Termination w.r.t. Q proof of /tmp/tmpMxmYhp/LengthOfFiniteLists_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, L, N) → a__U62(a__isNat(N), L)
a__U62(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, 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, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__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, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(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) → 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, L, N) → a__U62(a__isNat(N), L)
a__U62(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, 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, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__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, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)

Q is empty.

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

MARK(U62(X1, X2)) → A__U62(mark(X1), X2)
A__U41(tt, V2) → A__ISNATILIST(V2)
MARK(zeros) → A__ZEROS
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → A__ISNATILIST(X)
A__ISNAT(s(V1)) → A__U21(a__isNat(V1))
MARK(cons(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U61(X1, X2, X3)) → MARK(X1)
MARK(U21(X)) → A__U21(mark(X))
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__U61(tt, L, N) → A__ISNAT(N)
MARK(length(X)) → MARK(X)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__isNat(V1), V2)
MARK(U61(X1, X2, X3)) → A__U61(mark(X1), X2, X3)
MARK(U52(X)) → MARK(X)
A__ISNAT(length(V1)) → A__U11(a__isNatList(V1))
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U51(X1, X2)) → A__U51(mark(X1), X2)
A__ISNATILIST(V) → A__ISNATLIST(V)
MARK(U31(X)) → MARK(X)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__U51(tt, V2) → A__ISNATLIST(V2)
MARK(U52(X)) → A__U52(mark(X))
MARK(U11(X)) → A__U11(mark(X))
MARK(U62(X1, X2)) → MARK(X1)
MARK(U42(X)) → A__U42(mark(X))
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__U51(tt, V2) → A__U52(a__isNatList(V2))
A__U62(tt, L) → MARK(L)
MARK(U21(X)) → MARK(X)
MARK(U31(X)) → A__U31(mark(X))
A__U61(tt, L, N) → A__U62(a__isNat(N), L)
MARK(U41(X1, X2)) → MARK(X1)
A__ISNATILIST(V) → A__U31(a__isNatList(V))
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__U62(tt, L) → A__LENGTH(mark(L))
MARK(isNat(X)) → A__ISNAT(X)
MARK(U42(X)) → MARK(X)
A__ISNATILIST(cons(V1, V2)) → A__ISNAT(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)
A__U41(tt, V2) → A__U42(a__isNatIList(V2))
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNat(V1), V2)
A__LENGTH(cons(N, L)) → A__U61(a__isNatList(L), L, N)
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, L, N) → a__U62(a__isNat(N), L)
a__U62(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, 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, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__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, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

MARK(U62(X1, X2)) → A__U62(mark(X1), X2)
A__U41(tt, V2) → A__ISNATILIST(V2)
MARK(zeros) → A__ZEROS
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → A__ISNATILIST(X)
A__ISNAT(s(V1)) → A__U21(a__isNat(V1))
MARK(cons(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U61(X1, X2, X3)) → MARK(X1)
MARK(U21(X)) → A__U21(mark(X))
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__U61(tt, L, N) → A__ISNAT(N)
MARK(length(X)) → MARK(X)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__isNat(V1), V2)
MARK(U61(X1, X2, X3)) → A__U61(mark(X1), X2, X3)
MARK(U52(X)) → MARK(X)
A__ISNAT(length(V1)) → A__U11(a__isNatList(V1))
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U51(X1, X2)) → A__U51(mark(X1), X2)
A__ISNATILIST(V) → A__ISNATLIST(V)
MARK(U31(X)) → MARK(X)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__U51(tt, V2) → A__ISNATLIST(V2)
MARK(U52(X)) → A__U52(mark(X))
MARK(U11(X)) → A__U11(mark(X))
MARK(U62(X1, X2)) → MARK(X1)
MARK(U42(X)) → A__U42(mark(X))
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__U51(tt, V2) → A__U52(a__isNatList(V2))
A__U62(tt, L) → MARK(L)
MARK(U21(X)) → MARK(X)
MARK(U31(X)) → A__U31(mark(X))
A__U61(tt, L, N) → A__U62(a__isNat(N), L)
MARK(U41(X1, X2)) → MARK(X1)
A__ISNATILIST(V) → A__U31(a__isNatList(V))
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__U62(tt, L) → A__LENGTH(mark(L))
MARK(isNat(X)) → A__ISNAT(X)
MARK(U42(X)) → MARK(X)
A__ISNATILIST(cons(V1, V2)) → A__ISNAT(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)
A__U41(tt, V2) → A__U42(a__isNatIList(V2))
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNat(V1), V2)
A__LENGTH(cons(N, L)) → A__U61(a__isNatList(L), L, N)
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, L, N) → a__U62(a__isNat(N), L)
a__U62(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, 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, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__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, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 3 SCCs with 20 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ 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__ISNAT(s(V1)) → A__ISNAT(V1)
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)

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, L, N) → a__U62(a__isNat(N), L)
a__U62(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, 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, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__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, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(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__ISNATLIST(cons(V1, V2)) → A__U51(a__isNat(V1), V2)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)

The remaining pairs can at least be oriented weakly.

A__U51(tt, V2) → A__ISNATLIST(V2)
A__ISNAT(length(V1)) → A__ISNATLIST(V1)

Used ordering: Polynomial interpretation [25,35]:

POL(a__U52(x₁)) = 0
POL(A__ISNAT(x₁)) = (1/4)x_1
POL(0) = 0
POL(U51(x₁, x₂)) = 0
POL(A__ISNATLIST(x₁)) = (1/4)x_1
POL(a__U21(x₁)) = 0
POL(cons(x₁, x₂)) = 1/4 + (2)x_1 + (4)x_2
POL(tt) = 0
POL(A__U51(x₁, x₂)) = x_2
POL(isNatList(x₁)) = 1 + (1/4)x_1
POL(U52(x₁)) = 0
POL(a__U11(x₁)) = 0
POL(U11(x₁)) = 0
POL(s(x₁)) = 1/4 + (2)x_1
POL(a__isNat(x₁)) = 0
POL(isNat(x₁)) = 0
POL(length(x₁)) = (2)x_1
POL(a__isNatList(x₁)) = 1 + (1/4)x_1
POL(nil) = 0
POL(a__U51(x₁, x₂)) = 0
POL(U21(x₁)) = 0

The value of delta used in the strict ordering is 1/16.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

          ↳ QDP

          ↳ QDP

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

A__U51(tt, V2) → A__ISNATLIST(V2)
A__ISNAT(length(V1)) → A__ISNATLIST(V1)

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, L, N) → a__U62(a__isNat(N), L)
a__U62(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, 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, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__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, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(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 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

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

A__U41(tt, V2) → A__ISNATILIST(V2)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNat(V1), V2)

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, L, N) → a__U62(a__isNat(N), L)
a__U62(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, 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, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__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, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(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__ISNATILIST(cons(V1, V2)) → A__U41(a__isNat(V1), V2)

The remaining pairs can at least be oriented weakly.

A__U41(tt, V2) → A__ISNATILIST(V2)

Used ordering: Polynomial interpretation [25,35]:

POL(a__U52(x₁)) = 0
POL(A__U41(x₁, x₂)) = (1/2)x_2
POL(0) = 0
POL(U51(x₁, x₂)) = 0
POL(a__U21(x₁)) = 0
POL(cons(x₁, x₂)) = 1/4 + (2)x_2
POL(tt) = 0
POL(A__ISNATILIST(x₁)) = (1/4)x_1
POL(isNatList(x₁)) = 0
POL(U52(x₁)) = 0
POL(a__U11(x₁)) = 0
POL(s(x₁)) = 0
POL(a__isNat(x₁)) = 0
POL(U11(x₁)) = 0
POL(isNat(x₁)) = 0
POL(length(x₁)) = 0
POL(a__isNatList(x₁)) = 0
POL(nil) = 0
POL(a__U51(x₁, x₂)) = 0
POL(U21(x₁)) = 0

The value of delta used in the strict ordering is 1/16.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

          ↳ QDP

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

A__U41(tt, V2) → A__ISNATILIST(V2)

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, L, N) → a__U62(a__isNat(N), L)
a__U62(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, 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, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__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, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(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 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

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

MARK(U62(X1, X2)) → A__U62(mark(X1), X2)
MARK(U31(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(U62(X1, X2)) → MARK(X1)
MARK(cons(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
A__U62(tt, L) → MARK(L)
MARK(U21(X)) → MARK(X)
MARK(U61(X1, X2, X3)) → MARK(X1)
A__U61(tt, L, N) → A__U62(a__isNat(N), L)
MARK(length(X)) → MARK(X)
MARK(U61(X1, X2, X3)) → A__U61(mark(X1), X2, X3)
MARK(U52(X)) → MARK(X)
MARK(U41(X1, X2)) → MARK(X1)
A__U62(tt, L) → A__LENGTH(mark(L))
MARK(U42(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U11(X)) → MARK(X)
A__LENGTH(cons(N, L)) → A__U61(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, L, N) → a__U62(a__isNat(N), L)
a__U62(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, 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, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__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, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(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(U31(X)) → MARK(X)
MARK(U41(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.

MARK(U62(X1, X2)) → A__U62(mark(X1), X2)
MARK(s(X)) → MARK(X)
MARK(U62(X1, X2)) → MARK(X1)
MARK(cons(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
A__U62(tt, L) → MARK(L)
MARK(U21(X)) → MARK(X)
MARK(U61(X1, X2, X3)) → MARK(X1)
A__U61(tt, L, N) → A__U62(a__isNat(N), L)
MARK(length(X)) → MARK(X)
MARK(U61(X1, X2, X3)) → A__U61(mark(X1), X2, X3)
MARK(U52(X)) → MARK(X)
A__U62(tt, L) → A__LENGTH(mark(L))
MARK(U42(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U11(X)) → MARK(X)
A__LENGTH(cons(N, L)) → A__U61(a__isNatList(L), L, N)

Used ordering: Polynomial interpretation [25,35]:

POL(U62(x₁, x₂)) = (4)x_1 + (4)x_2
POL(a__U31(x₁)) = 2 + (4)x_1
POL(a__U52(x₁)) = x_1
POL(a__zeros) = 0
POL(U61(x₁, x₂, x₃)) = x_1 + (4)x_2
POL(mark(x₁)) = x_1
POL(a__isNatIList(x₁)) = 4 + (2)x_1
POL(a__U61(x₁, x₂, x₃)) = x_1 + (4)x_2
POL(U51(x₁, x₂)) = (4)x_1
POL(a__U21(x₁)) = (2)x_1
POL(A__U61(x₁, x₂, x₃)) = (1/2)x_2
POL(a__U62(x₁, x₂)) = (4)x_1 + (4)x_2
POL(a__length(x₁)) = x_1
POL(tt) = 0
POL(a__U42(x₁)) = x_1
POL(isNatList(x₁)) = 0
POL(a__U11(x₁)) = (4)x_1
POL(U52(x₁)) = x_1
POL(zeros) = 0
POL(isNatIList(x₁)) = 4 + (2)x_1
POL(s(x₁)) = (4)x_1
POL(a__isNat(x₁)) = 0
POL(U11(x₁)) = (4)x_1
POL(isNat(x₁)) = 0
POL(a__isNatList(x₁)) = 0
POL(nil) = 0
POL(a__U51(x₁, x₂)) = (4)x_1
POL(A__LENGTH(x₁)) = (1/4)x_1
POL(a__U41(x₁, x₂)) = 4 + x_1 + (4)x_2
POL(0) = 0
POL(cons(x₁, x₂)) = x_1 + (4)x_2
POL(MARK(x₁)) = (1/4)x_1
POL(U31(x₁)) = 2 + (4)x_1
POL(U41(x₁, x₂)) = 4 + x_1 + (4)x_2
POL(U42(x₁)) = x_1
POL(length(x₁)) = x_1
POL(A__U62(x₁, x₂)) = (1/2)x_2
POL(U21(x₁)) = (2)x_1

The value of delta used in the strict ordering is 1/2.
The following usable rules [17] were oriented:

a__zeros → cons(0, zeros)
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U31(tt) → tt
a__U21(tt) → tt
a__U11(tt) → tt
a__U61(tt, L, N) → a__U62(a__isNat(N), L)
a__U52(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U42(tt) → tt
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(0) → tt
a__U62(tt, L) → s(a__length(mark(L)))
a__isNatList(nil) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatIList(zeros) → tt
a__isNatIList(V) → a__U31(a__isNatList(V))
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)
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(isNatList(X)) → a__isNatList(X)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
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(nil) → nil
mark(s(X)) → s(mark(X))
a__U11(X) → U11(X)
a__zeros → zeros
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(tt) → tt
mark(0) → 0
a__U51(X1, X2) → U51(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__U52(X) → U52(X)
a__U31(X) → U31(X)
a__U21(X) → U21(X)
a__U42(X) → U42(X)
a__U41(X1, X2) → U41(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

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

MARK(U62(X1, X2)) → A__U62(mark(X1), X2)
MARK(s(X)) → MARK(X)
MARK(U62(X1, X2)) → MARK(X1)
MARK(cons(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
A__U62(tt, L) → MARK(L)
MARK(U21(X)) → MARK(X)
MARK(U61(X1, X2, X3)) → MARK(X1)
A__U61(tt, L, N) → A__U62(a__isNat(N), L)
MARK(length(X)) → MARK(X)
MARK(U61(X1, X2, X3)) → A__U61(mark(X1), X2, X3)
MARK(U52(X)) → MARK(X)
A__U62(tt, L) → A__LENGTH(mark(L))
MARK(U42(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
A__LENGTH(cons(N, L)) → A__U61(a__isNatList(L), L, N)
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, L, N) → a__U62(a__isNat(N), L)
a__U62(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, 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, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__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, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(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(U62(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U61(X1, X2, X3)) → MARK(X1)
MARK(length(X)) → MARK(X)
MARK(U61(X1, X2, X3)) → A__U61(mark(X1), X2, X3)

The remaining pairs can at least be oriented weakly.

MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
A__U62(tt, L) → MARK(L)
MARK(U21(X)) → MARK(X)
A__U61(tt, L, N) → A__U62(a__isNat(N), L)
MARK(U52(X)) → MARK(X)
A__U62(tt, L) → A__LENGTH(mark(L))
MARK(U42(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
A__LENGTH(cons(N, L)) → A__U61(a__isNatList(L), L, N)
MARK(U11(X)) → MARK(X)

Used ordering: Polynomial interpretation [25,35]:

POL(U62(x₁, x₂)) = 1/4 + (4)x_1 + (2)x_2
POL(a__U31(x₁)) = 0
POL(a__U52(x₁)) = x_1
POL(a__zeros) = 0
POL(U61(x₁, x₂, x₃)) = 1/4 + (4)x_1 + (2)x_2
POL(mark(x₁)) = x_1
POL(a__isNatIList(x₁)) = 0
POL(a__U61(x₁, x₂, x₃)) = 1/4 + (4)x_1 + (2)x_2
POL(U51(x₁, x₂)) = (4)x_1
POL(a__U21(x₁)) = (4)x_1
POL(A__U61(x₁, x₂, x₃)) = x_2
POL(a__U62(x₁, x₂)) = 1/4 + (4)x_1 + (2)x_2
POL(a__length(x₁)) = 1/4 + (2)x_1
POL(tt) = 0
POL(a__U42(x₁)) = x_1
POL(isNatList(x₁)) = 0
POL(a__U11(x₁)) = x_1
POL(U52(x₁)) = x_1
POL(zeros) = 0
POL(isNatIList(x₁)) = 0
POL(a__isNat(x₁)) = 0
POL(s(x₁)) = x_1
POL(U11(x₁)) = x_1
POL(isNat(x₁)) = 0
POL(a__isNatList(x₁)) = 0
POL(nil) = 0
POL(a__U51(x₁, x₂)) = (4)x_1
POL(A__LENGTH(x₁)) = (1/4)x_1
POL(a__U41(x₁, x₂)) = 0
POL(0) = 0
POL(cons(x₁, x₂)) = x_1 + (4)x_2
POL(MARK(x₁)) = (1/2)x_1
POL(U31(x₁)) = 0
POL(U41(x₁, x₂)) = 0
POL(U42(x₁)) = x_1
POL(length(x₁)) = 1/4 + (2)x_1
POL(A__U62(x₁, x₂)) = x_2
POL(U21(x₁)) = (4)x_1

The value of delta used in the strict ordering is 1/8.
The following usable rules [17] were oriented:

a__zeros → cons(0, zeros)
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U31(tt) → tt
a__U21(tt) → tt
a__U11(tt) → tt
a__U61(tt, L, N) → a__U62(a__isNat(N), L)
a__U52(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U42(tt) → tt
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(0) → tt
a__U62(tt, L) → s(a__length(mark(L)))
a__isNatList(nil) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatIList(zeros) → tt
a__isNatIList(V) → a__U31(a__isNatList(V))
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)
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(isNatList(X)) → a__isNatList(X)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
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(nil) → nil
mark(s(X)) → s(mark(X))
a__U11(X) → U11(X)
a__zeros → zeros
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(tt) → tt
mark(0) → 0
a__U51(X1, X2) → U51(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__U52(X) → U52(X)
a__U31(X) → U31(X)
a__U21(X) → U21(X)
a__U42(X) → U42(X)
a__U41(X1, X2) → U41(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ DependencyGraphProof

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

MARK(U52(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
A__U62(tt, L) → A__LENGTH(mark(L))
MARK(U42(X)) → MARK(X)
A__U62(tt, L) → MARK(L)
MARK(U21(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U11(X)) → MARK(X)
A__LENGTH(cons(N, L)) → A__U61(a__isNatList(L), L, N)
A__U61(tt, L, N) → A__U62(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, L, N) → a__U62(a__isNat(N), L)
a__U62(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, 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, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__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, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                          ↳ QDPOrderProof

                        ↳ QDP

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

MARK(s(X)) → MARK(X)
MARK(U52(X)) → MARK(X)
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)

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, L, N) → a__U62(a__isNat(N), L)
a__U62(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, 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, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__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, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(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(s(X)) → MARK(X)
MARK(U52(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U11(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.

MARK(U42(X)) → MARK(X)
MARK(U21(X)) → MARK(X)

Used ordering: Polynomial interpretation [25,35]:

POL(cons(x₁, x₂)) = 1 + x_1 + (1/4)x_2
POL(MARK(x₁)) = (1/4)x_1
POL(U52(x₁)) = 1/2 + (4)x_1
POL(U42(x₁)) = (2)x_1
POL(s(x₁)) = 1/4 + (4)x_1
POL(U11(x₁)) = 1/4 + x_1
POL(U51(x₁, x₂)) = 1/4 + (2)x_1 + x_2
POL(U21(x₁)) = (2)x_1

The value of delta used in the strict ordering is 1/16.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ QDPOrderProof

                        ↳ QDP

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

MARK(U42(X)) → MARK(X)
MARK(U21(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, L, N) → a__U62(a__isNat(N), L)
a__U62(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, 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, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__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, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(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(U42(X)) → MARK(X)
MARK(U21(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(MARK(x₁)) = (4)x_1
POL(U42(x₁)) = 1 + x_1
POL(U21(x₁)) = 4 + (4)x_1

The value of delta used in the strict ordering is 4.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ PisEmptyProof

                        ↳ QDP

Q DP problem:
P is empty.
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, L, N) → a__U62(a__isNat(N), L)
a__U62(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, 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, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__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, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                        ↳ QDP

                          ↳ QDPOrderProof

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

A__U62(tt, L) → A__LENGTH(mark(L))
A__U61(tt, L, N) → A__U62(a__isNat(N), L)
A__LENGTH(cons(N, L)) → A__U61(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, L, N) → a__U62(a__isNat(N), L)
a__U62(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, 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, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__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, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(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__U61(tt, L, N) → A__U62(a__isNat(N), L)

The remaining pairs can at least be oriented weakly.

A__U62(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__U61(a__isNatList(L), L, N)

Used ordering: Polynomial interpretation [25,35]:

POL(U62(x₁, x₂)) = 0
POL(a__U52(x₁)) = (2)x_1
POL(a__U31(x₁)) = x_1
POL(a__zeros) = 0
POL(U61(x₁, x₂, x₃)) = 0
POL(mark(x₁)) = x_1
POL(a__isNatIList(x₁)) = 4 + (1/2)x_1
POL(a__U61(x₁, x₂, x₃)) = 0
POL(U51(x₁, x₂)) = (1/2)x_2
POL(a__U21(x₁)) = 1/4
POL(A__U61(x₁, x₂, x₃)) = (1/4)x_1 + (1/4)x_2
POL(a__U62(x₁, x₂)) = 0
POL(a__length(x₁)) = 0
POL(tt) = 1/4
POL(a__U42(x₁)) = 1/4
POL(isNatList(x₁)) = (1/4)x_1
POL(U52(x₁)) = (2)x_1
POL(zeros) = 0
POL(a__U11(x₁)) = 1/4
POL(isNatIList(x₁)) = 4 + (1/2)x_1
POL(U11(x₁)) = 1/4
POL(a__isNat(x₁)) = 1/4
POL(s(x₁)) = 0
POL(isNat(x₁)) = 1/4
POL(a__isNatList(x₁)) = (1/4)x_1
POL(nil) = 2
POL(a__U51(x₁, x₂)) = (1/2)x_2
POL(A__LENGTH(x₁)) = (1/4)x_1
POL(a__U41(x₁, x₂)) = x_1
POL(0) = 0
POL(cons(x₁, x₂)) = (2)x_2
POL(U31(x₁)) = x_1
POL(U41(x₁, x₂)) = x_1
POL(U42(x₁)) = 1/4
POL(length(x₁)) = 0
POL(A__U62(x₁, x₂)) = (1/4)x_2
POL(U21(x₁)) = 1/4

The value of delta used in the strict ordering is 1/16.
The following usable rules [17] were oriented:

a__zeros → cons(0, zeros)
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U31(tt) → tt
a__U21(tt) → tt
a__U11(tt) → tt
a__U61(tt, L, N) → a__U62(a__isNat(N), L)
a__U52(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U42(tt) → tt
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(0) → tt
a__U62(tt, L) → s(a__length(mark(L)))
a__isNatList(nil) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatIList(zeros) → tt
a__isNatIList(V) → a__U31(a__isNatList(V))
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)
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(isNatList(X)) → a__isNatList(X)
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
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(nil) → nil
mark(s(X)) → s(mark(X))
a__U11(X) → U11(X)
a__zeros → zeros
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(tt) → tt
mark(0) → 0
a__U51(X1, X2) → U51(X1, X2)
a__isNatIList(X) → isNatIList(X)
a__isNatList(X) → isNatList(X)
a__U52(X) → U52(X)
a__U31(X) → U31(X)
a__U21(X) → U21(X)
a__U42(X) → U42(X)
a__U41(X1, X2) → U41(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__U61(X1, X2, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ AND

                        ↳ QDP

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

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

A__U62(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__U61(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, L, N) → a__U62(a__isNat(N), L)
a__U62(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, 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, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__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, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(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 2 less nodes.