Termination w.r.t. Q proof of /tmp/tmpx_2fQr/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

  ↳ RRRPoloQTRSProof

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.

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) → 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.
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)) = x₁
POL(U21(x₁)) = x₁
POL(U31(x₁)) = x₁
POL(U41(x₁, x₂)) = x₁ + x₂
POL(U42(x₁)) = x₁
POL(U51(x₁, x₂)) = x₁ + x₂
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = x₁ + 2·x₂ + x₃
POL(U62(x₁, x₂)) = x₁ + 2·x₂
POL(a__U11(x₁)) = x₁
POL(a__U21(x₁)) = x₁
POL(a__U31(x₁)) = x₁
POL(a__U41(x₁, x₂)) = x₁ + x₂
POL(a__U42(x₁)) = x₁
POL(a__U51(x₁, x₂)) = x₁ + x₂
POL(a__U52(x₁)) = x₁
POL(a__U61(x₁, x₂, x₃)) = x₁ + 2·x₂ + x₃
POL(a__U62(x₁, x₂)) = 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(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) = 1
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) → 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(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
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.

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) → 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(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
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.
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__U31(a__isNatList(V))
a__isNatIList(zeros) → tt

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁)) = x₁
POL(U21(x₁)) = x₁
POL(U31(x₁)) = x₁
POL(U41(x₁, x₂)) = 1 + x₁ + x₂
POL(U42(x₁)) = x₁
POL(U51(x₁, x₂)) = x₁ + x₂
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(U62(x₁, x₂)) = x₁ + 2·x₂
POL(a__U11(x₁)) = x₁
POL(a__U21(x₁)) = x₁
POL(a__U31(x₁)) = x₁
POL(a__U41(x₁, x₂)) = 1 + x₁ + x₂
POL(a__U42(x₁)) = x₁
POL(a__U51(x₁, x₂)) = x₁ + x₂
POL(a__U52(x₁)) = x₁
POL(a__U61(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(a__U62(x₁, x₂)) = x₁ + 2·x₂
POL(a__isNat(x₁)) = 2·x₁
POL(a__isNatIList(x₁)) = 1 + x₁
POL(a__isNatList(x₁)) = x₁
POL(a__length(x₁)) = 2·x₁
POL(a__zeros) = 0
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = 2·x₁
POL(isNatIList(x₁)) = 1 + 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

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(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
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.

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) → 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(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
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.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

a__U31(tt) → tt

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁)) = 2·x₁
POL(U21(x₁)) = x₁
POL(U31(x₁)) = 1 + x₁
POL(U41(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U42(x₁)) = 2·x₁
POL(U51(x₁, x₂)) = x₁ + 2·x₂
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + x₃
POL(U62(x₁, x₂)) = x₁ + 2·x₂
POL(a__U11(x₁)) = 2·x₁
POL(a__U21(x₁)) = x₁
POL(a__U31(x₁)) = 1 + x₁
POL(a__U41(x₁, x₂)) = 2·x₁ + 2·x₂
POL(a__U42(x₁)) = 2·x₁
POL(a__U51(x₁, x₂)) = x₁ + 2·x₂
POL(a__U52(x₁)) = x₁
POL(a__U61(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + x₃
POL(a__U62(x₁, x₂)) = 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(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) = 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) → tt
a__U21(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(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
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.

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) → tt
a__U21(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(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
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.
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__U11(a__isNatList(V1))

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁)) = x₁
POL(U21(x₁)) = x₁
POL(U31(x₁)) = x₁
POL(U41(x₁, x₂)) = x₁ + 2·x₂
POL(U42(x₁)) = x₁
POL(U51(x₁, x₂)) = x₁ + x₂
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = 1 + 2·x₁ + 2·x₂ + 2·x₃
POL(U62(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(a__U11(x₁)) = x₁
POL(a__U21(x₁)) = x₁
POL(a__U31(x₁)) = x₁
POL(a__U41(x₁, x₂)) = x₁ + 2·x₂
POL(a__U42(x₁)) = x₁
POL(a__U51(x₁, x₂)) = x₁ + x₂
POL(a__U52(x₁)) = x₁
POL(a__U61(x₁, x₂, x₃)) = 1 + 2·x₁ + 2·x₂ + 2·x₃
POL(a__U62(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(a__isNat(x₁)) = x₁
POL(a__isNatIList(x₁)) = 2·x₁
POL(a__isNatList(x₁)) = x₁
POL(a__length(x₁)) = 1 + 2·x₁
POL(a__zeros) = 0
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 2·x₁
POL(isNatList(x₁)) = 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

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

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__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(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
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.

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) → tt
a__U21(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(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
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.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

a__U11(tt) → tt

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁)) = 1 + x₁
POL(U21(x₁)) = x₁
POL(U31(x₁)) = 2·x₁
POL(U41(x₁, x₂)) = x₁ + 2·x₂
POL(U42(x₁)) = x₁
POL(U51(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(U62(x₁, x₂)) = 2·x₁ + 2·x₂
POL(a__U11(x₁)) = 1 + x₁
POL(a__U21(x₁)) = x₁
POL(a__U31(x₁)) = 2·x₁
POL(a__U41(x₁, x₂)) = x₁ + 2·x₂
POL(a__U42(x₁)) = x₁
POL(a__U51(x₁, x₂)) = 2·x₁ + 2·x₂
POL(a__U52(x₁)) = x₁
POL(a__U61(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(a__U62(x₁, x₂)) = 2·x₁ + 2·x₂
POL(a__isNat(x₁)) = x₁
POL(a__isNatIList(x₁)) = 2·x₁
POL(a__isNatList(x₁)) = 2·x₁
POL(a__length(x₁)) = 2·x₁
POL(a__zeros) = 0
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 2·x₁
POL(isNatList(x₁)) = 2·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

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

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

a__zeros → cons(0, zeros)
a__U21(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(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
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.

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

a__zeros → cons(0, zeros)
a__U21(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(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
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.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U61(tt, L, N) → a__U62(a__isNat(N), L)
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(0) → 0
mark(tt) → tt
mark(nil) → nil
a__zeros → zeros

Used ordering:
Polynomial interpretation [25]:

POL(0) = 1
POL(U11(x₁)) = x₁
POL(U21(x₁)) = x₁
POL(U31(x₁)) = x₁
POL(U41(x₁, x₂)) = 1 + x₁ + x₂
POL(U42(x₁)) = x₁
POL(U51(x₁, x₂)) = x₁ + x₂
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U62(x₁, x₂)) = x₁ + x₂
POL(a__U11(x₁)) = x₁
POL(a__U21(x₁)) = x₁
POL(a__U31(x₁)) = x₁
POL(a__U41(x₁, x₂)) = 1 + x₁ + x₂
POL(a__U42(x₁)) = x₁
POL(a__U51(x₁, x₂)) = x₁ + x₂
POL(a__U52(x₁)) = x₁
POL(a__U61(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U62(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(cons(x₁, x₂)) = x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 2 + 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

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

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

a__zeros → cons(0, zeros)
a__U21(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U52(tt) → tt
a__U62(tt, L) → s(a__length(mark(L)))
a__isNat(0) → tt
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
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(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
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.

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

a__zeros → cons(0, zeros)
a__U21(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U52(tt) → tt
a__U62(tt, L) → s(a__length(mark(L)))
a__isNat(0) → tt
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
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(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
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.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

a__U62(tt, L) → s(a__length(mark(L)))

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁)) = 2·x₁
POL(U21(x₁)) = x₁
POL(U31(x₁)) = 2·x₁
POL(U41(x₁, x₂)) = x₁ + 2·x₂
POL(U42(x₁)) = x₁
POL(U51(x₁, x₂)) = x₁ + 2·x₂
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(U62(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(a__U11(x₁)) = 2·x₁
POL(a__U21(x₁)) = x₁
POL(a__U31(x₁)) = 2·x₁
POL(a__U41(x₁, x₂)) = x₁ + 2·x₂
POL(a__U42(x₁)) = x₁
POL(a__U51(x₁, x₂)) = x₁ + 2·x₂
POL(a__U52(x₁)) = x₁
POL(a__U61(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(a__U62(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(a__isNat(x₁)) = 2·x₁
POL(a__isNatIList(x₁)) = x₁
POL(a__isNatList(x₁)) = 2·x₁
POL(a__length(x₁)) = 2·x₁
POL(a__zeros) = 0
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = 2·x₁
POL(isNatIList(x₁)) = x₁
POL(isNatList(x₁)) = 2·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

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

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

a__zeros → cons(0, zeros)
a__U21(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U52(tt) → tt
a__isNat(0) → tt
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
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(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
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.

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

a__zeros → cons(0, zeros)
a__U21(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U52(tt) → tt
a__isNat(0) → tt
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
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(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
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.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

a__U21(tt) → tt
a__U42(tt) → tt
a__U52(tt) → tt
mark(s(X)) → s(mark(X))
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)

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁)) = 1 + x₁
POL(U21(x₁)) = 1 + x₁
POL(U31(x₁)) = 1 + x₁
POL(U41(x₁, x₂)) = 1 + x₁ + x₂
POL(U42(x₁)) = 1 + x₁
POL(U51(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(U52(x₁)) = 1 + x₁
POL(U61(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U62(x₁, x₂)) = 1 + x₁ + x₂
POL(a__U11(x₁)) = 2 + x₁
POL(a__U21(x₁)) = 2 + x₁
POL(a__U31(x₁)) = 2 + x₁
POL(a__U41(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(a__U42(x₁)) = 2 + x₁
POL(a__U51(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(a__U52(x₁)) = 2 + x₁
POL(a__U61(x₁, x₂, x₃)) = 2 + x₁ + x₂ + x₃
POL(a__U62(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(a__isNat(x₁)) = 2 + 2·x₁
POL(a__isNatIList(x₁)) = 2 + 2·x₁
POL(a__isNatList(x₁)) = 2 + x₁
POL(a__length(x₁)) = 2 + x₁
POL(a__zeros) = 2
POL(cons(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(isNat(x₁)) = 1 + x₁
POL(isNatIList(x₁)) = 1 + x₁
POL(isNatList(x₁)) = 1 + x₁
POL(length(x₁)) = 1 + x₁
POL(mark(x₁)) = 2 + 2·x₁
POL(s(x₁)) = 1 + 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

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

a__zeros → cons(0, zeros)
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__isNat(0) → tt
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
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(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(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__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__isNat(0) → tt
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
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(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(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:

a__zeros → cons(0, zeros)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
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(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁)) = 2 + x₁
POL(U21(x₁)) = 2 + x₁
POL(U31(x₁)) = 2 + 2·x₁
POL(U41(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(U42(x₁)) = 2 + 2·x₁
POL(U51(x₁, x₂)) = 1 + x₁ + x₂
POL(U52(x₁)) = 2 + 2·x₁
POL(U61(x₁, x₂, x₃)) = 2 + x₁ + 2·x₂ + 2·x₃
POL(U62(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(a__U11(x₁)) = 2 + x₁
POL(a__U21(x₁)) = x₁
POL(a__U31(x₁)) = 2 + 2·x₁
POL(a__U41(x₁, x₂)) = x₁ + 2·x₂
POL(a__U42(x₁)) = x₁
POL(a__U51(x₁, x₂)) = x₁ + 2·x₂
POL(a__U52(x₁)) = 1 + x₁
POL(a__U61(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(a__U62(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(cons(x₁, x₂)) = x₁ + 2·x₂
POL(length(x₁)) = 2·x₁
POL(mark(x₁)) = 1 + 2·x₁
POL(s(x₁)) = 1 + 2·x₁
POL(tt) = 0
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__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__isNat(0) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)

Q is empty.

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

a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__isNat(0) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(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:

a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__isNat(0) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)

Used ordering:
Polynomial interpretation [25]:

POL(0) = 1
POL(a__U41(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(a__U42(x₁)) = x₁
POL(a__U51(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(a__U61(x₁, x₂, x₃)) = 1 + x₁ + 2·x₂ + 2·x₃
POL(a__isNat(x₁)) = x₁
POL(a__isNatIList(x₁)) = x₁
POL(a__isNatList(x₁)) = 2·x₁
POL(a__length(x₁)) = 2·x₁
POL(a__zeros) = 0
POL(cons(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(length(x₁)) = 2 + 2·x₁
POL(mark(x₁)) = 1 + 2·x₁
POL(tt) = 0
POL(zeros) = 0

↳ 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.