Termination w.r.t. Q proof of /tmp/tmpAZCisV/PALINDROME_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____(__(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

  ↳ DependencyPairsProof

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.

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

A__ISNEPAL(__(I, __(P, I))) → A__U71(a__isQid(I), P)
A__ISNEPAL(__(I, __(P, I))) → A__ISQID(I)
A__ISLIST(V) → A__U11(a__isNeList(V))
A__ISNEPAL(V) → A__U61(a__isQid(V))
MARK(U81(X)) → A__U81(mark(X))
A____(__(X, Y), Z) → MARK(X)
A__U51(tt, V2) → A__ISLIST(V2)
A__U71(tt, P) → A__ISPAL(P)
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
MARK(U51(X1, X2)) → MARK(X1)
A__ISLIST(__(V1, V2)) → A__ISLIST(V1)
A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
A__ISLIST(V) → A__ISNELIST(V)
A__ISNELIST(__(V1, V2)) → A__U51(a__isNeList(V1), V2)
A__ISPAL(V) → A__U81(a__isNePal(V))
MARK(U11(X)) → A__U11(mark(X))
A__U41(tt, V2) → A__U42(a__isNeList(V2))
MARK(isNeList(X)) → A__ISNELIST(X)
A__ISNEPAL(V) → A__ISQID(V)
MARK(U22(X)) → MARK(X)
A__ISNELIST(__(V1, V2)) → A__ISNELIST(V1)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)
A____(X, nil) → MARK(X)
A____(nil, X) → MARK(X)
A____(__(X, Y), Z) → MARK(Y)
MARK(U42(X)) → MARK(X)
A__ISPAL(V) → A__ISNEPAL(V)
A__ISNELIST(__(V1, V2)) → A__U41(a__isList(V1), V2)
A__U51(tt, V2) → A__U52(a__isList(V2))
A____(__(X, Y), Z) → MARK(Z)
A__ISNELIST(V) → A__U31(a__isQid(V))
A__ISLIST(__(V1, V2)) → A__U21(a__isList(V1), V2)
MARK(U81(X)) → MARK(X)
MARK(isList(X)) → A__ISLIST(X)
A__U21(tt, V2) → A__U22(a__isList(V2))
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
MARK(__(X1, X2)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U71(X1, X2)) → A__U71(mark(X1), X2)
A__U41(tt, V2) → A__ISNELIST(V2)
MARK(U22(X)) → A__U22(mark(X))
A__U71(tt, P) → A__U72(a__isPal(P))
MARK(U51(X1, X2)) → A__U51(mark(X1), X2)
A__U21(tt, V2) → A__ISLIST(V2)
MARK(U72(X)) → A__U72(mark(X))
MARK(U31(X)) → MARK(X)
MARK(U52(X)) → A__U52(mark(X))
MARK(isNePal(X)) → A__ISNEPAL(X)
MARK(U42(X)) → A__U42(mark(X))
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
MARK(isPal(X)) → A__ISPAL(X)
MARK(U72(X)) → MARK(X)
MARK(U71(X1, X2)) → MARK(X1)
A__ISNELIST(V) → A__ISQID(V)
MARK(U31(X)) → A__U31(mark(X))
MARK(U61(X)) → MARK(X)
MARK(isQid(X)) → A__ISQID(X)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U61(X)) → A__U61(mark(X))
A__ISNELIST(__(V1, V2)) → A__ISLIST(V1)
MARK(__(X1, X2)) → MARK(X2)
MARK(U11(X)) → MARK(X)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

A__ISNEPAL(__(I, __(P, I))) → A__U71(a__isQid(I), P)
A__ISNEPAL(__(I, __(P, I))) → A__ISQID(I)
A__ISLIST(V) → A__U11(a__isNeList(V))
A__ISNEPAL(V) → A__U61(a__isQid(V))
MARK(U81(X)) → A__U81(mark(X))
A____(__(X, Y), Z) → MARK(X)
A__U51(tt, V2) → A__ISLIST(V2)
A__U71(tt, P) → A__ISPAL(P)
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
MARK(U51(X1, X2)) → MARK(X1)
A__ISLIST(__(V1, V2)) → A__ISLIST(V1)
A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
A__ISLIST(V) → A__ISNELIST(V)
A__ISNELIST(__(V1, V2)) → A__U51(a__isNeList(V1), V2)
A__ISPAL(V) → A__U81(a__isNePal(V))
MARK(U11(X)) → A__U11(mark(X))
A__U41(tt, V2) → A__U42(a__isNeList(V2))
MARK(isNeList(X)) → A__ISNELIST(X)
A__ISNEPAL(V) → A__ISQID(V)
MARK(U22(X)) → MARK(X)
A__ISNELIST(__(V1, V2)) → A__ISNELIST(V1)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)
A____(X, nil) → MARK(X)
A____(nil, X) → MARK(X)
A____(__(X, Y), Z) → MARK(Y)
MARK(U42(X)) → MARK(X)
A__ISPAL(V) → A__ISNEPAL(V)
A__ISNELIST(__(V1, V2)) → A__U41(a__isList(V1), V2)
A__U51(tt, V2) → A__U52(a__isList(V2))
A____(__(X, Y), Z) → MARK(Z)
A__ISNELIST(V) → A__U31(a__isQid(V))
A__ISLIST(__(V1, V2)) → A__U21(a__isList(V1), V2)
MARK(U81(X)) → MARK(X)
MARK(isList(X)) → A__ISLIST(X)
A__U21(tt, V2) → A__U22(a__isList(V2))
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
MARK(__(X1, X2)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U71(X1, X2)) → A__U71(mark(X1), X2)
A__U41(tt, V2) → A__ISNELIST(V2)
MARK(U22(X)) → A__U22(mark(X))
A__U71(tt, P) → A__U72(a__isPal(P))
MARK(U51(X1, X2)) → A__U51(mark(X1), X2)
A__U21(tt, V2) → A__ISLIST(V2)
MARK(U72(X)) → A__U72(mark(X))
MARK(U31(X)) → MARK(X)
MARK(U52(X)) → A__U52(mark(X))
MARK(isNePal(X)) → A__ISNEPAL(X)
MARK(U42(X)) → A__U42(mark(X))
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
MARK(isPal(X)) → A__ISPAL(X)
MARK(U72(X)) → MARK(X)
MARK(U71(X1, X2)) → MARK(X1)
A__ISNELIST(V) → A__ISQID(V)
MARK(U31(X)) → A__U31(mark(X))
MARK(U61(X)) → MARK(X)
MARK(isQid(X)) → A__ISQID(X)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U61(X)) → A__U61(mark(X))
A__ISNELIST(__(V1, V2)) → A__ISLIST(V1)
MARK(__(X1, X2)) → MARK(X2)
MARK(U11(X)) → MARK(X)

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.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 3 SCCs with 28 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

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

A__ISNEPAL(__(I, __(P, I))) → A__U71(a__isQid(I), P)
A__U71(tt, P) → A__ISPAL(P)
A__ISPAL(V) → A__ISNEPAL(V)

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.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A__ISPAL(V) → A__ISNEPAL(V)

The remaining pairs can at least be oriented weakly.

A__ISNEPAL(__(I, __(P, I))) → A__U71(a__isQid(I), P)
A__U71(tt, P) → A__ISPAL(P)

Used ordering: Polynomial interpretation [25,35]:

POL(i) = 15/4
POL(A__U71(x₁, x₂)) = (1/2)x_1 + (3)x_2
POL(A__ISPAL(x₁)) = 1/4 + (3)x_1
POL(a__isQid(x₁)) = (9/4)x_1
POL(a) = 3/4
POL(__(x₁, x₂)) = (9/4)x_1 + (2)x_2
POL(tt) = 1/2
POL(u) = 1/2
POL(e) = 9/4
POL(A__ISNEPAL(x₁)) = (2)x_1
POL(o) = 15/4
POL(isQid(x₁)) = (1/2)x_1

The value of delta used in the strict ordering is 1/4.
The following usable rules [17] were oriented:

a__isQid(u) → tt
a__isQid(o) → tt
a__isQid(a) → tt
a__isQid(X) → isQid(X)
a__isQid(i) → tt
a__isQid(e) → tt

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

          ↳ QDP

          ↳ QDP

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

A__ISNEPAL(__(I, __(P, I))) → A__U71(a__isQid(I), P)
A__U71(tt, P) → A__ISPAL(P)

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.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

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

A__ISNELIST(__(V1, V2)) → A__U41(a__isList(V1), V2)
A__ISLIST(__(V1, V2)) → A__ISLIST(V1)
A__ISNELIST(__(V1, V2)) → A__ISNELIST(V1)
A__ISLIST(V) → A__ISNELIST(V)
A__ISNELIST(__(V1, V2)) → A__U51(a__isNeList(V1), V2)
A__U51(tt, V2) → A__ISLIST(V2)
A__U41(tt, V2) → A__ISNELIST(V2)
A__ISLIST(__(V1, V2)) → A__U21(a__isList(V1), V2)
A__ISNELIST(__(V1, V2)) → A__ISLIST(V1)
A__U21(tt, V2) → A__ISLIST(V2)

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.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A__ISNELIST(__(V1, V2)) → A__U41(a__isList(V1), V2)
A__ISLIST(__(V1, V2)) → A__ISLIST(V1)
A__ISNELIST(__(V1, V2)) → A__ISNELIST(V1)
A__ISLIST(V) → A__ISNELIST(V)
A__ISNELIST(__(V1, V2)) → A__U51(a__isNeList(V1), V2)
A__U41(tt, V2) → A__ISNELIST(V2)
A__ISLIST(__(V1, V2)) → A__U21(a__isList(V1), V2)
A__ISNELIST(__(V1, V2)) → A__ISLIST(V1)

The remaining pairs can at least be oriented weakly.

A__U51(tt, V2) → A__ISLIST(V2)
A__U21(tt, V2) → A__ISLIST(V2)

Used ordering: Polynomial interpretation [25,35]:

POL(i) = 13/4
POL(a__U52(x₁)) = 4 + (4)x_1
POL(a__U31(x₁)) = 9/4
POL(__(x₁, x₂)) = 4 + x_1 + (2)x_2
POL(A__ISNELIST(x₁)) = (4)x_1
POL(A__U41(x₁, x₂)) = 4 + (4)x_2
POL(U51(x₁, x₂)) = 7/2 + (7/4)x_1 + (4)x_2
POL(U21(x₁, x₂)) = 5/4 + (3/2)x_1 + (11/4)x_2
POL(a__isQid(x₁)) = 4
POL(tt) = 0
POL(a__U42(x₁)) = 3/2 + (1/2)x_1
POL(U52(x₁)) = 1 + (5/2)x_1
POL(a__U11(x₁)) = 4
POL(a__U21(x₁, x₂)) = 4 + (1/2)x_2
POL(a__isList(x₁)) = 0
POL(U11(x₁)) = 5/4 + (1/2)x_1
POL(nil) = 0
POL(a__U51(x₁, x₂)) = 4 + (2)x_1 + x_2
POL(a__U41(x₁, x₂)) = 4 + x_1 + (4)x_2
POL(a) = 1/4
POL(isList(x₁)) = 9/4 + (5/4)x_1
POL(e) = 0
POL(a__isNeList(x₁)) = (1/4)x_1
POL(o) = 4
POL(isQid(x₁)) = 4 + (9/4)x_1
POL(A__U21(x₁, x₂)) = 3 + (4)x_2
POL(U22(x₁)) = 4 + (5/4)x_1
POL(A__U51(x₁, x₂)) = 3 + (4)x_2
POL(U31(x₁)) = 3 + (7/2)x_1
POL(u) = 4
POL(U41(x₁, x₂)) = 4 + (7/4)x_1 + (3)x_2
POL(isNeList(x₁)) = 11/4 + (1/4)x_1
POL(U42(x₁)) = 5/2 + (7/2)x_1
POL(A__ISLIST(x₁)) = 3 + (4)x_1
POL(a__U22(x₁)) = 7/4

The value of delta used in the strict ordering is 3.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

          ↳ QDP

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

A__U51(tt, V2) → A__ISLIST(V2)
A__U21(tt, V2) → A__ISLIST(V2)

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.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

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

MARK(U31(X)) → MARK(X)
A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
A____(__(X, Y), Z) → MARK(Z)
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
MARK(U81(X)) → MARK(X)
MARK(U72(X)) → MARK(X)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U61(X)) → MARK(X)
MARK(U22(X)) → MARK(X)
MARK(__(X1, X2)) → MARK(X1)
MARK(U52(X)) → MARK(X)
A____(__(X, Y), Z) → MARK(X)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)
A____(nil, X) → MARK(X)
A____(X, nil) → MARK(X)
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
A____(__(X, Y), Z) → MARK(Y)
MARK(U42(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(__(X1, X2)) → MARK(X2)
MARK(U11(X)) → MARK(X)

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.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A____(nil, X) → MARK(X)
A____(X, nil) → MARK(X)

The remaining pairs can at least be oriented weakly.

MARK(U31(X)) → MARK(X)
A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
A____(__(X, Y), Z) → MARK(Z)
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
MARK(U81(X)) → MARK(X)
MARK(U72(X)) → MARK(X)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U61(X)) → MARK(X)
MARK(U22(X)) → MARK(X)
MARK(__(X1, X2)) → MARK(X1)
MARK(U52(X)) → MARK(X)
A____(__(X, Y), Z) → MARK(X)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
A____(__(X, Y), Z) → MARK(Y)
MARK(U42(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(__(X1, X2)) → MARK(X2)
MARK(U11(X)) → MARK(X)

Used ordering: Polynomial interpretation [25,35]:

POL(i) = 0
POL(a__U52(x₁)) = x_1
POL(A____(x₁, x₂)) = 4 + (1/2)x_1 + (1/4)x_2
POL(a__U31(x₁)) = (4)x_1
POL(__(x₁, x₂)) = (2)x_1 + x_2
POL(U81(x₁)) = (4)x_1
POL(a__isNePal(x₁)) = 0
POL(mark(x₁)) = x_1
POL(U51(x₁, x₂)) = (4)x_1
POL(U21(x₁, x₂)) = (4)x_1
POL(U71(x₁, x₂)) = (4)x_1
POL(a__U61(x₁)) = x_1
POL(a__isQid(x₁)) = 0
POL(tt) = 0
POL(a__U42(x₁)) = (5/2)x_1
POL(a__U11(x₁)) = (4)x_1
POL(U52(x₁)) = x_1
POL(a__U21(x₁, x₂)) = (4)x_1
POL(a__U72(x₁)) = (5/2)x_1
POL(U72(x₁)) = (5/2)x_1
POL(a__isList(x₁)) = 0
POL(U11(x₁)) = (4)x_1
POL(nil) = 4
POL(a__U51(x₁, x₂)) = (4)x_1
POL(a____(x₁, x₂)) = (2)x_1 + x_2
POL(a__U41(x₁, x₂)) = x_1
POL(a) = 1/4
POL(U61(x₁)) = x_1
POL(isList(x₁)) = 0
POL(a__isNeList(x₁)) = 0
POL(e) = 0
POL(isNePal(x₁)) = 0
POL(o) = 0
POL(isQid(x₁)) = 0
POL(a__U71(x₁, x₂)) = (4)x_1
POL(a__U81(x₁)) = (4)x_1
POL(isPal(x₁)) = 0
POL(MARK(x₁)) = 4 + (1/4)x_1
POL(U22(x₁)) = (5/4)x_1
POL(U31(x₁)) = (4)x_1
POL(U41(x₁, x₂)) = x_1
POL(u) = 1/2
POL(U42(x₁)) = (5/2)x_1
POL(isNeList(x₁)) = 0
POL(a__isPal(x₁)) = 0
POL(a__U22(x₁)) = (5/4)x_1

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

a__isNePal(X) → isNePal(X)
a__U52(X) → U52(X)
a__U51(X1, X2) → U51(X1, X2)
a__U71(X1, X2) → U71(X1, X2)
a__U61(X) → U61(X)
a__isPal(X) → isPal(X)
a__U72(X) → U72(X)
a__isQid(X) → isQid(X)
a__U81(X) → U81(X)
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))
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
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__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__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__isNePal(V) → a__U61(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isPal(V) → a__U81(a__isNePal(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isList(nil) → tt
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(V) → a__U31(a__isQid(V))
a__isQid(u) → tt
a__isQid(o) → tt
mark(U11(X)) → a__U11(mark(X))
a__isQid(a) → tt
a__isPal(nil) → tt
a__isQid(i) → tt
a__isQid(e) → tt
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(U81(X)) → a__U81(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X)) → a__U72(mark(X))
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
a____(X1, X2) → __(X1, X2)
mark(u) → u
mark(o) → o
mark(i) → i
mark(e) → e
mark(a) → a
mark(tt) → tt
mark(nil) → nil
a__isNeList(X) → isNeList(X)
a__U42(X) → U42(X)
a__U41(X1, X2) → U41(X1, X2)
a__U31(X) → U31(X)
a__isList(X) → isList(X)
a__U22(X) → U22(X)
a__U21(X1, X2) → U21(X1, X2)
a__U11(X) → U11(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

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

MARK(U31(X)) → MARK(X)
A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
A____(__(X, Y), Z) → MARK(Z)
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
MARK(U81(X)) → MARK(X)
MARK(U72(X)) → MARK(X)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U61(X)) → MARK(X)
MARK(U22(X)) → MARK(X)
MARK(__(X1, X2)) → MARK(X1)
MARK(U52(X)) → MARK(X)
A____(__(X, Y), Z) → MARK(X)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
MARK(U42(X)) → MARK(X)
A____(__(X, Y), Z) → MARK(Y)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U11(X)) → MARK(X)
MARK(__(X1, X2)) → MARK(X2)

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.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
A____(__(X, Y), Z) → MARK(Z)
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
MARK(__(X1, X2)) → MARK(X1)
A____(__(X, Y), Z) → MARK(X)
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
A____(__(X, Y), Z) → MARK(Y)
MARK(__(X1, X2)) → MARK(X2)

The remaining pairs can at least be oriented weakly.

MARK(U31(X)) → MARK(X)
MARK(U81(X)) → MARK(X)
MARK(U72(X)) → MARK(X)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U61(X)) → MARK(X)
MARK(U22(X)) → MARK(X)
MARK(U52(X)) → MARK(X)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U11(X)) → MARK(X)

Used ordering: Polynomial interpretation [25,35]:

POL(i) = 9/4
POL(a__U52(x₁)) = x_1
POL(A____(x₁, x₂)) = 1/2 + (4)x_1 + (2)x_2
POL(a__U31(x₁)) = x_1
POL(__(x₁, x₂)) = 1 + (2)x_1 + x_2
POL(U81(x₁)) = x_1
POL(a__isNePal(x₁)) = 0
POL(mark(x₁)) = x_1
POL(U51(x₁, x₂)) = (2)x_1
POL(U21(x₁, x₂)) = (4)x_1
POL(U71(x₁, x₂)) = (2)x_1
POL(a__U61(x₁)) = (4)x_1
POL(a__isQid(x₁)) = 0
POL(tt) = 0
POL(a__U42(x₁)) = (4)x_1
POL(a__U11(x₁)) = (5/2)x_1
POL(U52(x₁)) = x_1
POL(a__U21(x₁, x₂)) = (4)x_1
POL(a__U72(x₁)) = (5/2)x_1
POL(U72(x₁)) = (5/2)x_1
POL(a__isList(x₁)) = 0
POL(U11(x₁)) = (5/2)x_1
POL(nil) = 4
POL(a__U51(x₁, x₂)) = (2)x_1
POL(a____(x₁, x₂)) = 1 + (2)x_1 + x_2
POL(a__U41(x₁, x₂)) = x_1
POL(a) = 0
POL(U61(x₁)) = (4)x_1
POL(isList(x₁)) = 0
POL(a__isNeList(x₁)) = 0
POL(e) = 0
POL(isNePal(x₁)) = 0
POL(o) = 0
POL(isQid(x₁)) = 0
POL(a__U71(x₁, x₂)) = (2)x_1
POL(a__U81(x₁)) = x_1
POL(isPal(x₁)) = 0
POL(MARK(x₁)) = 1/4 + (2)x_1
POL(U22(x₁)) = x_1
POL(U31(x₁)) = x_1
POL(U41(x₁, x₂)) = x_1
POL(u) = 1/2
POL(U42(x₁)) = (4)x_1
POL(isNeList(x₁)) = 0
POL(a__isPal(x₁)) = 0
POL(a__U22(x₁)) = x_1

The value of delta used in the strict ordering is 7/4.
The following usable rules [17] were oriented:

a__isNePal(X) → isNePal(X)
a__U52(X) → U52(X)
a__U51(X1, X2) → U51(X1, X2)
a__U71(X1, X2) → U71(X1, X2)
a__U61(X) → U61(X)
a__isPal(X) → isPal(X)
a__U72(X) → U72(X)
a__isQid(X) → isQid(X)
a__U81(X) → U81(X)
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))
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
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__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__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__isNePal(V) → a__U61(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isPal(V) → a__U81(a__isNePal(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isList(nil) → tt
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(V) → a__U31(a__isQid(V))
a__isQid(u) → tt
a__isQid(o) → tt
mark(U11(X)) → a__U11(mark(X))
a__isQid(a) → tt
a__isPal(nil) → tt
a__isQid(i) → tt
a__isQid(e) → tt
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(X)
mark(isPal(X)) → a__isPal(X)
mark(U81(X)) → a__U81(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X)) → a__U72(mark(X))
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(mark(X))
a____(X1, X2) → __(X1, X2)
mark(u) → u
mark(o) → o
mark(i) → i
mark(e) → e
mark(a) → a
mark(tt) → tt
mark(nil) → nil
a__isNeList(X) → isNeList(X)
a__U42(X) → U42(X)
a__U41(X1, X2) → U41(X1, X2)
a__U31(X) → U31(X)
a__isList(X) → isList(X)
a__U22(X) → U22(X)
a__U21(X1, X2) → U21(X1, X2)
a__U11(X) → U11(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

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

MARK(U22(X)) → MARK(X)
MARK(U31(X)) → MARK(X)
MARK(U52(X)) → MARK(X)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
MARK(U81(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X)) → MARK(X)
MARK(U11(X)) → MARK(X)
MARK(U61(X)) → MARK(X)

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.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U52(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.

MARK(U22(X)) → MARK(X)
MARK(U31(X)) → MARK(X)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
MARK(U81(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X)) → MARK(X)
MARK(U11(X)) → MARK(X)
MARK(U61(X)) → MARK(X)

Used ordering: Polynomial interpretation [25,35]:

POL(U61(x₁)) = (11/4)x_1
POL(U81(x₁)) = (5/2)x_1
POL(U21(x₁, x₂)) = (15/4)x_1
POL(U51(x₁, x₂)) = (4)x_1
POL(U71(x₁, x₂)) = (4)x_1
POL(MARK(x₁)) = (4)x_1
POL(U22(x₁)) = (7/4)x_1
POL(U31(x₁)) = (5/4)x_1
POL(U41(x₁, x₂)) = (3)x_1
POL(U52(x₁)) = 4 + (3/2)x_1
POL(U42(x₁)) = (2)x_1
POL(U72(x₁)) = (3/2)x_1
POL(U11(x₁)) = (4)x_1

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

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

MARK(U31(X)) → MARK(X)
MARK(U22(X)) → MARK(X)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U81(X)) → MARK(X)
MARK(U42(X)) → MARK(X)
MARK(U72(X)) → MARK(X)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U11(X)) → MARK(X)
MARK(U61(X)) → MARK(X)

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.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U31(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U81(X)) → MARK(X)
MARK(U42(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U11(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.

MARK(U22(X)) → MARK(X)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U72(X)) → MARK(X)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U61(X)) → MARK(X)

Used ordering: Polynomial interpretation [25,35]:

POL(MARK(x₁)) = (1/2)x_1
POL(U22(x₁)) = (4)x_1
POL(U61(x₁)) = (9/4)x_1
POL(U31(x₁)) = 4 + (3/2)x_1
POL(U41(x₁, x₂)) = (3/2)x_1
POL(U81(x₁)) = 4 + x_1
POL(U42(x₁)) = 4 + x_1
POL(U72(x₁)) = x_1
POL(U11(x₁)) = 4 + (4)x_1
POL(U21(x₁, x₂)) = 4 + x_1
POL(U71(x₁, x₂)) = (3/2)x_1
POL(U51(x₁, x₂)) = 3/2 + (4)x_1

The value of delta used in the strict ordering is 3/4.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

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

MARK(U22(X)) → MARK(X)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X)) → MARK(X)
MARK(U61(X)) → MARK(X)

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.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(U22(X)) → MARK(X)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X)) → MARK(X)
MARK(U61(X)) → MARK(X)

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

POL(MARK(x₁)) = (1/2)x_1
POL(U22(x₁)) = 4 + (4)x_1
POL(U61(x₁)) = 7/2 + (15/4)x_1
POL(U41(x₁, x₂)) = 1/4 + (5/4)x_1
POL(U72(x₁)) = 5/4 + (4)x_1
POL(U71(x₁, x₂)) = 15/4 + (4)x_1

The value of delta used in the strict ordering is 1/8.
The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ PisEmptyProof

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