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

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____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__U11(tt) → tt
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt, P) → a__U72(a__isPal(P))
a__U72(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
a__isQid(a) → tt
a__isQid(e) → tt
a__isQid(i) → tt
a__isQid(o) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(U11(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(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(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X)) → a__U72(mark(X))
mark(isPal(X)) → a__isPal(X)
mark(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__U11(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isPal(X) → isPal(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)

Q is empty.

↳ QTRS

  ↳ RRRPoloQTRSProof

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

a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__U11(tt) → tt
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt, P) → a__U72(a__isPal(P))
a__U72(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
a__isQid(a) → tt
a__isQid(e) → tt
a__isQid(i) → tt
a__isQid(o) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(U11(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(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(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X)) → a__U72(mark(X))
mark(isPal(X)) → a__isPal(X)
mark(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__U11(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isPal(X) → isPal(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)

Q is empty.

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

a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__U11(tt) → tt
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt, P) → a__U72(a__isPal(P))
a__U72(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
a__isQid(a) → tt
a__isQid(e) → tt
a__isQid(i) → tt
a__isQid(o) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(U11(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(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(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X)) → a__U72(mark(X))
mark(isPal(X)) → a__isPal(X)
mark(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__U11(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isPal(X) → isPal(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(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____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isPal(nil) → tt
a__isQid(a) → tt
a__isQid(i) → tt
a__isQid(o) → tt

Used ordering:
Polynomial interpretation [25]:

POL(U11(x₁)) = x₁
POL(U21(x₁, x₂)) = x₁ + 2·x₂
POL(U22(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)) = 2·x₁
POL(U71(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U72(x₁)) = x₁
POL(U81(x₁)) = x₁
POL(__(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(a) = 1
POL(a__U11(x₁)) = x₁
POL(a__U21(x₁, x₂)) = x₁ + 2·x₂
POL(a__U22(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)) = 2·x₁
POL(a__U71(x₁, x₂)) = 2·x₁ + 2·x₂
POL(a__U72(x₁)) = x₁
POL(a__U81(x₁)) = x₁
POL(a____(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(a__isList(x₁)) = 2·x₁
POL(a__isNeList(x₁)) = 2·x₁
POL(a__isNePal(x₁)) = 2·x₁
POL(a__isPal(x₁)) = 2·x₁
POL(a__isQid(x₁)) = x₁
POL(e) = 0
POL(i) = 1
POL(isList(x₁)) = 2·x₁
POL(isNeList(x₁)) = 2·x₁
POL(isNePal(x₁)) = 2·x₁
POL(isPal(x₁)) = 2·x₁
POL(isQid(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(nil) = 2
POL(o) = 2
POL(tt) = 0
POL(u) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

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

a__U11(tt) → tt
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt, P) → a__U72(a__isPal(P))
a__U72(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isNeList(V) → a__U31(a__isQid(V))
a__isNePal(V) → a__U61(a__isQid(V))
a__isPal(V) → a__U81(a__isNePal(V))
a__isQid(e) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(U11(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(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(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X)) → a__U72(mark(X))
mark(isPal(X)) → a__isPal(X)
mark(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__U11(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isPal(X) → isPal(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)

Q is empty.

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

a__U11(tt) → tt
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt, P) → a__U72(a__isPal(P))
a__U72(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isNeList(V) → a__U31(a__isQid(V))
a__isNePal(V) → a__U61(a__isQid(V))
a__isPal(V) → a__U81(a__isNePal(V))
a__isQid(e) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(U11(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(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(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X)) → a__U72(mark(X))
mark(isPal(X)) → a__isPal(X)
mark(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__U11(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isPal(X) → isPal(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(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, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U72(tt) → tt
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X)) → a__U72(mark(X))
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__U42(X) → U42(X)
a__U61(X) → U61(X)
a__isPal(X) → isPal(X)
a__isNePal(X) → isNePal(X)

Used ordering:
Polynomial interpretation [25]:

POL(U11(x₁)) = x₁
POL(U21(x₁, x₂)) = 1 + x₁ + x₂
POL(U22(x₁)) = 2·x₁
POL(U31(x₁)) = x₁
POL(U41(x₁, x₂)) = x₁ + x₂
POL(U42(x₁)) = 1 + x₁
POL(U51(x₁, x₂)) = x₁ + x₂
POL(U52(x₁)) = 1 + x₁
POL(U61(x₁)) = 1 + x₁
POL(U71(x₁, x₂)) = 2 + x₁ + x₂
POL(U72(x₁)) = 2 + x₁
POL(U81(x₁)) = x₁
POL(__(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(a) = 2
POL(a__U11(x₁)) = x₁
POL(a__U21(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(a__U22(x₁)) = 2·x₁
POL(a__U31(x₁)) = x₁
POL(a__U41(x₁, x₂)) = x₁ + x₂
POL(a__U42(x₁)) = 2 + x₁
POL(a__U51(x₁, x₂)) = x₁ + 2·x₂
POL(a__U52(x₁)) = 1 + x₁
POL(a__U61(x₁)) = 2 + x₁
POL(a__U71(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(a__U72(x₁)) = 2 + x₁
POL(a__U81(x₁)) = x₁
POL(a____(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(a__isList(x₁)) = x₁
POL(a__isNeList(x₁)) = x₁
POL(a__isNePal(x₁)) = 2 + 2·x₁
POL(a__isPal(x₁)) = 2 + 2·x₁
POL(a__isQid(x₁)) = x₁
POL(e) = 2
POL(i) = 2
POL(isList(x₁)) = x₁
POL(isNeList(x₁)) = x₁
POL(isNePal(x₁)) = 1 + x₁
POL(isPal(x₁)) = 1 + 2·x₁
POL(isQid(x₁)) = x₁
POL(mark(x₁)) = 2·x₁
POL(nil) = 1
POL(o) = 2
POL(tt) = 2
POL(u) = 2

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

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

a__U11(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U71(tt, P) → a__U72(a__isPal(P))
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isNeList(V) → a__U31(a__isQid(V))
a__isNePal(V) → a__U61(a__isQid(V))
a__isPal(V) → a__U81(a__isNePal(V))
a__isQid(e) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(U11(X)) → a__U11(mark(X))
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(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(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U61(X)) → a__U61(mark(X))
mark(isPal(X)) → a__isPal(X)
mark(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
a__U11(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)

Q is empty.

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

a__U11(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U71(tt, P) → a__U72(a__isPal(P))
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isNeList(V) → a__U31(a__isQid(V))
a__isNePal(V) → a__U61(a__isQid(V))
a__isPal(V) → a__U81(a__isNePal(V))
a__isQid(e) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(U11(X)) → a__U11(mark(X))
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(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(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U61(X)) → a__U61(mark(X))
mark(isPal(X)) → a__isPal(X)
mark(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
a__U11(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(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__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U71(tt, P) → a__U72(a__isPal(P))
a__isNePal(V) → a__U61(a__isQid(V))
a__isPal(V) → a__U81(a__isNePal(V))
mark(isPal(X)) → a__isPal(X)
mark(isNePal(X)) → a__isNePal(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U52(X) → U52(X)
a__U71(X1, X2) → U71(X1, X2)

Used ordering:
Polynomial interpretation [25]:

POL(U11(x₁)) = x₁
POL(U21(x₁, x₂)) = 1 + 2·x₁ + x₂
POL(U22(x₁)) = 1 + x₁
POL(U31(x₁)) = x₁
POL(U41(x₁, x₂)) = 2·x₁ + x₂
POL(U42(x₁)) = 1 + 2·x₁
POL(U51(x₁, x₂)) = x₁ + x₂
POL(U52(x₁)) = x₁
POL(U61(x₁)) = x₁
POL(U71(x₁, x₂)) = x₁ + x₂
POL(U72(x₁)) = 1 + x₁
POL(U81(x₁)) = x₁
POL(__(x₁, x₂)) = 2·x₁ + 2·x₂
POL(a__U11(x₁)) = x₁
POL(a__U21(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(a__U22(x₁)) = 2 + x₁
POL(a__U31(x₁)) = x₁
POL(a__U41(x₁, x₂)) = 2·x₁ + 2·x₂
POL(a__U42(x₁)) = 2 + 2·x₁
POL(a__U51(x₁, x₂)) = x₁ + x₂
POL(a__U52(x₁)) = 1 + 2·x₁
POL(a__U61(x₁)) = x₁
POL(a__U71(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(a__U72(x₁)) = 1 + x₁
POL(a__U81(x₁)) = x₁
POL(a____(x₁, x₂)) = x₁ + x₂
POL(a__isList(x₁)) = x₁
POL(a__isNeList(x₁)) = x₁
POL(a__isNePal(x₁)) = 1 + x₁
POL(a__isPal(x₁)) = 2 + x₁
POL(a__isQid(x₁)) = x₁
POL(e) = 2
POL(isList(x₁)) = x₁
POL(isNeList(x₁)) = x₁
POL(isNePal(x₁)) = 1 + 2·x₁
POL(isPal(x₁)) = 2 + x₁
POL(isQid(x₁)) = x₁
POL(mark(x₁)) = 2·x₁
POL(tt) = 2
POL(u) = 2

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

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

a__U11(tt) → tt
a__U31(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isNeList(V) → a__U31(a__isQid(V))
a__isQid(e) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(U11(X)) → a__U11(mark(X))
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(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(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U61(X)) → a__U61(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
a__U11(X) → U11(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U72(X) → U72(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)

Q is empty.

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

a__U11(tt) → tt
a__U31(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isNeList(V) → a__U31(a__isQid(V))
a__isQid(e) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(U11(X)) → a__U11(mark(X))
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(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(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U61(X)) → a__U61(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
a__U11(X) → U11(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U72(X) → U72(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(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__isQid(e) → tt
a__isQid(u) → tt
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
mark(U22(X)) → a__U22(mark(X))
mark(U42(X)) → a__U42(mark(X))
mark(U61(X)) → a__U61(mark(X))
a__U51(X1, X2) → U51(X1, X2)
a__U72(X) → U72(X)

Used ordering:
Polynomial interpretation [25]:

POL(U11(x₁)) = x₁
POL(U22(x₁)) = 2 + 2·x₁
POL(U31(x₁)) = x₁
POL(U41(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U42(x₁)) = 1 + 2·x₁
POL(U51(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(U61(x₁)) = 2 + 2·x₁
POL(U72(x₁)) = x₁
POL(U81(x₁)) = x₁
POL(__(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(a__U11(x₁)) = x₁
POL(a__U22(x₁)) = 2 + x₁
POL(a__U31(x₁)) = x₁
POL(a__U41(x₁, x₂)) = 2·x₁ + 2·x₂
POL(a__U42(x₁)) = x₁
POL(a__U51(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(a__U61(x₁)) = 1 + 2·x₁
POL(a__U72(x₁)) = 2 + 2·x₁
POL(a__U81(x₁)) = x₁
POL(a____(x₁, x₂)) = 2 + x₁ + x₂
POL(a__isList(x₁)) = 2·x₁
POL(a__isNeList(x₁)) = x₁
POL(a__isQid(x₁)) = x₁
POL(e) = 1
POL(isList(x₁)) = 2·x₁
POL(isNeList(x₁)) = x₁
POL(isQid(x₁)) = x₁
POL(mark(x₁)) = 2·x₁
POL(tt) = 0
POL(u) = 2

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

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

a__U11(tt) → tt
a__U31(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isNeList(V) → a__U31(a__isQid(V))
mark(U11(X)) → a__U11(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
a__U11(X) → U11(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__isNeList(X) → isNeList(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)

Q is empty.

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

a__U11(tt) → tt
a__U31(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isNeList(V) → a__U31(a__isQid(V))
mark(U11(X)) → a__U11(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
a__U11(X) → U11(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__isNeList(X) → isNeList(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(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
a__U81(tt) → tt
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
a__U11(X) → U11(X)
a__isList(X) → isList(X)
a__U81(X) → U81(X)

Used ordering:
Polynomial interpretation [25]:

POL(U11(x₁)) = 1 + 2·x₁
POL(U31(x₁)) = x₁
POL(U41(x₁, x₂)) = 2 + x₁ + x₂
POL(U51(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(U81(x₁)) = 1 + 2·x₁
POL(a__U11(x₁)) = 2 + 2·x₁
POL(a__U31(x₁)) = x₁
POL(a__U41(x₁, x₂)) = 2 + x₁ + x₂
POL(a__U51(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(a__U81(x₁)) = 2 + 2·x₁
POL(a__isList(x₁)) = 2 + 2·x₁
POL(a__isNeList(x₁)) = x₁
POL(a__isQid(x₁)) = x₁
POL(isList(x₁)) = 1 + x₁
POL(isNeList(x₁)) = x₁
POL(isQid(x₁)) = x₁
POL(mark(x₁)) = 2·x₁
POL(tt) = 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__U31(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isNeList(V) → a__U31(a__isQid(V))
mark(U11(X)) → a__U11(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)

Q is empty.

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

a__U31(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isNeList(V) → a__U31(a__isQid(V))
mark(U11(X)) → a__U11(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(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
a__isList(V) → a__U11(a__isNeList(V))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U81(X)) → a__U81(mark(X))
a__U41(X1, X2) → U41(X1, X2)

Used ordering:
Polynomial interpretation [25]:

POL(U11(x₁)) = 2·x₁
POL(U31(x₁)) = 1 + x₁
POL(U41(x₁, x₂)) = x₁ + x₂
POL(U81(x₁)) = 1 + 2·x₁
POL(a__U11(x₁)) = x₁
POL(a__U31(x₁)) = 1 + x₁
POL(a__U41(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(a__U81(x₁)) = x₁
POL(a__isList(x₁)) = 2 + 2·x₁
POL(a__isNeList(x₁)) = 1 + 2·x₁
POL(a__isQid(x₁)) = 2·x₁
POL(isList(x₁)) = 2 + 2·x₁
POL(isNeList(x₁)) = 1 + x₁
POL(isQid(x₁)) = 2·x₁
POL(mark(x₁)) = 2·x₁
POL(tt) = 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__isNeList(V) → a__U31(a__isQid(V))
mark(U11(X)) → a__U11(mark(X))
mark(isQid(X)) → a__isQid(X)
a__U31(X) → U31(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)

Q is empty.

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

a__isNeList(V) → a__U31(a__isQid(V))
mark(U11(X)) → a__U11(mark(X))
mark(isQid(X)) → a__isQid(X)
a__U31(X) → U31(X)
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(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(U11(X)) → a__U11(mark(X))
a__isNeList(X) → isNeList(X)
a__isQid(X) → isQid(X)

Used ordering:
Polynomial interpretation [25]:

POL(U11(x₁)) = 2 + 2·x₁
POL(U31(x₁)) = x₁
POL(a__U11(x₁)) = 2 + x₁
POL(a__U31(x₁)) = x₁
POL(a__isNeList(x₁)) = 2 + 2·x₁
POL(a__isQid(x₁)) = 2 + 2·x₁
POL(isNeList(x₁)) = 1 + x₁
POL(isQid(x₁)) = 1 + x₁
POL(mark(x₁)) = 2·x₁

↳ 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__isNeList(V) → a__U31(a__isQid(V))
mark(isQid(X)) → a__isQid(X)
a__U31(X) → U31(X)

Q is empty.

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

a__isNeList(V) → a__U31(a__isQid(V))
mark(isQid(X)) → a__isQid(X)
a__U31(X) → U31(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(isQid(X)) → a__isQid(X)
a__U31(X) → U31(X)

Used ordering:
Polynomial interpretation [25]:

POL(U31(x₁)) = 1 + x₁
POL(a__U31(x₁)) = 2 + 2·x₁
POL(a__isNeList(x₁)) = 2 + 2·x₁
POL(a__isQid(x₁)) = x₁
POL(isQid(x₁)) = 2 + 2·x₁
POL(mark(x₁)) = 1 + 2·x₁

↳ 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__isNeList(V) → a__U31(a__isQid(V))

Q is empty.

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

a__isNeList(V) → a__U31(a__isQid(V))

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__isNeList(V) → a__U31(a__isQid(V))

Used ordering:
Polynomial interpretation [25]:

POL(a__U31(x₁)) = 1 + x₁
POL(a__isNeList(x₁)) = 2 + 2·x₁
POL(a__isQid(x₁)) = x₁

↳ 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.