Termination w.r.t. Q proof of /tmp/tmpJ-sq-_/LengthOfFiniteLists_nokinds

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, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)

Q is empty.

↳ QTRS

  ↳ RRRPoloQTRSProof

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

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)

Q is empty.

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

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

a__isNatList(nil) → tt
a__length(nil) → 0

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁, x₂)) = x₁ + 2·x₂
POL(a__U11(x₁, x₂)) = x₁ + 2·x₂
POL(a__and(x₁, x₂)) = 2·x₁ + 2·x₂
POL(a__isNat(x₁)) = x₁
POL(a__isNatIList(x₁)) = x₁
POL(a__isNatList(x₁)) = x₁
POL(a__length(x₁)) = 2·x₁
POL(a__zeros) = 0
POL(and(x₁, x₂)) = 2·x₁ + 2·x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 2·x₁
POL(mark(x₁)) = x₁
POL(nil) = 2
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

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

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)

Q is empty.

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

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__isNatList(V1)
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

a__isNat(length(V1)) → a__isNatList(V1)

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(a__U11(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(a__and(x₁, x₂)) = x₁ + 2·x₂
POL(a__isNat(x₁)) = x₁
POL(a__isNatIList(x₁)) = 2·x₁
POL(a__isNatList(x₁)) = 2·x₁
POL(a__length(x₁)) = 1 + 2·x₁
POL(a__zeros) = 0
POL(and(x₁, x₂)) = x₁ + 2·x₂
POL(cons(x₁, x₂)) = x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 2·x₁
POL(isNatList(x₁)) = 2·x₁
POL(length(x₁)) = 1 + 2·x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

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

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)

Q is empty.

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

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

a__isNatIList(V) → a__isNatList(V)
a__isNatIList(zeros) → tt

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁, x₂)) = x₁ + 2·x₂
POL(a__U11(x₁, x₂)) = x₁ + 2·x₂
POL(a__and(x₁, x₂)) = 2·x₁ + x₂
POL(a__isNat(x₁)) = x₁
POL(a__isNatIList(x₁)) = 2 + x₁
POL(a__isNatList(x₁)) = x₁
POL(a__length(x₁)) = 2·x₁
POL(a__zeros) = 0
POL(and(x₁, x₂)) = 2·x₁ + x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 2 + x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 2·x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

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

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)

Q is empty.

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

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(s(V1)) → a__isNat(V1)
a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)
a__isNatIList(X) → isNatIList(X)

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

a__isNatIList(cons(V1, V2)) → a__and(a__isNat(V1), isNatIList(V2))
mark(isNat(X)) → a__isNat(X)
mark(isNatList(X)) → a__isNatList(X)
mark(0) → 0
mark(tt) → tt
mark(nil) → nil
a__zeros → zeros
a__isNatIList(X) → isNatIList(X)

Used ordering:
Polynomial interpretation [25]:

POL(0) = 1
POL(U11(x₁, x₂)) = x₁ + x₂
POL(a__U11(x₁, x₂)) = x₁ + x₂
POL(a__and(x₁, x₂)) = x₁ + x₂
POL(a__isNat(x₁)) = x₁
POL(a__isNatIList(x₁)) = 2 + x₁
POL(a__isNatList(x₁)) = x₁
POL(a__length(x₁)) = x₁
POL(a__zeros) = 1
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 1 + x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = x₁
POL(mark(x₁)) = 1 + x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 1
POL(zeros) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

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

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(s(V1)) → a__isNat(V1)
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)

Q is empty.

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

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(s(V1)) → a__isNat(V1)
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIList(X)) → a__isNatIList(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

mark(isNatIList(X)) → a__isNatIList(X)

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁, x₂)) = 2·x₁ + 2·x₂
POL(a__U11(x₁, x₂)) = 2·x₁ + 2·x₂
POL(a__and(x₁, x₂)) = x₁ + x₂
POL(a__isNat(x₁)) = x₁
POL(a__isNatIList(x₁)) = x₁
POL(a__isNatList(x₁)) = x₁
POL(a__length(x₁)) = 2·x₁
POL(a__zeros) = 0
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 2 + 2·x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 2·x₁
POL(mark(x₁)) = x₁
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

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

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(s(V1)) → a__isNat(V1)
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)

Q is empty.

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

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(s(V1)) → a__isNat(V1)
a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

a__isNatList(cons(V1, V2)) → a__and(a__isNat(V1), isNatList(V2))
a__isNat(X) → isNat(X)
a__isNatList(X) → isNatList(X)

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁, x₂)) = x₁ + x₂
POL(a__U11(x₁, x₂)) = x₁ + x₂
POL(a__and(x₁, x₂)) = x₁ + x₂
POL(a__isNat(x₁)) = 1 + x₁
POL(a__isNatList(x₁)) = 1 + x₁
POL(a__length(x₁)) = x₁
POL(a__zeros) = 1
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = x₁
POL(mark(x₁)) = 1 + x₁
POL(s(x₁)) = x₁
POL(tt) = 1
POL(zeros) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

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

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(s(V1)) → a__isNat(V1)
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)

Q is empty.

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

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(s(V1)) → a__isNat(V1)
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

a__isNat(0) → tt

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁, x₂)) = x₁ + x₂
POL(a__U11(x₁, x₂)) = x₁ + x₂
POL(a__and(x₁, x₂)) = x₁ + x₂
POL(a__isNat(x₁)) = 1 + 2·x₁
POL(a__isNatList(x₁)) = x₁
POL(a__length(x₁)) = x₁
POL(a__zeros) = 0
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(length(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

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

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(s(V1)) → a__isNat(V1)
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)

Q is empty.

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

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(s(V1)) → a__isNat(V1)
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

a__and(tt, X) → mark(X)

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁, x₂)) = x₁ + x₂
POL(a__U11(x₁, x₂)) = x₁ + x₂
POL(a__and(x₁, x₂)) = x₁ + 2·x₂
POL(a__isNat(x₁)) = 2·x₁
POL(a__isNatList(x₁)) = 1 + x₁
POL(a__length(x₁)) = 1 + x₁
POL(a__zeros) = 0
POL(and(x₁, x₂)) = x₁ + 2·x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(length(x₁)) = 1 + x₁
POL(mark(x₁)) = x₁
POL(s(x₁)) = x₁
POL(tt) = 1
POL(zeros) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

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

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__isNat(s(V1)) → a__isNat(V1)
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)

Q is empty.

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

a__zeros → cons(0, zeros)
a__U11(tt, L) → s(a__length(mark(L)))
a__isNat(s(V1)) → a__isNat(V1)
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

a__U11(tt, L) → s(a__length(mark(L)))

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁, x₂)) = x₁ + x₂
POL(a__U11(x₁, x₂)) = x₁ + x₂
POL(a__and(x₁, x₂)) = x₁ + 2·x₂
POL(a__isNat(x₁)) = x₁
POL(a__isNatList(x₁)) = x₁
POL(a__length(x₁)) = x₁
POL(a__zeros) = 0
POL(and(x₁, x₂)) = x₁ + 2·x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(length(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(s(x₁)) = x₁
POL(tt) = 2
POL(zeros) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

                                    ↳ QTRS

                                      ↳ RRRPoloQTRSProof

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

a__zeros → cons(0, zeros)
a__isNat(s(V1)) → a__isNat(V1)
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)

Q is empty.

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

a__zeros → cons(0, zeros)
a__isNat(s(V1)) → a__isNat(V1)
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

a__isNat(s(V1)) → a__isNat(V1)
a__length(cons(N, L)) → a__U11(a__and(a__isNatList(L), isNat(N)), L)
mark(zeros) → a__zeros
mark(length(X)) → a__length(mark(X))
mark(s(X)) → s(mark(X))

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁, x₂)) = x₁ + x₂
POL(a__U11(x₁, x₂)) = x₁ + x₂
POL(a__and(x₁, x₂)) = x₁ + 2·x₂
POL(a__isNat(x₁)) = 2·x₁
POL(a__isNatList(x₁)) = x₁
POL(a__length(x₁)) = 1 + 2·x₁
POL(a__zeros) = 1
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 2·x₁ + x₂
POL(isNat(x₁)) = 2·x₁
POL(length(x₁)) = 1 + 2·x₁
POL(mark(x₁)) = 2·x₁
POL(s(x₁)) = 1 + 2·x₁
POL(zeros) = 1

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

                                    ↳ QTRS

                                      ↳ RRRPoloQTRSProof

                                        ↳ QTRS

                                          ↳ RRRPoloQTRSProof

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

a__zeros → cons(0, zeros)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)

Q is empty.

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

a__zeros → cons(0, zeros)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

a__zeros → cons(0, zeros)

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁, x₂)) = 2·x₁ + x₂
POL(a__U11(x₁, x₂)) = 2·x₁ + 2·x₂
POL(a__and(x₁, x₂)) = x₁ + 2·x₂
POL(a__length(x₁)) = 2 + 2·x₁
POL(a__zeros) = 2
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 2·x₁ + x₂
POL(length(x₁)) = 2 + x₁
POL(mark(x₁)) = 2·x₁
POL(zeros) = 1

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

                                    ↳ QTRS

                                      ↳ RRRPoloQTRSProof

                                        ↳ QTRS

                                          ↳ RRRPoloQTRSProof

                                            ↳ QTRS

                                              ↳ RRRPoloQTRSProof

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

mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)

Q is empty.

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

mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

mark(cons(X1, X2)) → cons(mark(X1), X2)
a__U11(X1, X2) → U11(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)

Used ordering:
Polynomial interpretation [25]:

POL(U11(x₁, x₂)) = 1 + x₁ + x₂
POL(a__U11(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(a__and(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(a__length(x₁)) = 2 + 2·x₁
POL(and(x₁, x₂)) = 1 + x₁ + x₂
POL(cons(x₁, x₂)) = 1 + x₁ + x₂
POL(length(x₁)) = 1 + 2·x₁
POL(mark(x₁)) = 2 + 2·x₁

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

                                    ↳ QTRS

                                      ↳ RRRPoloQTRSProof

                                        ↳ QTRS

                                          ↳ RRRPoloQTRSProof

                                            ↳ QTRS

                                              ↳ RRRPoloQTRSProof

                                                ↳ QTRS

                                                  ↳ RRRPoloQTRSProof

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

mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(and(X1, X2)) → a__and(mark(X1), X2)

Q is empty.

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

mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(and(X1, X2)) → a__and(mark(X1), X2)

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(and(X1, X2)) → a__and(mark(X1), X2)

Used ordering:
Polynomial interpretation [25]:

POL(U11(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(a__U11(x₁, x₂)) = 1 + x₁ + x₂
POL(a__and(x₁, x₂)) = 1 + x₁ + x₂
POL(and(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(mark(x₁)) = 2 + 2·x₁

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

                                    ↳ QTRS

                                      ↳ RRRPoloQTRSProof

                                        ↳ QTRS

                                          ↳ RRRPoloQTRSProof

                                            ↳ QTRS

                                              ↳ RRRPoloQTRSProof

                                                ↳ QTRS

                                                  ↳ RRRPoloQTRSProof

                                                    ↳ QTRS

                                                      ↳ RisEmptyProof

Q restricted rewrite system:
R is empty.
Q is empty.

The TRS R is empty. Hence, termination is trivially proven.