Termination w.r.t. Q proof of /tmp/tmpefehtv/OvConsOS_nokinds_noand

Termination w.r.t. Q of the following Term Rewriting System could be proven:

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

a__zeros → cons(0, zeros)
a__U11(tt) → tt
a__U21(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIList(V2))
a__U62(tt) → tt
a__U71(tt, L, N) → a__U72(a__isNat(N), L)
a__U72(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → a__U92(a__isNat(M), IL, M, N)
a__U92(tt, IL, M, N) → a__U93(a__isNat(N), IL, M, N)
a__U93(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(V) → a__U31(a__isNatList(V))
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__isNatList(take(V1, V2)) → a__U61(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__isNatList(L), L, N)
a__take(0, IL) → a__U81(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U91(a__isNatIList(IL), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X)) → a__U11(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
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(U92(X1, X2, X3, X4)) → a__U92(mark(X1), X2, X3, X4)
mark(U93(X1, X2, X3, X4)) → a__U93(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(X) → U11(X)
a__U21(X) → U21(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__isNatList(X) → isNatList(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__U92(X1, X2, X3, X4) → U92(X1, X2, X3, X4)
a__U93(X1, X2, X3, X4) → U93(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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) → tt
a__U21(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIList(V2))
a__U62(tt) → tt
a__U71(tt, L, N) → a__U72(a__isNat(N), L)
a__U72(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → a__U92(a__isNat(M), IL, M, N)
a__U92(tt, IL, M, N) → a__U93(a__isNat(N), IL, M, N)
a__U93(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(V) → a__U31(a__isNatList(V))
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__isNatList(take(V1, V2)) → a__U61(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__isNatList(L), L, N)
a__take(0, IL) → a__U81(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U91(a__isNatIList(IL), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X)) → a__U11(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
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(U92(X1, X2, X3, X4)) → a__U92(mark(X1), X2, X3, X4)
mark(U93(X1, X2, X3, X4)) → a__U93(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(X) → U11(X)
a__U21(X) → U21(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__isNatList(X) → isNatList(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__U92(X1, X2, X3, X4) → U92(X1, X2, X3, X4)
a__U93(X1, X2, X3, X4) → U93(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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__LENGTH(cons(N, L)) → A__U71(a__isNatList(L), L, N)
A__U41(tt, V2) → A__ISNATILIST(V2)
A__U71(tt, L, N) → A__U72(a__isNat(N), L)
A__ISNAT(s(V1)) → A__U21(a__isNat(V1))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(U93(X1, X2, X3, X4)) → A__U93(mark(X1), X2, X3, X4)
A__U71(tt, L, N) → A__ISNAT(N)
MARK(length(X)) → MARK(X)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__isNat(V1), V2)
MARK(U81(X)) → A__U81(mark(X))
A__U93(tt, IL, M, N) → MARK(N)
MARK(take(X1, X2)) → MARK(X1)
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
MARK(U51(X1, X2)) → MARK(X1)
A__ISNATILIST(V) → A__ISNATLIST(V)
A__ISNATLIST(take(V1, V2)) → A__U61(a__isNat(V1), V2)
A__U51(tt, V2) → A__ISNATLIST(V2)
MARK(U11(X)) → A__U11(mark(X))
A__TAKE(0, IL) → A__ISNATILIST(IL)
MARK(U62(X)) → A__U62(mark(X))
A__U92(tt, IL, M, N) → A__ISNAT(N)
MARK(U41(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNATILIST(cons(V1, V2)) → A__ISNAT(V1)
A__ISNATLIST(take(V1, V2)) → A__ISNAT(V1)
MARK(U42(X)) → MARK(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
A__U41(tt, V2) → A__U42(a__isNatIList(V2))
MARK(U91(X1, X2, X3, X4)) → A__U91(mark(X1), X2, X3, X4)
MARK(take(X1, X2)) → MARK(X2)
MARK(U72(X1, X2)) → A__U72(mark(X1), X2)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U62(X)) → MARK(X)
MARK(zeros) → A__ZEROS
MARK(s(X)) → MARK(X)
A__TAKE(s(M), cons(N, IL)) → A__ISNATILIST(IL)
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(U92(X1, X2, X3, X4)) → MARK(X1)
A__TAKE(s(M), cons(N, IL)) → A__U91(a__isNatIList(IL), IL, M, N)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U81(X)) → MARK(X)
MARK(U72(X1, X2)) → MARK(X1)
MARK(U21(X)) → A__U21(mark(X))
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__U91(tt, IL, M, N) → A__U92(a__isNat(M), IL, M, N)
MARK(U93(X1, X2, X3, X4)) → MARK(X1)
MARK(U52(X)) → MARK(X)
A__ISNAT(length(V1)) → A__U11(a__isNatList(V1))
MARK(U61(X1, X2)) → A__U61(mark(X1), X2)
MARK(U51(X1, X2)) → A__U51(mark(X1), X2)
A__U72(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
MARK(U31(X)) → MARK(X)
MARK(U52(X)) → A__U52(mark(X))
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__U72(tt, L) → MARK(L)
MARK(U42(X)) → A__U42(mark(X))
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__U61(tt, V2) → A__U62(a__isNatIList(V2))
A__U51(tt, V2) → A__U52(a__isNatList(V2))
MARK(U21(X)) → MARK(X)
MARK(U31(X)) → A__U31(mark(X))
MARK(U61(X1, X2)) → MARK(X1)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U92(X1, X2, X3, X4)) → A__U92(mark(X1), X2, X3, X4)
A__U91(tt, IL, M, N) → A__ISNAT(M)
A__ISNATILIST(V) → A__U31(a__isNatList(V))
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__U92(tt, IL, M, N) → A__U93(a__isNat(N), IL, M, N)
A__TAKE(0, IL) → A__U81(a__isNatIList(IL))
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNat(V1), V2)
A__U61(tt, V2) → A__ISNATILIST(V2)
MARK(U11(X)) → MARK(X)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt) → tt
a__U21(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIList(V2))
a__U62(tt) → tt
a__U71(tt, L, N) → a__U72(a__isNat(N), L)
a__U72(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → a__U92(a__isNat(M), IL, M, N)
a__U92(tt, IL, M, N) → a__U93(a__isNat(N), IL, M, N)
a__U93(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(V) → a__U31(a__isNatList(V))
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__isNatList(take(V1, V2)) → a__U61(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__isNatList(L), L, N)
a__take(0, IL) → a__U81(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U91(a__isNatIList(IL), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X)) → a__U11(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
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(U92(X1, X2, X3, X4)) → a__U92(mark(X1), X2, X3, X4)
mark(U93(X1, X2, X3, X4)) → a__U93(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(X) → U11(X)
a__U21(X) → U21(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__isNatList(X) → isNatList(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__U92(X1, X2, X3, X4) → U92(X1, X2, X3, X4)
a__U93(X1, X2, X3, X4) → U93(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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__LENGTH(cons(N, L)) → A__U71(a__isNatList(L), L, N)
A__U41(tt, V2) → A__ISNATILIST(V2)
A__U71(tt, L, N) → A__U72(a__isNat(N), L)
A__ISNAT(s(V1)) → A__U21(a__isNat(V1))
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(U93(X1, X2, X3, X4)) → A__U93(mark(X1), X2, X3, X4)
A__U71(tt, L, N) → A__ISNAT(N)
MARK(length(X)) → MARK(X)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__isNat(V1), V2)
MARK(U81(X)) → A__U81(mark(X))
A__U93(tt, IL, M, N) → MARK(N)
MARK(take(X1, X2)) → MARK(X1)
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
MARK(U51(X1, X2)) → MARK(X1)
A__ISNATILIST(V) → A__ISNATLIST(V)
A__ISNATLIST(take(V1, V2)) → A__U61(a__isNat(V1), V2)
A__U51(tt, V2) → A__ISNATLIST(V2)
MARK(U11(X)) → A__U11(mark(X))
A__TAKE(0, IL) → A__ISNATILIST(IL)
MARK(U62(X)) → A__U62(mark(X))
A__U92(tt, IL, M, N) → A__ISNAT(N)
MARK(U41(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNATILIST(cons(V1, V2)) → A__ISNAT(V1)
A__ISNATLIST(take(V1, V2)) → A__ISNAT(V1)
MARK(U42(X)) → MARK(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
A__U41(tt, V2) → A__U42(a__isNatIList(V2))
MARK(U91(X1, X2, X3, X4)) → A__U91(mark(X1), X2, X3, X4)
MARK(take(X1, X2)) → MARK(X2)
MARK(U72(X1, X2)) → A__U72(mark(X1), X2)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U62(X)) → MARK(X)
MARK(zeros) → A__ZEROS
MARK(s(X)) → MARK(X)
A__TAKE(s(M), cons(N, IL)) → A__ISNATILIST(IL)
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(U92(X1, X2, X3, X4)) → MARK(X1)
A__TAKE(s(M), cons(N, IL)) → A__U91(a__isNatIList(IL), IL, M, N)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U81(X)) → MARK(X)
MARK(U72(X1, X2)) → MARK(X1)
MARK(U21(X)) → A__U21(mark(X))
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__U91(tt, IL, M, N) → A__U92(a__isNat(M), IL, M, N)
MARK(U93(X1, X2, X3, X4)) → MARK(X1)
MARK(U52(X)) → MARK(X)
A__ISNAT(length(V1)) → A__U11(a__isNatList(V1))
MARK(U61(X1, X2)) → A__U61(mark(X1), X2)
MARK(U51(X1, X2)) → A__U51(mark(X1), X2)
A__U72(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
MARK(U31(X)) → MARK(X)
MARK(U52(X)) → A__U52(mark(X))
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__U72(tt, L) → MARK(L)
MARK(U42(X)) → A__U42(mark(X))
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__U61(tt, V2) → A__U62(a__isNatIList(V2))
A__U51(tt, V2) → A__U52(a__isNatList(V2))
MARK(U21(X)) → MARK(X)
MARK(U31(X)) → A__U31(mark(X))
MARK(U61(X1, X2)) → MARK(X1)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U92(X1, X2, X3, X4)) → A__U92(mark(X1), X2, X3, X4)
A__U91(tt, IL, M, N) → A__ISNAT(M)
A__ISNATILIST(V) → A__U31(a__isNatList(V))
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__U92(tt, IL, M, N) → A__U93(a__isNat(N), IL, M, N)
A__TAKE(0, IL) → A__U81(a__isNatIList(IL))
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNat(V1), V2)
A__U61(tt, V2) → A__ISNATILIST(V2)
MARK(U11(X)) → MARK(X)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt) → tt
a__U21(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIList(V2))
a__U62(tt) → tt
a__U71(tt, L, N) → a__U72(a__isNat(N), L)
a__U72(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → a__U92(a__isNat(M), IL, M, N)
a__U92(tt, IL, M, N) → a__U93(a__isNat(N), IL, M, N)
a__U93(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(V) → a__U31(a__isNatList(V))
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__isNatList(take(V1, V2)) → a__U61(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__isNatList(L), L, N)
a__take(0, IL) → a__U81(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U91(a__isNatIList(IL), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X)) → a__U11(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
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(U92(X1, X2, X3, X4)) → a__U92(mark(X1), X2, X3, X4)
mark(U93(X1, X2, X3, X4)) → a__U93(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(X) → U11(X)
a__U21(X) → U21(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__isNatList(X) → isNatList(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__U92(X1, X2, X3, X4) → U92(X1, X2, X3, X4)
a__U93(X1, X2, X3, X4) → U93(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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 27 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

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

A__U51(tt, V2) → A__ISNATLIST(V2)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__isNat(V1), V2)
A__U41(tt, V2) → A__ISNATILIST(V2)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__ISNATLIST(take(V1, V2)) → A__ISNAT(V1)
A__ISNATILIST(cons(V1, V2)) → A__ISNAT(V1)
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNat(V1), V2)
A__U61(tt, V2) → A__ISNATILIST(V2)
A__ISNATLIST(take(V1, V2)) → A__U61(a__isNat(V1), V2)
A__ISNATILIST(V) → A__ISNATLIST(V)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt) → tt
a__U21(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIList(V2))
a__U62(tt) → tt
a__U71(tt, L, N) → a__U72(a__isNat(N), L)
a__U72(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → a__U92(a__isNat(M), IL, M, N)
a__U92(tt, IL, M, N) → a__U93(a__isNat(N), IL, M, N)
a__U93(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(V) → a__U31(a__isNatList(V))
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__isNatList(take(V1, V2)) → a__U61(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__isNatList(L), L, N)
a__take(0, IL) → a__U81(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U91(a__isNatIList(IL), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X)) → a__U11(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
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(U92(X1, X2, X3, X4)) → a__U92(mark(X1), X2, X3, X4)
mark(U93(X1, X2, X3, X4)) → a__U93(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(X) → U11(X)
a__U21(X) → U21(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__isNatList(X) → isNatList(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__U92(X1, X2, X3, X4) → U92(X1, X2, X3, X4)
a__U93(X1, X2, X3, X4) → U93(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

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

A__U51(tt, V2) → A__ISNATLIST(V2)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__isNat(V1), V2)
A__U41(tt, V2) → A__ISNATILIST(V2)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
A__ISNATILIST(cons(V1, V2)) → A__ISNAT(V1)
A__ISNATLIST(take(V1, V2)) → A__ISNAT(V1)
A__U61(tt, V2) → A__ISNATILIST(V2)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNat(V1), V2)
A__ISNATILIST(V) → A__ISNATLIST(V)
A__ISNATLIST(take(V1, V2)) → A__U61(a__isNat(V1), V2)

The TRS R consists of the following rules:

a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(X) → isNat(X)
a__U21(tt) → tt
a__U21(X) → U21(X)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__isNatList(take(V1, V2)) → a__U61(a__isNat(V1), V2)
a__isNatList(X) → isNatList(X)
a__U11(tt) → tt
a__U11(X) → U11(X)
a__U61(tt, V2) → a__U62(a__isNatIList(V2))
a__U61(X1, X2) → U61(X1, X2)
a__isNatIList(V) → a__U31(a__isNatList(V))
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatIList(X) → isNatIList(X)
a__U62(tt) → tt
a__U62(X) → U62(X)
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U41(X1, X2) → U41(X1, X2)
a__U42(tt) → tt
a__U42(X) → U42(X)
a__U31(tt) → tt
a__U31(X) → U31(X)
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U51(X1, X2) → U51(X1, X2)
a__U52(tt) → tt
a__U52(X) → U52(X)

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

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

A__ISNATLIST(cons(V1, V2)) → A__U51(a__isNat(V1), V2)
The graph contains the following edges 1 > 2
A__U51(tt, V2) → A__ISNATLIST(V2)
The graph contains the following edges 2 >= 1
A__ISNATILIST(V) → A__ISNATLIST(V)
The graph contains the following edges 1 >= 1
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
The graph contains the following edges 1 > 1
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNat(V1), V2)
The graph contains the following edges 1 > 2
A__ISNATILIST(cons(V1, V2)) → A__ISNAT(V1)
The graph contains the following edges 1 > 1
A__ISNATLIST(take(V1, V2)) → A__U61(a__isNat(V1), V2)
The graph contains the following edges 1 > 2
A__ISNAT(s(V1)) → A__ISNAT(V1)
The graph contains the following edges 1 > 1
A__U41(tt, V2) → A__ISNATILIST(V2)
The graph contains the following edges 2 >= 1
A__U61(tt, V2) → A__ISNATILIST(V2)
The graph contains the following edges 2 >= 1
A__ISNATLIST(take(V1, V2)) → A__ISNAT(V1)
The graph contains the following edges 1 > 1
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

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

MARK(take(X1, X2)) → MARK(X2)
MARK(U72(X1, X2)) → A__U72(mark(X1), X2)
A__LENGTH(cons(N, L)) → A__U71(a__isNatList(L), L, N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
A__U71(tt, L, N) → A__U72(a__isNat(N), L)
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(cons(X1, X2)) → MARK(X1)
A__TAKE(s(M), cons(N, IL)) → A__U91(a__isNatIList(IL), IL, M, N)
MARK(U92(X1, X2, X3, X4)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U81(X)) → MARK(X)
MARK(U93(X1, X2, X3, X4)) → A__U93(mark(X1), X2, X3, X4)
MARK(U72(X1, X2)) → MARK(X1)
A__U91(tt, IL, M, N) → A__U92(a__isNat(M), IL, M, N)
MARK(U93(X1, X2, X3, X4)) → MARK(X1)
MARK(length(X)) → MARK(X)
A__U93(tt, IL, M, N) → MARK(N)
MARK(U52(X)) → MARK(X)
MARK(take(X1, X2)) → MARK(X1)
MARK(U51(X1, X2)) → MARK(X1)
A__U72(tt, L) → A__LENGTH(mark(L))
MARK(U31(X)) → MARK(X)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__U72(tt, L) → MARK(L)
MARK(U21(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U92(X1, X2, X3, X4)) → A__U92(mark(X1), X2, X3, X4)
MARK(U41(X1, X2)) → MARK(X1)
A__U92(tt, IL, M, N) → A__U93(a__isNat(N), IL, M, N)
MARK(U42(X)) → MARK(X)
MARK(U11(X)) → MARK(X)
MARK(U91(X1, X2, X3, X4)) → A__U91(mark(X1), X2, X3, X4)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt) → tt
a__U21(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIList(V2))
a__U62(tt) → tt
a__U71(tt, L, N) → a__U72(a__isNat(N), L)
a__U72(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → a__U92(a__isNat(M), IL, M, N)
a__U92(tt, IL, M, N) → a__U93(a__isNat(N), IL, M, N)
a__U93(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(V) → a__U31(a__isNatList(V))
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__isNatList(take(V1, V2)) → a__U61(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__isNatList(L), L, N)
a__take(0, IL) → a__U81(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U91(a__isNatIList(IL), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X)) → a__U11(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
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(U92(X1, X2, X3, X4)) → a__U92(mark(X1), X2, X3, X4)
mark(U93(X1, X2, X3, X4)) → a__U93(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(X) → U11(X)
a__U21(X) → U21(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__isNatList(X) → isNatList(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__U92(X1, X2, X3, X4) → U92(X1, X2, X3, X4)
a__U93(X1, X2, X3, X4) → U93(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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(X2)
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(U92(X1, X2, X3, X4)) → MARK(X1)
MARK(U81(X)) → MARK(X)
MARK(U93(X1, X2, X3, X4)) → A__U93(mark(X1), X2, X3, X4)
MARK(U93(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
MARK(U92(X1, X2, X3, X4)) → A__U92(mark(X1), X2, X3, X4)
MARK(U91(X1, X2, X3, X4)) → A__U91(mark(X1), X2, X3, X4)

The remaining pairs can at least be oriented weakly.

MARK(U72(X1, X2)) → A__U72(mark(X1), X2)
A__LENGTH(cons(N, L)) → A__U71(a__isNatList(L), L, N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
A__U71(tt, L, N) → A__U72(a__isNat(N), L)
MARK(cons(X1, X2)) → MARK(X1)
A__TAKE(s(M), cons(N, IL)) → A__U91(a__isNatIList(IL), IL, M, N)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U72(X1, X2)) → MARK(X1)
A__U91(tt, IL, M, N) → A__U92(a__isNat(M), IL, M, N)
MARK(length(X)) → MARK(X)
A__U93(tt, IL, M, N) → MARK(N)
MARK(U52(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
A__U72(tt, L) → A__LENGTH(mark(L))
MARK(U31(X)) → MARK(X)
A__U72(tt, L) → MARK(L)
MARK(U21(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U41(X1, X2)) → MARK(X1)
A__U92(tt, IL, M, N) → A__U93(a__isNat(N), IL, M, N)
MARK(U42(X)) → MARK(X)
MARK(U11(X)) → MARK(X)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__LENGTH(x₁)) = x₁
POL(A__TAKE(x₁, x₂)) = x₂
POL(A__U71(x₁, x₂, x₃)) = x₂
POL(A__U72(x₁, x₂)) = x₂
POL(A__U91(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(A__U92(x₁, x₂, x₃, x₄)) = x₄
POL(A__U93(x₁, x₂, x₃, x₄)) = x₄
POL(MARK(x₁)) = x₁
POL(U11(x₁)) = x₁
POL(U21(x₁)) = x₁
POL(U31(x₁)) = x₁
POL(U41(x₁, x₂)) = x₁
POL(U42(x₁)) = x₁
POL(U51(x₁, x₂)) = x₁
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂)) = x₁
POL(U62(x₁)) = x₁
POL(U71(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U72(x₁, x₂)) = x₁ + x₂
POL(U81(x₁)) = 1 + x₁
POL(U91(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₂ + x₃ + x₄
POL(U92(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₂ + x₃ + x₄
POL(U93(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₂ + x₃ + x₄
POL(a__U11(x₁)) = x₁
POL(a__U21(x₁)) = x₁
POL(a__U31(x₁)) = x₁
POL(a__U41(x₁, x₂)) = x₁
POL(a__U42(x₁)) = x₁
POL(a__U51(x₁, x₂)) = x₁
POL(a__U52(x₁)) = x₁
POL(a__U61(x₁, x₂)) = x₁
POL(a__U62(x₁)) = x₁
POL(a__U71(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U72(x₁, x₂)) = x₁ + x₂
POL(a__U81(x₁)) = 1 + x₁
POL(a__U91(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₂ + x₃ + x₄
POL(a__U92(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₂ + x₃ + x₄
POL(a__U93(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₂ + x₃ + x₄
POL(a__isNat(x₁)) = 0
POL(a__isNatIList(x₁)) = 0
POL(a__isNatList(x₁)) = 0
POL(a__length(x₁)) = x₁
POL(a__take(x₁, x₂)) = 1 + x₁ + x₂
POL(a__zeros) = 0
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = x₁
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__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U42(X) → U42(X)
a__isNatIList(X) → isNatIList(X)
a__U62(X) → U62(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__isNatList(X) → isNatList(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U72(X1, X2) → U72(X1, X2)
a__isNat(X) → isNat(X)
a__U93(X1, X2, X3, X4) → U93(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__U92(X1, X2, X3, X4) → U92(X1, X2, X3, X4)
a__U62(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIList(V2))
a__U72(tt, L) → s(a__length(mark(L)))
a__U71(tt, L, N) → a__U72(a__isNat(N), L)
a__U42(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U52(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U11(tt) → tt
a__zeros → cons(0, zeros)
a__U31(tt) → tt
a__U21(tt) → tt
a__isNatList(take(V1, V2)) → a__U61(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__isNatList(L), L, N)
a__take(0, IL) → a__U81(a__isNatIList(IL))
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(V) → a__U31(a__isNatList(V))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → a__U92(a__isNat(M), IL, M, N)
a__U92(tt, IL, M, N) → a__U93(a__isNat(N), IL, M, N)
a__U93(tt, IL, M, N) → cons(mark(N), take(M, IL))
mark(isNat(X)) → a__isNat(X)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U62(X)) → a__U62(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(isNatList(X)) → a__isNatList(X)
mark(U52(X)) → a__U52(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(isNatIList(X)) → a__isNatIList(X)
mark(U42(X)) → a__U42(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U31(X)) → a__U31(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U11(X)) → a__U11(mark(X))
mark(zeros) → a__zeros
a__take(s(M), cons(N, IL)) → a__U91(a__isNatIList(IL), IL, M, N)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U11(X) → U11(X)
a__U21(X) → U21(X)
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
mark(U93(X1, X2, X3, X4)) → a__U93(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(U92(X1, X2, X3, X4)) → a__U92(mark(X1), X2, X3, X4)
mark(length(X)) → a__length(mark(X))
mark(U81(X)) → a__U81(mark(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

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

MARK(U72(X1, X2)) → A__U72(mark(X1), X2)
MARK(U71(X1, X2, X3)) → MARK(X1)
A__LENGTH(cons(N, L)) → A__U71(a__isNatList(L), L, N)
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
A__U71(tt, L, N) → A__U72(a__isNat(N), L)
A__TAKE(s(M), cons(N, IL)) → A__U91(a__isNatIList(IL), IL, M, N)
MARK(cons(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U72(X1, X2)) → MARK(X1)
A__U91(tt, IL, M, N) → A__U92(a__isNat(M), IL, M, N)
MARK(length(X)) → MARK(X)
A__U93(tt, IL, M, N) → MARK(N)
MARK(U52(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
A__U72(tt, L) → A__LENGTH(mark(L))
MARK(U31(X)) → MARK(X)
A__U72(tt, L) → MARK(L)
MARK(U21(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U41(X1, X2)) → MARK(X1)
A__U92(tt, IL, M, N) → A__U93(a__isNat(N), IL, M, N)
MARK(U42(X)) → MARK(X)
MARK(U11(X)) → MARK(X)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt) → tt
a__U21(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIList(V2))
a__U62(tt) → tt
a__U71(tt, L, N) → a__U72(a__isNat(N), L)
a__U72(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → a__U92(a__isNat(M), IL, M, N)
a__U92(tt, IL, M, N) → a__U93(a__isNat(N), IL, M, N)
a__U93(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(V) → a__U31(a__isNatList(V))
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__isNatList(take(V1, V2)) → a__U61(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__isNatList(L), L, N)
a__take(0, IL) → a__U81(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U91(a__isNatIList(IL), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X)) → a__U11(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
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(U92(X1, X2, X3, X4)) → a__U92(mark(X1), X2, X3, X4)
mark(U93(X1, X2, X3, X4)) → a__U93(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(X) → U11(X)
a__U21(X) → U21(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__isNatList(X) → isNatList(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__U92(X1, X2, X3, X4) → U92(X1, X2, X3, X4)
a__U93(X1, X2, X3, X4) → U93(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

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

A__U72(tt, L) → A__LENGTH(mark(L))
MARK(U31(X)) → MARK(X)
MARK(U72(X1, X2)) → A__U72(mark(X1), X2)
MARK(U71(X1, X2, X3)) → MARK(X1)
A__LENGTH(cons(N, L)) → A__U71(a__isNatList(L), L, N)
A__U72(tt, L) → MARK(L)
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
A__U71(tt, L, N) → A__U72(a__isNat(N), L)
MARK(cons(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U21(X)) → MARK(X)
MARK(U72(X1, X2)) → MARK(X1)
MARK(U61(X1, X2)) → MARK(X1)
MARK(length(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U52(X)) → MARK(X)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U11(X)) → MARK(X)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt) → tt
a__U21(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIList(V2))
a__U62(tt) → tt
a__U71(tt, L, N) → a__U72(a__isNat(N), L)
a__U72(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → a__U92(a__isNat(M), IL, M, N)
a__U92(tt, IL, M, N) → a__U93(a__isNat(N), IL, M, N)
a__U93(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(V) → a__U31(a__isNatList(V))
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__isNatList(take(V1, V2)) → a__U61(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__isNatList(L), L, N)
a__take(0, IL) → a__U81(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U91(a__isNatIList(IL), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X)) → a__U11(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
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(U92(X1, X2, X3, X4)) → a__U92(mark(X1), X2, X3, X4)
mark(U93(X1, X2, X3, X4)) → a__U93(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(X) → U11(X)
a__U21(X) → U21(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__isNatList(X) → isNatList(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__U92(X1, X2, X3, X4) → U92(X1, X2, X3, X4)
a__U93(X1, X2, X3, X4) → U93(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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(U72(X1, X2)) → A__U72(mark(X1), X2)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U72(X1, X2)) → MARK(X1)
MARK(length(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)

The remaining pairs can at least be oriented weakly.

A__U72(tt, L) → A__LENGTH(mark(L))
MARK(U31(X)) → MARK(X)
A__LENGTH(cons(N, L)) → A__U71(a__isNatList(L), L, N)
A__U72(tt, L) → MARK(L)
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
A__U71(tt, L, N) → A__U72(a__isNat(N), L)
MARK(cons(X1, X2)) → MARK(X1)
MARK(U21(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U11(X)) → MARK(X)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__LENGTH(x₁)) = x₁
POL(A__U71(x₁, x₂, x₃)) = x₂
POL(A__U72(x₁, x₂)) = x₂
POL(MARK(x₁)) = x₁
POL(U11(x₁)) = x₁
POL(U21(x₁)) = x₁
POL(U31(x₁)) = x₁
POL(U41(x₁, x₂)) = x₁
POL(U42(x₁)) = x₁
POL(U51(x₁, x₂)) = x₁
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂)) = x₁
POL(U62(x₁)) = x₁
POL(U71(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U72(x₁, x₂)) = 1 + x₁ + x₂
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(U92(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(U93(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₄
POL(a__U11(x₁)) = x₁
POL(a__U21(x₁)) = x₁
POL(a__U31(x₁)) = x₁
POL(a__U41(x₁, x₂)) = x₁
POL(a__U42(x₁)) = x₁
POL(a__U51(x₁, x₂)) = x₁
POL(a__U52(x₁)) = x₁
POL(a__U61(x₁, x₂)) = x₁
POL(a__U62(x₁)) = x₁
POL(a__U71(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(a__U72(x₁, x₂)) = 1 + x₁ + x₂
POL(a__U81(x₁)) = 0
POL(a__U91(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(a__U92(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(a__U93(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₄
POL(a__isNat(x₁)) = 0
POL(a__isNatIList(x₁)) = 0
POL(a__isNatList(x₁)) = 0
POL(a__length(x₁)) = 1 + x₁
POL(a__take(x₁, x₂)) = x₂
POL(a__zeros) = 0
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 1 + x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = x₂
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U42(X) → U42(X)
a__isNatIList(X) → isNatIList(X)
a__U62(X) → U62(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__isNatList(X) → isNatList(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U72(X1, X2) → U72(X1, X2)
a__isNat(X) → isNat(X)
a__U93(X1, X2, X3, X4) → U93(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__U92(X1, X2, X3, X4) → U92(X1, X2, X3, X4)
a__U62(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIList(V2))
a__U72(tt, L) → s(a__length(mark(L)))
a__U71(tt, L, N) → a__U72(a__isNat(N), L)
a__U42(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U52(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U11(tt) → tt
a__zeros → cons(0, zeros)
a__U31(tt) → tt
a__U21(tt) → tt
a__isNatList(take(V1, V2)) → a__U61(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__isNatList(L), L, N)
a__take(0, IL) → a__U81(a__isNatIList(IL))
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(V) → a__U31(a__isNatList(V))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → a__U92(a__isNat(M), IL, M, N)
a__U92(tt, IL, M, N) → a__U93(a__isNat(N), IL, M, N)
a__U93(tt, IL, M, N) → cons(mark(N), take(M, IL))
mark(isNat(X)) → a__isNat(X)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U62(X)) → a__U62(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(isNatList(X)) → a__isNatList(X)
mark(U52(X)) → a__U52(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(isNatIList(X)) → a__isNatIList(X)
mark(U42(X)) → a__U42(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U31(X)) → a__U31(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U11(X)) → a__U11(mark(X))
mark(zeros) → a__zeros
a__take(s(M), cons(N, IL)) → a__U91(a__isNatIList(IL), IL, M, N)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U11(X) → U11(X)
a__U21(X) → U21(X)
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
mark(U93(X1, X2, X3, X4)) → a__U93(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(U92(X1, X2, X3, X4)) → a__U92(mark(X1), X2, X3, X4)
mark(length(X)) → a__length(mark(X))
mark(U81(X)) → a__U81(mark(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

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

A__U72(tt, L) → A__LENGTH(mark(L))
MARK(U31(X)) → MARK(X)
A__LENGTH(cons(N, L)) → A__U71(a__isNatList(L), L, N)
A__U72(tt, L) → MARK(L)
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
A__U71(tt, L, N) → A__U72(a__isNat(N), L)
MARK(cons(X1, X2)) → MARK(X1)
MARK(U21(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U11(X)) → MARK(X)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt) → tt
a__U21(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIList(V2))
a__U62(tt) → tt
a__U71(tt, L, N) → a__U72(a__isNat(N), L)
a__U72(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → a__U92(a__isNat(M), IL, M, N)
a__U92(tt, IL, M, N) → a__U93(a__isNat(N), IL, M, N)
a__U93(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(V) → a__U31(a__isNatList(V))
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__isNatList(take(V1, V2)) → a__U61(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__isNatList(L), L, N)
a__take(0, IL) → a__U81(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U91(a__isNatIList(IL), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X)) → a__U11(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
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(U92(X1, X2, X3, X4)) → a__U92(mark(X1), X2, X3, X4)
mark(U93(X1, X2, X3, X4)) → a__U93(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(X) → U11(X)
a__U21(X) → U21(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__isNatList(X) → isNatList(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__U92(X1, X2, X3, X4) → U92(X1, X2, X3, X4)
a__U93(X1, X2, X3, X4) → U93(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ UsableRulesProof

                            ↳ QDP

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

MARK(U31(X)) → MARK(X)
MARK(U62(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U52(X)) → MARK(X)
MARK(U41(X1, X2)) → MARK(X1)
MARK(cons(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
MARK(U21(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U11(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt) → tt
a__U21(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIList(V2))
a__U62(tt) → tt
a__U71(tt, L, N) → a__U72(a__isNat(N), L)
a__U72(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → a__U92(a__isNat(M), IL, M, N)
a__U92(tt, IL, M, N) → a__U93(a__isNat(N), IL, M, N)
a__U93(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(V) → a__U31(a__isNatList(V))
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__isNatList(take(V1, V2)) → a__U61(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__isNatList(L), L, N)
a__take(0, IL) → a__U81(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U91(a__isNatIList(IL), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X)) → a__U11(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
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(U92(X1, X2, X3, X4)) → a__U92(mark(X1), X2, X3, X4)
mark(U93(X1, X2, X3, X4)) → a__U93(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(X) → U11(X)
a__U21(X) → U21(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__isNatList(X) → isNatList(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__U92(X1, X2, X3, X4) → U92(X1, X2, X3, X4)
a__U93(X1, X2, X3, X4) → U93(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ UsableRulesProof

                                ↳ QDP

                                  ↳ QDPSizeChangeProof

                            ↳ QDP

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

MARK(U31(X)) → MARK(X)
MARK(U62(X)) → MARK(X)
MARK(U52(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U41(X1, X2)) → MARK(X1)
MARK(cons(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
MARK(U21(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U11(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)

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

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

MARK(U31(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(U62(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(s(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(U52(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(U41(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(cons(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U42(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(U21(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(U51(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
MARK(U11(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(U61(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                            ↳ QDP

                              ↳ QDPOrderProof

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

A__U72(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__U71(a__isNatList(L), L, N)
A__U71(tt, L, N) → A__U72(a__isNat(N), L)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt) → tt
a__U21(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIList(V2))
a__U62(tt) → tt
a__U71(tt, L, N) → a__U72(a__isNat(N), L)
a__U72(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → a__U92(a__isNat(M), IL, M, N)
a__U92(tt, IL, M, N) → a__U93(a__isNat(N), IL, M, N)
a__U93(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(V) → a__U31(a__isNatList(V))
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__isNatList(take(V1, V2)) → a__U61(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__isNatList(L), L, N)
a__take(0, IL) → a__U81(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U91(a__isNatIList(IL), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X)) → a__U11(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
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(U92(X1, X2, X3, X4)) → a__U92(mark(X1), X2, X3, X4)
mark(U93(X1, X2, X3, X4)) → a__U93(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(X) → U11(X)
a__U21(X) → U21(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__isNatList(X) → isNatList(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__U92(X1, X2, X3, X4) → U92(X1, X2, X3, X4)
a__U93(X1, X2, X3, X4) → U93(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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, L, N) → A__U72(a__isNat(N), L)

The remaining pairs can at least be oriented weakly.

A__U72(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__U71(a__isNatList(L), L, N)

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( a__U52(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	1	0	/

x₁

M( a__U31(x₁) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( a__U91(x₁, ..., x₄) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	1	1	\
\	1	1	/

x₃

/	0	0	\
\	0	0	/

x₄

M( U81(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( a__zeros ) =

/	0	\
\	0	/

M( a__U92(x₁, ..., x₄) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	1	1	\
\	1	1	/

x₃

/	0	0	\
\	0	0	/

x₄

M( U92(x₁, ..., x₄) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	1	1	\
\	1	1	/

x₃

/	0	0	\
\	0	0	/

x₄

M( mark(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	1	0	/

x₁

M( a__isNatIList(x₁) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	1	\
\	1	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( U51(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	1	0	/

x₂

M( a__U72(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	1	1	/

x₂

M( a__U21(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( a__length(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	1	0	/

x₁

M( tt ) =

/	1	\
\	1	/

M( a__U42(x₁) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	1	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( a__U11(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	1	0	/

x₁

M( U52(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	1	0	/

x₁

M( isNatIList(x₁) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( U11(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	1	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	1	1	/

x₁

M( a__isNat(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	1	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	1	/

x₁

M( a__isNatList(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	1	0	/

x₁

M( nil ) =

/	1	\
\	1	/

M( a__U51(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	1	0	/

x₂

M( a__U62(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( U91(x₁, ..., x₄) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	1	1	\
\	1	1	/

x₃

/	0	0	\
\	0	0	/

x₄

M( a__U41(x₁, x₂) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( a__U61(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

/	0	0	\
\	0	0	/

x₂

M( U93(x₁, ..., x₄) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	1	1	\
\	1	1	/

x₃

/	0	0	\
\	0	0	/

x₄

M( U72(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	1	1	/

x₂

M( a__take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	1	\
\	1	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( 0 ) =

/	1	\
\	1	/

M( a__U81(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( U62(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( a__U93(x₁, ..., x₄) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	1	1	\
\	1	1	/

x₃

/	0	0	\
\	0	0	/

x₄

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	1	1	/

x₂

M( U61(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

/	0	0	\
\	0	0	/

x₂

M( U31(x₁) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( U41(x₁, x₂) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( U42(x₁) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	1	0	/

x₁

M( a__U71(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	1	1	/

x₂

/	0	0	\
\	0	0	/

x₃

M( U71(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	1	1	/

x₂

/	0	0	\
\	0	0	/

x₃

M( U21(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

Tuple symbols:

M( A__LENGTH(x₁) ) =

[

]

x₁

M( A__U71(x₁, ..., x₃) ) =

[

]

x₁

[

]

x₂

[

]

x₃

M( A__U72(x₁, x₂) ) =

[

]

x₁

[

]

x₂

Matrix type:
We used a basic matrix type which is not further parametrizeable.

As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:

a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U42(X) → U42(X)
a__isNatIList(X) → isNatIList(X)
a__U62(X) → U62(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__isNatList(X) → isNatList(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U72(X1, X2) → U72(X1, X2)
a__isNat(X) → isNat(X)
a__U93(X1, X2, X3, X4) → U93(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__U92(X1, X2, X3, X4) → U92(X1, X2, X3, X4)
a__U62(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIList(V2))
a__U72(tt, L) → s(a__length(mark(L)))
a__U71(tt, L, N) → a__U72(a__isNat(N), L)
a__U42(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U52(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U11(tt) → tt
a__zeros → cons(0, zeros)
a__U31(tt) → tt
a__U21(tt) → tt
a__isNatList(take(V1, V2)) → a__U61(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__isNatList(L), L, N)
a__take(0, IL) → a__U81(a__isNatIList(IL))
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(V) → a__U31(a__isNatList(V))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → a__U92(a__isNat(M), IL, M, N)
a__U92(tt, IL, M, N) → a__U93(a__isNat(N), IL, M, N)
a__U93(tt, IL, M, N) → cons(mark(N), take(M, IL))
mark(isNat(X)) → a__isNat(X)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U62(X)) → a__U62(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(isNatList(X)) → a__isNatList(X)
mark(U52(X)) → a__U52(mark(X))
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(isNatIList(X)) → a__isNatIList(X)
mark(U42(X)) → a__U42(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U31(X)) → a__U31(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U11(X)) → a__U11(mark(X))
mark(zeros) → a__zeros
a__take(s(M), cons(N, IL)) → a__U91(a__isNatIList(IL), IL, M, N)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U11(X) → U11(X)
a__U21(X) → U21(X)
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
mark(U93(X1, X2, X3, X4)) → a__U93(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(U92(X1, X2, X3, X4)) → a__U92(mark(X1), X2, X3, X4)
mark(length(X)) → a__length(mark(X))
mark(U81(X)) → a__U81(mark(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ DependencyGraphProof

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

A__U72(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__U71(a__isNatList(L), L, N)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt) → tt
a__U21(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIList(V2))
a__U62(tt) → tt
a__U71(tt, L, N) → a__U72(a__isNat(N), L)
a__U72(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → a__U92(a__isNat(M), IL, M, N)
a__U92(tt, IL, M, N) → a__U93(a__isNat(N), IL, M, N)
a__U93(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(V) → a__U31(a__isNatList(V))
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__isNatList(take(V1, V2)) → a__U61(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__isNatList(L), L, N)
a__take(0, IL) → a__U81(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U91(a__isNatIList(IL), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X)) → a__U11(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
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(U92(X1, X2, X3, X4)) → a__U92(mark(X1), X2, X3, X4)
mark(U93(X1, X2, X3, X4)) → a__U93(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
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(X) → U11(X)
a__U21(X) → U21(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__isNatList(X) → isNatList(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__U92(X1, X2, X3, X4) → U92(X1, X2, X3, X4)
a__U93(X1, X2, X3, X4) → U93(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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 2 less nodes.