) = 1
POL(a__isNatList(x₁)) = 1
POL(a__length(x₁)) = 0
POL(a__take(x₁, x₂)) = 1
POL(a__zeros) = 1
POL(and(x₁, x₂)) = 0
POL(cons(x₁, x₂)) = 1
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 1
POL(length(x₁)) = 0
POL(mark(x₁)) = 1
POL(nil) = 1
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 0
POL(tt) = 1
POL(zeros) = 0

The following usable rules [17] were oriented:

a__isNat(0) → tt
a__U91(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__U81(tt) → nil
a__isNatIList(zeros) → tt
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNatIListKind(zeros) → tt
a__isNatIListKind(nil) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U63(tt) → tt
a__U71(tt, L) → s(a__length(mark(L)))
a__U61(tt, V1, V2) → a__U62(a__isNat(V1), V2)
a__U62(tt, V2) → a__U63(a__isNatIList(V2))
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(isNatIList(X)) → a__isNatIList(X)
mark(U43(X)) → a__U43(mark(X))
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U53(X)) → a__U53(mark(X))
mark(U63(X)) → a__U63(mark(X))
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(U81(X)) → a__U81(mark(X))
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__and(tt, X) → mark(X)
a__isNatIListKind(take(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatList(take(V1, V2)) → a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
a__take(0, IL) → a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL)))
a__take(s(M), cons(N, IL)) → a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U53(X) → U53(X)
a__U52(X1, X2) → U52(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__isNatIList(X) → isNatIList(X)
a__U43(X) → U43(X)
a__U42(X1, X2) → U42(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__and(X1, X2) → and(X1, X2)
a__take(X1, X2) → take(X1, X2)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__U81(X) → U81(X)
a__length(X) → length(X)
a__U71(X1, X2) → U71(X1, X2)
a__U63(X) → U63(X)
a__U62(X1, X2) → U62(X1, X2)
mark(nil) → nil
a__zeros → zeros
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__zeros → cons(0, zeros)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

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

A__U71(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__U71(tt, zeros) → A__LENGTH(cons(0, zeros))
A__LENGTH(cons(N, L)) → A__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
A__U71(tt, isNatIListKind(x0)) → A__LENGTH(a__isNatIListKind(x0))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, V1, V2) → a__U62(a__isNat(V1), V2)
a__U62(tt, V2) → a__U63(a__isNatIList(V2))
a__U63(tt) → tt
a__U71(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
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__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatIListKind(take(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatList(take(V1, V2)) → a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
a__take(0, IL) → a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL)))
a__take(s(M), cons(N, IL)) → a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2) → U71(X1, X2)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A__U71(tt, zeros) → A__LENGTH(cons(0, zeros))

The remaining pairs can at least be oriented weakly.

A__U71(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__LENGTH(cons(N, L)) → A__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
A__U71(tt, isNatIListKind(x0)) → A__LENGTH(a__isNatIListKind(x0))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__LENGTH(x₁)) = x₁
POL(A__U71(x₁, x₂)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 1
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 1
POL(U41(x₁, x₂, x₃)) = x₁
POL(U42(x₁, x₂)) = 0
POL(U43(x₁)) = x₁
POL(U51(x₁, x₂, x₃)) = x₃
POL(U52(x₁, x₂)) = x₂
POL(U53(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = 1
POL(U62(x₁, x₂)) = 1
POL(U63(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U81(x₁)) = x₁
POL(U91(x₁, x₂, x₃, x₄)) = 0
POL(a__U11(x₁, x₂)) = 1
POL(a__U12(x₁)) = 1
POL(a__U21(x₁, x₂)) = 1
POL(a__U22(x₁)) = 1
POL(a__U31(x₁, x₂)) = 1
POL(a__U32(x₁)) = 1
POL(a__U41(x₁, x₂, x₃)) = x₁
POL(a__U42(x₁, x₂)) = 1
POL(a__U43(x₁)) = x₁
POL(a__U51(x₁, x₂, x₃)) = x₃
POL(a__U52(x₁, x₂)) = x₂
POL(a__U53(x₁)) = x₁
POL(a__U61(x₁, x₂, x₃)) = 1
POL(a__U62(x₁, x₂)) = 1
POL(a__U63(x₁)) = 1
POL(a__U71(x₁, x₂)) = 0
POL(a__U81(x₁)) = x₁
POL(a__U91(x₁, x₂, x₃, x₄)) = 1
POL(a__and(x₁, x₂)) = x₁ + x₂
POL(a__isNat(x₁)) = 1
POL(a__isNatIList(x₁)) = 1
POL(a__isNatIListKind(x₁)) = 1
POL(a__isNatKind(x₁)) = 1
POL(a__isNatList(x₁)) = x₁
POL(a__length(x₁)) = x₁
POL(a__take(x₁, x₂)) = 1
POL(a__zeros) = 1
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = x₁
POL(mark(x₁)) = 1 + x₁
POL(nil) = 1
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 1
POL(tt) = 1
POL(zeros) = 0

The following usable rules [17] were oriented:

a__isNat(0) → tt
a__U91(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__U81(tt) → nil
a__isNatIList(zeros) → tt
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNatIListKind(zeros) → tt
a__isNatIListKind(nil) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U63(tt) → tt
a__U71(tt, L) → s(a__length(mark(L)))
a__U61(tt, V1, V2) → a__U62(a__isNat(V1), V2)
a__U62(tt, V2) → a__U63(a__isNatIList(V2))
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(isNatIList(X)) → a__isNatIList(X)
mark(U43(X)) → a__U43(mark(X))
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U53(X)) → a__U53(mark(X))
mark(U63(X)) → a__U63(mark(X))
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(U81(X)) → a__U81(mark(X))
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__and(tt, X) → mark(X)
a__isNatIListKind(take(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatList(take(V1, V2)) → a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
a__take(0, IL) → a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL)))
a__take(s(M), cons(N, IL)) → a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U53(X) → U53(X)
a__U52(X1, X2) → U52(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__isNatIList(X) → isNatIList(X)
a__U43(X) → U43(X)
a__U42(X1, X2) → U42(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__and(X1, X2) → and(X1, X2)
a__take(X1, X2) → take(X1, X2)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__U81(X) → U81(X)
a__length(X) → length(X)
a__U71(X1, X2) → U71(X1, X2)
a__U63(X) → U63(X)
a__U62(X1, X2) → U62(X1, X2)
mark(nil) → nil
a__zeros → zeros
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__zeros → cons(0, zeros)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

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

A__U71(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__LENGTH(cons(N, L)) → A__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
A__U71(tt, isNatIListKind(x0)) → A__LENGTH(a__isNatIListKind(x0))

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, V1, V2) → a__U62(a__isNat(V1), V2)
a__U62(tt, V2) → a__U63(a__isNatIList(V2))
a__U63(tt) → tt
a__U71(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
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__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatIListKind(take(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatList(take(V1, V2)) → a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
a__take(0, IL) → a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL)))
a__take(s(M), cons(N, IL)) → a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2) → U71(X1, X2)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A__U71(tt, isNatIListKind(x0)) → A__LENGTH(a__isNatIListKind(x0))

The remaining pairs can at least be oriented weakly.

A__U71(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__LENGTH(cons(N, L)) → A__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__LENGTH(x₁)) = x₁
POL(A__U71(x₁, x₂)) = 1
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 0
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = x₁
POL(U62(x₁, x₂)) = 0
POL(U63(x₁)) = x₁
POL(U71(x₁, x₂)) = 0
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃, x₄)) = 1
POL(a__U11(x₁, x₂)) = 0
POL(a__U12(x₁)) = 0
POL(a__U21(x₁, x₂)) = x₁
POL(a__U22(x₁)) = x₁
POL(a__U31(x₁, x₂)) = x₁
POL(a__U32(x₁)) = 0
POL(a__U41(x₁, x₂, x₃)) = 0
POL(a__U42(x₁, x₂)) = 0
POL(a__U43(x₁)) = 0
POL(a__U51(x₁, x₂, x₃)) = 0
POL(a__U52(x₁, x₂)) = 0
POL(a__U53(x₁)) = 0
POL(a__U61(x₁, x₂, x₃)) = x₁
POL(a__U62(x₁, x₂)) = 0
POL(a__U63(x₁)) = x₁
POL(a__U71(x₁, x₂)) = 0
POL(a__U81(x₁)) = 0
POL(a__U91(x₁, x₂, x₃, x₄)) = 1
POL(a__and(x₁, x₂)) = x₂
POL(a__isNat(x₁)) = 0
POL(a__isNatIList(x₁)) = 0
POL(a__isNatIListKind(x₁)) = 0
POL(a__isNatKind(x₁)) = 0
POL(a__isNatList(x₁)) = 0
POL(a__length(x₁)) = x₁
POL(a__take(x₁, x₂)) = 1
POL(a__zeros) = 1
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = 1
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 1
POL(tt) = 0
POL(zeros) = 1

The following usable rules [17] were oriented:

a__isNat(0) → tt
a__U91(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__U81(tt) → nil
a__isNatIList(zeros) → tt
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNatIListKind(zeros) → tt
a__isNatIListKind(nil) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatKind(0) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U63(tt) → tt
a__U71(tt, L) → s(a__length(mark(L)))
a__U61(tt, V1, V2) → a__U62(a__isNat(V1), V2)
a__U62(tt, V2) → a__U63(a__isNatIList(V2))
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(isNatIList(X)) → a__isNatIList(X)
mark(U43(X)) → a__U43(mark(X))
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U53(X)) → a__U53(mark(X))
mark(U63(X)) → a__U63(mark(X))
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(U81(X)) → a__U81(mark(X))
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__and(tt, X) → mark(X)
a__isNatIListKind(take(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatList(take(V1, V2)) → a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
a__take(0, IL) → a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL)))
a__take(s(M), cons(N, IL)) → a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U53(X) → U53(X)
a__U52(X1, X2) → U52(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__isNatIList(X) → isNatIList(X)
a__U43(X) → U43(X)
a__U42(X1, X2) → U42(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__and(X1, X2) → and(X1, X2)
a__take(X1, X2) → take(X1, X2)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__U81(X) → U81(X)
a__length(X) → length(X)
a__U71(X1, X2) → U71(X1, X2)
a__U63(X) → U63(X)
a__U62(X1, X2) → U62(X1, X2)
mark(nil) → nil
a__zeros → zeros
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__zeros → cons(0, zeros)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

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

A__U71(tt, take(x0, x1)) → A__LENGTH(a__take(mark(x0), mark(x1)))
A__LENGTH(cons(N, L)) → A__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, V1, V2) → a__U62(a__isNat(V1), V2)
a__U62(tt, V2) → a__U63(a__isNatIList(V2))
a__U63(tt) → tt
a__U71(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__and(tt, X) → mark(X)
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__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatIListKind(take(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatList(take(V1, V2)) → a__U61(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
a__take(0, IL) → a__U81(a__and(a__isNatIList(IL), isNatIListKind(IL)))
a__take(s(M), cons(N, IL)) → a__U91(a__and(a__and(a__isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2) → U71(X1, X2)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)

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