Termination w.r.t. Q proof of /tmp/tmpIH1mDn/PALINDROME_complete

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, V) → a__U12(a__isNeList(V))
a__U12(tt) → tt
a__U21(tt, V1, V2) → a__U22(a__isList(V1), V2)
a__U22(tt, V2) → a__U23(a__isList(V2))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isQid(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isList(V1), V2)
a__U42(tt, V2) → a__U43(a__isNeList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNeList(V1), V2)
a__U52(tt, V2) → a__U53(a__isList(V2))
a__U53(tt) → tt
a__U61(tt, V) → a__U62(a__isQid(V))
a__U62(tt) → tt
a__U71(tt, V) → a__U72(a__isNePal(V))
a__U72(tt) → tt
a__and(tt, X) → mark(X)
a__isList(V) → a__U11(a__isPalListKind(V), V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNeList(V) → a__U31(a__isPalListKind(V), V)
a__isNeList(__(V1, V2)) → a__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNeList(__(V1, V2)) → a__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNePal(V) → a__U61(a__isPalListKind(V), V)
a__isNePal(__(I, __(P, I))) → a__and(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))
a__isPal(V) → a__U71(a__isPalListKind(V), V)
a__isPal(nil) → tt
a__isPalListKind(a) → tt
a__isPalListKind(e) → tt
a__isPalListKind(i) → tt
a__isPalListKind(nil) → tt
a__isPalListKind(o) → tt
a__isPalListKind(u) → tt
a__isPalListKind(__(V1, V2)) → a__and(a__isPalListKind(V1), isPalListKind(V2))
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(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X)) → a__U72(mark(X))
mark(isNePal(X)) → a__isNePal(X)
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(X)
mark(isPal(X)) → a__isPal(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(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNeList(X) → isNeList(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2) → U22(X1, X2)
a__isList(X) → isList(X)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__isQid(X) → isQid(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isNePal(X) → isNePal(X)
a__and(X1, X2) → and(X1, X2)
a__isPalListKind(X) → isPalListKind(X)
a__isPal(X) → isPal(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, V) → a__U12(a__isNeList(V))
a__U12(tt) → tt
a__U21(tt, V1, V2) → a__U22(a__isList(V1), V2)
a__U22(tt, V2) → a__U23(a__isList(V2))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isQid(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isList(V1), V2)
a__U42(tt, V2) → a__U43(a__isNeList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNeList(V1), V2)
a__U52(tt, V2) → a__U53(a__isList(V2))
a__U53(tt) → tt
a__U61(tt, V) → a__U62(a__isQid(V))
a__U62(tt) → tt
a__U71(tt, V) → a__U72(a__isNePal(V))
a__U72(tt) → tt
a__and(tt, X) → mark(X)
a__isList(V) → a__U11(a__isPalListKind(V), V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNeList(V) → a__U31(a__isPalListKind(V), V)
a__isNeList(__(V1, V2)) → a__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNeList(__(V1, V2)) → a__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNePal(V) → a__U61(a__isPalListKind(V), V)
a__isNePal(__(I, __(P, I))) → a__and(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))
a__isPal(V) → a__U71(a__isPalListKind(V), V)
a__isPal(nil) → tt
a__isPalListKind(a) → tt
a__isPalListKind(e) → tt
a__isPalListKind(i) → tt
a__isPalListKind(nil) → tt
a__isPalListKind(o) → tt
a__isPalListKind(u) → tt
a__isPalListKind(__(V1, V2)) → a__and(a__isPalListKind(V1), isPalListKind(V2))
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(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X)) → a__U72(mark(X))
mark(isNePal(X)) → a__isNePal(X)
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(X)
mark(isPal(X)) → a__isPal(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(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNeList(X) → isNeList(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2) → U22(X1, X2)
a__isList(X) → isList(X)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__isQid(X) → isQid(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isNePal(X) → isNePal(X)
a__and(X1, X2) → and(X1, X2)
a__isPalListKind(X) → isPalListKind(X)
a__isPal(X) → isPal(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__ISPAL(V) → A__U71(a__isPalListKind(V), V)
MARK(U11(X1, X2)) → MARK(X1)
A__ISNEPAL(__(I, __(P, I))) → A__ISQID(I)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U42(tt, V2) → A__ISNELIST(V2)
A__ISNELIST(V) → A__ISPALLISTKIND(V)
A__AND(tt, X) → MARK(X)
A__ISLIST(V) → A__ISPALLISTKIND(V)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNELIST(V) → A__U31(a__isPalListKind(V), V)
A__ISNELIST(__(V1, V2)) → A__ISPALLISTKIND(V1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__U21(tt, V1, V2) → A__ISLIST(V1)
A____(__(X, Y), Z) → MARK(X)
A__U41(tt, V1, V2) → A__U42(a__isList(V1), V2)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
MARK(U43(X)) → A__U43(mark(X))
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
MARK(U41(X1, X2, X3)) → MARK(X1)
A__U61(tt, V) → A__ISQID(V)
MARK(U32(X)) → MARK(X)
MARK(isNeList(X)) → A__ISNELIST(X)
A__ISNEPAL(__(I, __(P, I))) → A__AND(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))
A__U22(tt, V2) → A__U23(a__isList(V2))
MARK(U62(X)) → A__U62(mark(X))
MARK(U21(X1, X2, X3)) → MARK(X1)
A__U11(tt, V) → A__U12(a__isNeList(V))
A__U71(tt, V) → A__ISNEPAL(V)
A____(X, nil) → MARK(X)
A____(nil, X) → MARK(X)
A____(__(X, Y), Z) → MARK(Y)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U62(X)) → MARK(X)
A__U51(tt, V1, V2) → A__ISNELIST(V1)
A__ISLIST(__(V1, V2)) → A__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
A__ISNELIST(__(V1, V2)) → A__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
MARK(U12(X)) → A__U12(mark(X))
A____(__(X, Y), Z) → MARK(Z)
A__ISPALLISTKIND(__(V1, V2)) → A__ISPALLISTKIND(V1)
A__U31(tt, V) → A__U32(a__isQid(V))
MARK(U31(X1, X2)) → MARK(X1)
A__U51(tt, V1, V2) → A__U52(a__isNeList(V1), V2)
MARK(isList(X)) → A__ISLIST(X)
MARK(U23(X)) → A__U23(mark(X))
MARK(__(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → A__U71(mark(X1), X2)
A__ISNELIST(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__ISPALLISTKIND(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__U21(tt, V1, V2) → A__U22(a__isList(V1), V2)
A__U52(tt, V2) → A__ISLIST(V2)
MARK(U61(X1, X2)) → A__U61(mark(X1), X2)
MARK(U22(X1, X2)) → A__U22(mark(X1), X2)
A__U52(tt, V2) → A__U53(a__isList(V2))
A__U42(tt, V2) → A__U43(a__isNeList(V2))
A__U41(tt, V1, V2) → A__ISLIST(V1)
MARK(U72(X)) → A__U72(mark(X))
A__ISNEPAL(__(I, __(P, I))) → A__AND(a__isQid(I), isPalListKind(I))
A__ISPAL(V) → A__ISPALLISTKIND(V)
MARK(isNePal(X)) → A__ISNEPAL(X)
A__ISNELIST(__(V1, V2)) → A__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
A__U11(tt, V) → A__ISNELIST(V)
A__ISNEPAL(V) → A__U61(a__isPalListKind(V), V)
A__U71(tt, V) → A__U72(a__isNePal(V))
A__ISNEPAL(V) → A__ISPALLISTKIND(V)
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
MARK(U21(X1, X2, X3)) → A__U21(mark(X1), X2, X3)
A__U22(tt, V2) → A__ISLIST(V2)
MARK(isPal(X)) → A__ISPAL(X)
MARK(U72(X)) → MARK(X)
MARK(U71(X1, X2)) → MARK(X1)
A__U31(tt, V) → A__ISQID(V)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U32(X)) → A__U32(mark(X))
MARK(U53(X)) → A__U53(mark(X))
MARK(U43(X)) → MARK(X)
MARK(isQid(X)) → A__ISQID(X)
A__U61(tt, V) → A__U62(a__isQid(V))
MARK(U23(X)) → MARK(X)
A__ISLIST(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__ISLIST(__(V1, V2)) → A__ISPALLISTKIND(V1)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(isPalListKind(X)) → A__ISPALLISTKIND(X)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__ISLIST(V) → A__U11(a__isPalListKind(V), V)
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, V) → a__U12(a__isNeList(V))
a__U12(tt) → tt
a__U21(tt, V1, V2) → a__U22(a__isList(V1), V2)
a__U22(tt, V2) → a__U23(a__isList(V2))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isQid(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isList(V1), V2)
a__U42(tt, V2) → a__U43(a__isNeList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNeList(V1), V2)
a__U52(tt, V2) → a__U53(a__isList(V2))
a__U53(tt) → tt
a__U61(tt, V) → a__U62(a__isQid(V))
a__U62(tt) → tt
a__U71(tt, V) → a__U72(a__isNePal(V))
a__U72(tt) → tt
a__and(tt, X) → mark(X)
a__isList(V) → a__U11(a__isPalListKind(V), V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNeList(V) → a__U31(a__isPalListKind(V), V)
a__isNeList(__(V1, V2)) → a__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNeList(__(V1, V2)) → a__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNePal(V) → a__U61(a__isPalListKind(V), V)
a__isNePal(__(I, __(P, I))) → a__and(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))
a__isPal(V) → a__U71(a__isPalListKind(V), V)
a__isPal(nil) → tt
a__isPalListKind(a) → tt
a__isPalListKind(e) → tt
a__isPalListKind(i) → tt
a__isPalListKind(nil) → tt
a__isPalListKind(o) → tt
a__isPalListKind(u) → tt
a__isPalListKind(__(V1, V2)) → a__and(a__isPalListKind(V1), isPalListKind(V2))
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(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X)) → a__U72(mark(X))
mark(isNePal(X)) → a__isNePal(X)
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(X)
mark(isPal(X)) → a__isPal(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(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNeList(X) → isNeList(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2) → U22(X1, X2)
a__isList(X) → isList(X)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__isQid(X) → isQid(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isNePal(X) → isNePal(X)
a__and(X1, X2) → and(X1, X2)
a__isPalListKind(X) → isPalListKind(X)
a__isPal(X) → isPal(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__ISPAL(V) → A__U71(a__isPalListKind(V), V)
MARK(U11(X1, X2)) → MARK(X1)
A__ISNEPAL(__(I, __(P, I))) → A__ISQID(I)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U42(tt, V2) → A__ISNELIST(V2)
A__ISNELIST(V) → A__ISPALLISTKIND(V)
A__AND(tt, X) → MARK(X)
A__ISLIST(V) → A__ISPALLISTKIND(V)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNELIST(V) → A__U31(a__isPalListKind(V), V)
A__ISNELIST(__(V1, V2)) → A__ISPALLISTKIND(V1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__U21(tt, V1, V2) → A__ISLIST(V1)
A____(__(X, Y), Z) → MARK(X)
A__U41(tt, V1, V2) → A__U42(a__isList(V1), V2)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
MARK(U43(X)) → A__U43(mark(X))
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
MARK(U41(X1, X2, X3)) → MARK(X1)
A__U61(tt, V) → A__ISQID(V)
MARK(U32(X)) → MARK(X)
MARK(isNeList(X)) → A__ISNELIST(X)
A__ISNEPAL(__(I, __(P, I))) → A__AND(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))
A__U22(tt, V2) → A__U23(a__isList(V2))
MARK(U62(X)) → A__U62(mark(X))
MARK(U21(X1, X2, X3)) → MARK(X1)
A__U11(tt, V) → A__U12(a__isNeList(V))
A__U71(tt, V) → A__ISNEPAL(V)
A____(X, nil) → MARK(X)
A____(nil, X) → MARK(X)
A____(__(X, Y), Z) → MARK(Y)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U62(X)) → MARK(X)
A__U51(tt, V1, V2) → A__ISNELIST(V1)
A__ISLIST(__(V1, V2)) → A__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
A__ISNELIST(__(V1, V2)) → A__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
MARK(U12(X)) → A__U12(mark(X))
A____(__(X, Y), Z) → MARK(Z)
A__ISPALLISTKIND(__(V1, V2)) → A__ISPALLISTKIND(V1)
A__U31(tt, V) → A__U32(a__isQid(V))
MARK(U31(X1, X2)) → MARK(X1)
A__U51(tt, V1, V2) → A__U52(a__isNeList(V1), V2)
MARK(isList(X)) → A__ISLIST(X)
MARK(U23(X)) → A__U23(mark(X))
MARK(__(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → A__U71(mark(X1), X2)
A__ISNELIST(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__ISPALLISTKIND(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__U21(tt, V1, V2) → A__U22(a__isList(V1), V2)
A__U52(tt, V2) → A__ISLIST(V2)
MARK(U61(X1, X2)) → A__U61(mark(X1), X2)
MARK(U22(X1, X2)) → A__U22(mark(X1), X2)
A__U52(tt, V2) → A__U53(a__isList(V2))
A__U42(tt, V2) → A__U43(a__isNeList(V2))
A__U41(tt, V1, V2) → A__ISLIST(V1)
MARK(U72(X)) → A__U72(mark(X))
A__ISNEPAL(__(I, __(P, I))) → A__AND(a__isQid(I), isPalListKind(I))
A__ISPAL(V) → A__ISPALLISTKIND(V)
MARK(isNePal(X)) → A__ISNEPAL(X)
A__ISNELIST(__(V1, V2)) → A__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
A__U11(tt, V) → A__ISNELIST(V)
A__ISNEPAL(V) → A__U61(a__isPalListKind(V), V)
A__U71(tt, V) → A__U72(a__isNePal(V))
A__ISNEPAL(V) → A__ISPALLISTKIND(V)
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
MARK(U21(X1, X2, X3)) → A__U21(mark(X1), X2, X3)
A__U22(tt, V2) → A__ISLIST(V2)
MARK(isPal(X)) → A__ISPAL(X)
MARK(U72(X)) → MARK(X)
MARK(U71(X1, X2)) → MARK(X1)
A__U31(tt, V) → A__ISQID(V)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U32(X)) → A__U32(mark(X))
MARK(U53(X)) → A__U53(mark(X))
MARK(U43(X)) → MARK(X)
MARK(isQid(X)) → A__ISQID(X)
A__U61(tt, V) → A__U62(a__isQid(V))
MARK(U23(X)) → MARK(X)
A__ISLIST(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__ISLIST(__(V1, V2)) → A__ISPALLISTKIND(V1)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(isPalListKind(X)) → A__ISPALLISTKIND(X)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__ISLIST(V) → A__U11(a__isPalListKind(V), V)
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, V) → a__U12(a__isNeList(V))
a__U12(tt) → tt
a__U21(tt, V1, V2) → a__U22(a__isList(V1), V2)
a__U22(tt, V2) → a__U23(a__isList(V2))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isQid(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isList(V1), V2)
a__U42(tt, V2) → a__U43(a__isNeList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNeList(V1), V2)
a__U52(tt, V2) → a__U53(a__isList(V2))
a__U53(tt) → tt
a__U61(tt, V) → a__U62(a__isQid(V))
a__U62(tt) → tt
a__U71(tt, V) → a__U72(a__isNePal(V))
a__U72(tt) → tt
a__and(tt, X) → mark(X)
a__isList(V) → a__U11(a__isPalListKind(V), V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNeList(V) → a__U31(a__isPalListKind(V), V)
a__isNeList(__(V1, V2)) → a__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNeList(__(V1, V2)) → a__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNePal(V) → a__U61(a__isPalListKind(V), V)
a__isNePal(__(I, __(P, I))) → a__and(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))
a__isPal(V) → a__U71(a__isPalListKind(V), V)
a__isPal(nil) → tt
a__isPalListKind(a) → tt
a__isPalListKind(e) → tt
a__isPalListKind(i) → tt
a__isPalListKind(nil) → tt
a__isPalListKind(o) → tt
a__isPalListKind(u) → tt
a__isPalListKind(__(V1, V2)) → a__and(a__isPalListKind(V1), isPalListKind(V2))
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(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X)) → a__U72(mark(X))
mark(isNePal(X)) → a__isNePal(X)
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(X)
mark(isPal(X)) → a__isPal(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(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNeList(X) → isNeList(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2) → U22(X1, X2)
a__isList(X) → isList(X)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__isQid(X) → isQid(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isNePal(X) → isNePal(X)
a__and(X1, X2) → and(X1, X2)
a__isPalListKind(X) → isPalListKind(X)
a__isPal(X) → isPal(X)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

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

A__ISPAL(V) → A__U71(a__isPalListKind(V), V)
MARK(U11(X1, X2)) → MARK(X1)
A__U42(tt, V2) → A__ISNELIST(V2)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__ISNELIST(V) → A__ISPALLISTKIND(V)
A__AND(tt, X) → MARK(X)
A__ISLIST(V) → A__ISPALLISTKIND(V)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNELIST(__(V1, V2)) → A__ISPALLISTKIND(V1)
A__U21(tt, V1, V2) → A__ISLIST(V1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A____(__(X, Y), Z) → MARK(X)
A__U41(tt, V1, V2) → A__U42(a__isList(V1), V2)
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(isNeList(X)) → A__ISNELIST(X)
MARK(U32(X)) → MARK(X)
A__ISNEPAL(__(I, __(P, I))) → A__AND(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))
A__U71(tt, V) → A__ISNEPAL(V)
MARK(U21(X1, X2, X3)) → MARK(X1)
A____(nil, X) → MARK(X)
A____(X, nil) → MARK(X)
A____(__(X, Y), Z) → MARK(Y)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U62(X)) → MARK(X)
A__U51(tt, V1, V2) → A__ISNELIST(V1)
A__ISNELIST(__(V1, V2)) → A__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
A__ISLIST(__(V1, V2)) → A__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
A____(__(X, Y), Z) → MARK(Z)
A__ISPALLISTKIND(__(V1, V2)) → A__ISPALLISTKIND(V1)
MARK(U31(X1, X2)) → MARK(X1)
A__U51(tt, V1, V2) → A__U52(a__isNeList(V1), V2)
MARK(isList(X)) → A__ISLIST(X)
MARK(__(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → A__U71(mark(X1), X2)
A__ISNELIST(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__ISPALLISTKIND(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__U21(tt, V1, V2) → A__U22(a__isList(V1), V2)
A__U52(tt, V2) → A__ISLIST(V2)
A__U41(tt, V1, V2) → A__ISLIST(V1)
MARK(U22(X1, X2)) → A__U22(mark(X1), X2)
A__ISNEPAL(__(I, __(P, I))) → A__AND(a__isQid(I), isPalListKind(I))
A__ISPAL(V) → A__ISPALLISTKIND(V)
MARK(isNePal(X)) → A__ISNEPAL(X)
A__ISNELIST(__(V1, V2)) → A__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
A__U11(tt, V) → A__ISNELIST(V)
A__ISNEPAL(V) → A__ISPALLISTKIND(V)
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
A__U22(tt, V2) → A__ISLIST(V2)
MARK(U21(X1, X2, X3)) → A__U21(mark(X1), X2, X3)
MARK(isPal(X)) → A__ISPAL(X)
MARK(U72(X)) → MARK(X)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
A__ISLIST(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__ISLIST(__(V1, V2)) → A__ISPALLISTKIND(V1)
MARK(U23(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(isPalListKind(X)) → A__ISPALLISTKIND(X)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
A__ISLIST(V) → A__U11(a__isPalListKind(V), V)
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, V) → a__U12(a__isNeList(V))
a__U12(tt) → tt
a__U21(tt, V1, V2) → a__U22(a__isList(V1), V2)
a__U22(tt, V2) → a__U23(a__isList(V2))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isQid(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isList(V1), V2)
a__U42(tt, V2) → a__U43(a__isNeList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNeList(V1), V2)
a__U52(tt, V2) → a__U53(a__isList(V2))
a__U53(tt) → tt
a__U61(tt, V) → a__U62(a__isQid(V))
a__U62(tt) → tt
a__U71(tt, V) → a__U72(a__isNePal(V))
a__U72(tt) → tt
a__and(tt, X) → mark(X)
a__isList(V) → a__U11(a__isPalListKind(V), V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNeList(V) → a__U31(a__isPalListKind(V), V)
a__isNeList(__(V1, V2)) → a__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNeList(__(V1, V2)) → a__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNePal(V) → a__U61(a__isPalListKind(V), V)
a__isNePal(__(I, __(P, I))) → a__and(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))
a__isPal(V) → a__U71(a__isPalListKind(V), V)
a__isPal(nil) → tt
a__isPalListKind(a) → tt
a__isPalListKind(e) → tt
a__isPalListKind(i) → tt
a__isPalListKind(nil) → tt
a__isPalListKind(o) → tt
a__isPalListKind(u) → tt
a__isPalListKind(__(V1, V2)) → a__and(a__isPalListKind(V1), isPalListKind(V2))
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(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X)) → a__U72(mark(X))
mark(isNePal(X)) → a__isNePal(X)
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(X)
mark(isPal(X)) → a__isPal(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(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNeList(X) → isNeList(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2) → U22(X1, X2)
a__isList(X) → isList(X)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__isQid(X) → isQid(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isNePal(X) → isNePal(X)
a__and(X1, X2) → and(X1, X2)
a__isPalListKind(X) → isPalListKind(X)
a__isPal(X) → isPal(X)

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

The following pairs can be oriented strictly and are deleted.

MARK(U62(X)) → MARK(X)
A__ISNEPAL(__(I, __(P, I))) → A__AND(a__isQid(I), isPalListKind(I))
A__ISPAL(V) → A__ISPALLISTKIND(V)
A__ISNEPAL(V) → A__ISPALLISTKIND(V)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U61(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.

A__ISPAL(V) → A__U71(a__isPalListKind(V), V)
MARK(U11(X1, X2)) → MARK(X1)
A__U42(tt, V2) → A__ISNELIST(V2)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__ISNELIST(V) → A__ISPALLISTKIND(V)
A__AND(tt, X) → MARK(X)
A__ISLIST(V) → A__ISPALLISTKIND(V)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNELIST(__(V1, V2)) → A__ISPALLISTKIND(V1)
A__U21(tt, V1, V2) → A__ISLIST(V1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A____(__(X, Y), Z) → MARK(X)
A__U41(tt, V1, V2) → A__U42(a__isList(V1), V2)
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(isNeList(X)) → A__ISNELIST(X)
MARK(U32(X)) → MARK(X)
A__ISNEPAL(__(I, __(P, I))) → A__AND(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))
A__U71(tt, V) → A__ISNEPAL(V)
MARK(U21(X1, X2, X3)) → MARK(X1)
A____(nil, X) → MARK(X)
A____(X, nil) → MARK(X)
A____(__(X, Y), Z) → MARK(Y)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
A__U51(tt, V1, V2) → A__ISNELIST(V1)
A__ISNELIST(__(V1, V2)) → A__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
A__ISLIST(__(V1, V2)) → A__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
A____(__(X, Y), Z) → MARK(Z)
A__ISPALLISTKIND(__(V1, V2)) → A__ISPALLISTKIND(V1)
MARK(U31(X1, X2)) → MARK(X1)
A__U51(tt, V1, V2) → A__U52(a__isNeList(V1), V2)
MARK(isList(X)) → A__ISLIST(X)
MARK(__(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → A__U71(mark(X1), X2)
A__ISNELIST(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__ISPALLISTKIND(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__U21(tt, V1, V2) → A__U22(a__isList(V1), V2)
A__U52(tt, V2) → A__ISLIST(V2)
A__U41(tt, V1, V2) → A__ISLIST(V1)
MARK(U22(X1, X2)) → A__U22(mark(X1), X2)
MARK(isNePal(X)) → A__ISNEPAL(X)
A__ISNELIST(__(V1, V2)) → A__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
A__U11(tt, V) → A__ISNELIST(V)
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
A__U22(tt, V2) → A__ISLIST(V2)
MARK(U21(X1, X2, X3)) → A__U21(mark(X1), X2, X3)
MARK(isPal(X)) → A__ISPAL(X)
MARK(U72(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
A__ISLIST(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__ISLIST(__(V1, V2)) → A__ISPALLISTKIND(V1)
MARK(U23(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(isPalListKind(X)) → A__ISPALLISTKIND(X)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
A__ISLIST(V) → A__U11(a__isPalListKind(V), V)
MARK(__(X1, X2)) → MARK(X2)

Used ordering: Polynomial interpretation [25]:

POL(A__AND(x₁, x₂)) = x₂
POL(A__ISLIST(x₁)) = 0
POL(A__ISNELIST(x₁)) = 0
POL(A__ISNEPAL(x₁)) = 1
POL(A__ISPAL(x₁)) = 1
POL(A__ISPALLISTKIND(x₁)) = 0
POL(A__U11(x₁, x₂)) = 0
POL(A__U21(x₁, x₂, x₃)) = 0
POL(A__U22(x₁, x₂)) = 0
POL(A__U41(x₁, x₂, x₃)) = 0
POL(A__U42(x₁, x₂)) = 0
POL(A__U51(x₁, x₂, x₃)) = 0
POL(A__U52(x₁, x₂)) = 0
POL(A__U71(x₁, x₂)) = 1
POL(A____(x₁, x₂)) = x₁ + x₂
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁
POL(U12(x₁)) = x₁
POL(U21(x₁, x₂, x₃)) = x₁
POL(U22(x₁, x₂)) = x₁
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = x₁
POL(U42(x₁, x₂)) = x₁
POL(U43(x₁)) = x₁
POL(U51(x₁, x₂, x₃)) = x₁
POL(U52(x₁, x₂)) = x₁
POL(U53(x₁)) = x₁
POL(U61(x₁, x₂)) = 1 + x₁
POL(U62(x₁)) = 1 + x₁
POL(U71(x₁, x₂)) = 1 + x₁
POL(U72(x₁)) = x₁
POL(__(x₁, x₂)) = x₁ + x₂
POL(a) = 0
POL(a__U11(x₁, x₂)) = x₁
POL(a__U12(x₁)) = x₁
POL(a__U21(x₁, x₂, x₃)) = x₁
POL(a__U22(x₁, x₂)) = x₁
POL(a__U23(x₁)) = x₁
POL(a__U31(x₁, x₂)) = x₁
POL(a__U32(x₁)) = x₁
POL(a__U41(x₁, x₂, x₃)) = x₁
POL(a__U42(x₁, x₂)) = x₁
POL(a__U43(x₁)) = x₁
POL(a__U51(x₁, x₂, x₃)) = x₁
POL(a__U52(x₁, x₂)) = x₁
POL(a__U53(x₁)) = x₁
POL(a__U61(x₁, x₂)) = 1 + x₁
POL(a__U62(x₁)) = 1 + x₁
POL(a__U71(x₁, x₂)) = 1 + x₁
POL(a__U72(x₁)) = x₁
POL(a____(x₁, x₂)) = x₁ + x₂
POL(a__and(x₁, x₂)) = x₁ + x₂
POL(a__isList(x₁)) = 0
POL(a__isNeList(x₁)) = 0
POL(a__isNePal(x₁)) = 1
POL(a__isPal(x₁)) = 1
POL(a__isPalListKind(x₁)) = 0
POL(a__isQid(x₁)) = 0
POL(and(x₁, x₂)) = x₁ + x₂
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 0
POL(isNeList(x₁)) = 0
POL(isNePal(x₁)) = 1
POL(isPal(x₁)) = 1
POL(isPalListKind(x₁)) = 0
POL(isQid(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(isNeList(X)) → a__isNeList(X)
mark(U12(X)) → a__U12(mark(X))
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(isList(X)) → a__isList(X)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(isQid(X)) → a__isQid(X)
mark(U32(X)) → a__U32(mark(X))
mark(U53(X)) → a__U53(mark(X))
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U43(X)) → a__U43(mark(X))
a__isPal(nil) → tt
a__isPalListKind(a) → tt
a__isPal(V) → a__U71(a__isPalListKind(V), V)
a__isPalListKind(nil) → tt
a__isPalListKind(o) → tt
a__isPalListKind(e) → tt
a__isPalListKind(i) → tt
a__isQid(a) → tt
a__isQid(e) → tt
a__isPalListKind(u) → tt
a__isQid(u) → tt
a__isQid(i) → tt
a__isQid(o) → tt
a__U12(X) → U12(X)
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__isNeList(X) → isNeList(X)
a__isList(X) → isList(X)
a__U22(X1, X2) → U22(X1, X2)
a__U31(X1, X2) → U31(X1, X2)
a__U23(X) → U23(X)
a__isQid(X) → isQid(X)
a__U32(X) → U32(X)
a__U42(X1, X2) → U42(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U43(X) → U43(X)
a__U53(X) → U53(X)
a__U52(X1, X2) → U52(X1, X2)
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(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(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__isPalListKind(X) → isPalListKind(X)
a__isPal(X) → isPal(X)
a__isNePal(X) → isNePal(X)
a__and(X1, X2) → and(X1, X2)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U42(tt, V2) → a__U43(a__isNeList(V2))
a__U41(tt, V1, V2) → a__U42(a__isList(V1), V2)
a__U32(tt) → tt
a__U31(tt, V) → a__U32(a__isQid(V))
a__U23(tt) → tt
a__U22(tt, V2) → a__U23(a__isList(V2))
a__U21(tt, V1, V2) → a__U22(a__isList(V1), V2)
a__U12(tt) → tt
a__U11(tt, V) → a__U12(a__isNeList(V))
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
a__isNePal(__(I, __(P, I))) → a__and(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(X)
a____(nil, X) → mark(X)
a____(X, nil) → mark(X)
mark(isNePal(X)) → a__isNePal(X)
a__and(tt, X) → mark(X)
a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a__isPalListKind(__(V1, V2)) → a__and(a__isPalListKind(V1), isPalListKind(V2))
a__isNeList(__(V1, V2)) → a__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNePal(V) → a__U61(a__isPalListKind(V), V)
a__isNeList(V) → a__U31(a__isPalListKind(V), V)
a__isNeList(__(V1, V2)) → a__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isList(V) → a__U11(a__isPalListKind(V), V)
a__U71(tt, V) → a__U72(a__isNePal(V))
a__U72(tt) → tt
a__U61(tt, V) → a__U62(a__isQid(V))
a__U62(tt) → tt
a__U52(tt, V2) → a__U53(a__isList(V2))
a__U53(tt) → tt
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNeList(V1), V2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

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

A__ISPAL(V) → A__U71(a__isPalListKind(V), V)
MARK(U11(X1, X2)) → MARK(X1)
A__U42(tt, V2) → A__ISNELIST(V2)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__ISNELIST(V) → A__ISPALLISTKIND(V)
A__AND(tt, X) → MARK(X)
A__ISLIST(V) → A__ISPALLISTKIND(V)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNELIST(__(V1, V2)) → A__ISPALLISTKIND(V1)
A__U21(tt, V1, V2) → A__ISLIST(V1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A____(__(X, Y), Z) → MARK(X)
A__U41(tt, V1, V2) → A__U42(a__isList(V1), V2)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(isNeList(X)) → A__ISNELIST(X)
A__ISNEPAL(__(I, __(P, I))) → A__AND(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))
MARK(U21(X1, X2, X3)) → MARK(X1)
A__U71(tt, V) → A__ISNEPAL(V)
A____(nil, X) → MARK(X)
A____(X, nil) → MARK(X)
A____(__(X, Y), Z) → MARK(Y)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
A__U51(tt, V1, V2) → A__ISNELIST(V1)
A__ISNELIST(__(V1, V2)) → A__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
A__ISLIST(__(V1, V2)) → A__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
A____(__(X, Y), Z) → MARK(Z)
A__ISPALLISTKIND(__(V1, V2)) → A__ISPALLISTKIND(V1)
MARK(U31(X1, X2)) → MARK(X1)
A__U51(tt, V1, V2) → A__U52(a__isNeList(V1), V2)
MARK(isList(X)) → A__ISLIST(X)
MARK(__(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → A__U71(mark(X1), X2)
A__ISNELIST(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__ISPALLISTKIND(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__U21(tt, V1, V2) → A__U22(a__isList(V1), V2)
A__U52(tt, V2) → A__ISLIST(V2)
A__U41(tt, V1, V2) → A__ISLIST(V1)
MARK(U22(X1, X2)) → A__U22(mark(X1), X2)
A__ISNELIST(__(V1, V2)) → A__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
MARK(isNePal(X)) → A__ISNEPAL(X)
A__U11(tt, V) → A__ISNELIST(V)
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
MARK(U21(X1, X2, X3)) → A__U21(mark(X1), X2, X3)
A__U22(tt, V2) → A__ISLIST(V2)
MARK(U72(X)) → MARK(X)
MARK(isPal(X)) → A__ISPAL(X)
MARK(U43(X)) → MARK(X)
MARK(U23(X)) → MARK(X)
A__ISLIST(__(V1, V2)) → A__ISPALLISTKIND(V1)
A__ISLIST(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(isPalListKind(X)) → A__ISPALLISTKIND(X)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
A__ISLIST(V) → A__U11(a__isPalListKind(V), V)
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, V) → a__U12(a__isNeList(V))
a__U12(tt) → tt
a__U21(tt, V1, V2) → a__U22(a__isList(V1), V2)
a__U22(tt, V2) → a__U23(a__isList(V2))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isQid(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isList(V1), V2)
a__U42(tt, V2) → a__U43(a__isNeList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNeList(V1), V2)
a__U52(tt, V2) → a__U53(a__isList(V2))
a__U53(tt) → tt
a__U61(tt, V) → a__U62(a__isQid(V))
a__U62(tt) → tt
a__U71(tt, V) → a__U72(a__isNePal(V))
a__U72(tt) → tt
a__and(tt, X) → mark(X)
a__isList(V) → a__U11(a__isPalListKind(V), V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNeList(V) → a__U31(a__isPalListKind(V), V)
a__isNeList(__(V1, V2)) → a__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNeList(__(V1, V2)) → a__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNePal(V) → a__U61(a__isPalListKind(V), V)
a__isNePal(__(I, __(P, I))) → a__and(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))
a__isPal(V) → a__U71(a__isPalListKind(V), V)
a__isPal(nil) → tt
a__isPalListKind(a) → tt
a__isPalListKind(e) → tt
a__isPalListKind(i) → tt
a__isPalListKind(nil) → tt
a__isPalListKind(o) → tt
a__isPalListKind(u) → tt
a__isPalListKind(__(V1, V2)) → a__and(a__isPalListKind(V1), isPalListKind(V2))
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(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X)) → a__U72(mark(X))
mark(isNePal(X)) → a__isNePal(X)
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(X)
mark(isPal(X)) → a__isPal(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(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNeList(X) → isNeList(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2) → U22(X1, X2)
a__isList(X) → isList(X)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__isQid(X) → isQid(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isNePal(X) → isNePal(X)
a__and(X1, X2) → and(X1, X2)
a__isPalListKind(X) → isPalListKind(X)
a__isPal(X) → isPal(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__U71(a__isPalListKind(V), V)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__ISNELIST(__(V1, V2)) → A__ISPALLISTKIND(V1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A____(__(X, Y), Z) → MARK(X)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
MARK(U53(X)) → MARK(X)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U41(X1, X2, X3)) → MARK(X1)
A__ISNEPAL(__(I, __(P, I))) → A__AND(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))
MARK(U21(X1, X2, X3)) → MARK(X1)
A____(nil, X) → MARK(X)
A____(X, nil) → MARK(X)
A____(__(X, Y), Z) → MARK(Y)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → MARK(X1)
A__ISNELIST(__(V1, V2)) → A__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
A__ISLIST(__(V1, V2)) → A__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
A____(__(X, Y), Z) → MARK(Z)
MARK(__(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → A__U71(mark(X1), X2)
A__ISNELIST(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
MARK(U22(X1, X2)) → A__U22(mark(X1), X2)
A__ISNELIST(__(V1, V2)) → A__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
MARK(isNePal(X)) → A__ISNEPAL(X)
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
MARK(U21(X1, X2, X3)) → A__U21(mark(X1), X2, X3)
MARK(U43(X)) → MARK(X)
MARK(U23(X)) → MARK(X)
A__ISLIST(__(V1, V2)) → A__ISPALLISTKIND(V1)
A__ISLIST(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(__(X1, X2)) → MARK(X2)

The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → MARK(X1)
A__U42(tt, V2) → A__ISNELIST(V2)
A__ISNELIST(V) → A__ISPALLISTKIND(V)
A__AND(tt, X) → MARK(X)
A__ISLIST(V) → A__ISPALLISTKIND(V)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__U21(tt, V1, V2) → A__ISLIST(V1)
A__U41(tt, V1, V2) → A__U42(a__isList(V1), V2)
A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
MARK(U32(X)) → MARK(X)
MARK(isNeList(X)) → A__ISNELIST(X)
A__U71(tt, V) → A__ISNEPAL(V)
MARK(U12(X)) → MARK(X)
A__U51(tt, V1, V2) → A__ISNELIST(V1)
A__ISPALLISTKIND(__(V1, V2)) → A__ISPALLISTKIND(V1)
MARK(U31(X1, X2)) → MARK(X1)
A__U51(tt, V1, V2) → A__U52(a__isNeList(V1), V2)
MARK(isList(X)) → A__ISLIST(X)
MARK(and(X1, X2)) → MARK(X1)
A__ISPALLISTKIND(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__U21(tt, V1, V2) → A__U22(a__isList(V1), V2)
A__U52(tt, V2) → A__ISLIST(V2)
A__U41(tt, V1, V2) → A__ISLIST(V1)
A__U11(tt, V) → A__ISNELIST(V)
A__U22(tt, V2) → A__ISLIST(V2)
MARK(U72(X)) → MARK(X)
MARK(isPal(X)) → A__ISPAL(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(isPalListKind(X)) → A__ISPALLISTKIND(X)
A__ISLIST(V) → A__U11(a__isPalListKind(V), V)

Used ordering: Polynomial interpretation [25]:

POL(A__AND(x₁, x₂)) = 1 + x₂
POL(A__ISLIST(x₁)) = 1 + x₁
POL(A__ISNELIST(x₁)) = 1 + x₁
POL(A__ISNEPAL(x₁)) = x₁
POL(A__ISPAL(x₁)) = 1 + x₁
POL(A__ISPALLISTKIND(x₁)) = 1
POL(A__U11(x₁, x₂)) = 1 + x₂
POL(A__U21(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(A__U22(x₁, x₂)) = 1 + x₂
POL(A__U41(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(A__U42(x₁, x₂)) = 1 + x₂
POL(A__U51(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(A__U52(x₁, x₂)) = 1 + x₂
POL(A__U71(x₁, x₂)) = x₂
POL(A____(x₁, x₂)) = 1 + x₁ + x₂
POL(MARK(x₁)) = 1 + x₁
POL(U11(x₁, x₂)) = x₁ + x₂
POL(U12(x₁)) = x₁
POL(U21(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U22(x₁, x₂)) = 1 + x₁ + x₂
POL(U23(x₁)) = 1 + x₁
POL(U31(x₁, x₂)) = x₁ + x₂
POL(U32(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U42(x₁, x₂)) = 1 + x₁ + x₂
POL(U43(x₁)) = 1 + x₁
POL(U51(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U52(x₁, x₂)) = 1 + x₁ + x₂
POL(U53(x₁)) = 1 + x₁
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = x₁ + x₂
POL(U72(x₁)) = x₁
POL(__(x₁, x₂)) = 1 + x₁ + x₂
POL(a) = 0
POL(a__U11(x₁, x₂)) = x₁ + x₂
POL(a__U12(x₁)) = x₁
POL(a__U21(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(a__U22(x₁, x₂)) = 1 + x₁ + x₂
POL(a__U23(x₁)) = 1 + x₁
POL(a__U31(x₁, x₂)) = x₁ + x₂
POL(a__U32(x₁)) = x₁
POL(a__U41(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(a__U42(x₁, x₂)) = 1 + x₁ + x₂
POL(a__U43(x₁)) = 1 + x₁
POL(a__U51(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(a__U52(x₁, x₂)) = 1 + x₁ + x₂
POL(a__U53(x₁)) = 1 + x₁
POL(a__U61(x₁, x₂)) = 0
POL(a__U62(x₁)) = 0
POL(a__U71(x₁, x₂)) = x₁ + x₂
POL(a__U72(x₁)) = x₁
POL(a____(x₁, x₂)) = 1 + x₁ + x₂
POL(a__and(x₁, x₂)) = x₁ + x₂
POL(a__isList(x₁)) = x₁
POL(a__isNeList(x₁)) = x₁
POL(a__isNePal(x₁)) = x₁
POL(a__isPal(x₁)) = x₁
POL(a__isPalListKind(x₁)) = 0
POL(a__isQid(x₁)) = 0
POL(and(x₁, x₂)) = x₁ + x₂
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = x₁
POL(isNeList(x₁)) = x₁
POL(isNePal(x₁)) = x₁
POL(isPal(x₁)) = x₁
POL(isPalListKind(x₁)) = 0
POL(isQid(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 1
POL(o) = 1
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(isNeList(X)) → a__isNeList(X)
mark(U12(X)) → a__U12(mark(X))
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(isList(X)) → a__isList(X)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(isQid(X)) → a__isQid(X)
mark(U32(X)) → a__U32(mark(X))
mark(U53(X)) → a__U53(mark(X))
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U43(X)) → a__U43(mark(X))
a__isPal(nil) → tt
a__isPalListKind(a) → tt
a__isPal(V) → a__U71(a__isPalListKind(V), V)
a__isPalListKind(nil) → tt
a__isPalListKind(o) → tt
a__isPalListKind(e) → tt
a__isPalListKind(i) → tt
a__isQid(a) → tt
a__isQid(e) → tt
a__isPalListKind(u) → tt
a__isQid(u) → tt
a__isQid(i) → tt
a__isQid(o) → tt
a__U12(X) → U12(X)
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__isNeList(X) → isNeList(X)
a__isList(X) → isList(X)
a__U22(X1, X2) → U22(X1, X2)
a__U31(X1, X2) → U31(X1, X2)
a__U23(X) → U23(X)
a__isQid(X) → isQid(X)
a__U32(X) → U32(X)
a__U42(X1, X2) → U42(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U43(X) → U43(X)
a__U53(X) → U53(X)
a__U52(X1, X2) → U52(X1, X2)
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(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(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__isPalListKind(X) → isPalListKind(X)
a__isPal(X) → isPal(X)
a__isNePal(X) → isNePal(X)
a__and(X1, X2) → and(X1, X2)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U42(tt, V2) → a__U43(a__isNeList(V2))
a__U41(tt, V1, V2) → a__U42(a__isList(V1), V2)
a__U32(tt) → tt
a__U31(tt, V) → a__U32(a__isQid(V))
a__U23(tt) → tt
a__U22(tt, V2) → a__U23(a__isList(V2))
a__U21(tt, V1, V2) → a__U22(a__isList(V1), V2)
a__U12(tt) → tt
a__U11(tt, V) → a__U12(a__isNeList(V))
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
a__isNePal(__(I, __(P, I))) → a__and(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(X)
a____(nil, X) → mark(X)
a____(X, nil) → mark(X)
mark(isNePal(X)) → a__isNePal(X)
a__and(tt, X) → mark(X)
a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a__isPalListKind(__(V1, V2)) → a__and(a__isPalListKind(V1), isPalListKind(V2))
a__isNeList(__(V1, V2)) → a__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNePal(V) → a__U61(a__isPalListKind(V), V)
a__isNeList(V) → a__U31(a__isPalListKind(V), V)
a__isNeList(__(V1, V2)) → a__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isList(V) → a__U11(a__isPalListKind(V), V)
a__U71(tt, V) → a__U72(a__isNePal(V))
a__U72(tt) → tt
a__U61(tt, V) → a__U62(a__isQid(V))
a__U62(tt) → tt
a__U52(tt, V2) → a__U53(a__isList(V2))
a__U53(tt) → tt
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNeList(V1), V2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

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

MARK(U12(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
A__U51(tt, V1, V2) → A__ISNELIST(V1)
A__U42(tt, V2) → A__ISNELIST(V2)
A__ISPALLISTKIND(__(V1, V2)) → A__ISPALLISTKIND(V1)
MARK(U31(X1, X2)) → MARK(X1)
A__U51(tt, V1, V2) → A__U52(a__isNeList(V1), V2)
A__ISNELIST(V) → A__ISPALLISTKIND(V)
MARK(isList(X)) → A__ISLIST(X)
A__AND(tt, X) → MARK(X)
A__ISLIST(V) → A__ISPALLISTKIND(V)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__U21(tt, V1, V2) → A__ISLIST(V1)
MARK(and(X1, X2)) → MARK(X1)
A__U41(tt, V1, V2) → A__U42(a__isList(V1), V2)
A__ISPALLISTKIND(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__U21(tt, V1, V2) → A__U22(a__isList(V1), V2)
A__U52(tt, V2) → A__ISLIST(V2)
A__U41(tt, V1, V2) → A__ISLIST(V1)
A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
A__U11(tt, V) → A__ISNELIST(V)
A__U22(tt, V2) → A__ISLIST(V2)
MARK(isNeList(X)) → A__ISNELIST(X)
MARK(U32(X)) → MARK(X)
MARK(isPal(X)) → A__ISPAL(X)
MARK(U72(X)) → MARK(X)
A__U71(tt, V) → A__ISNEPAL(V)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(isPalListKind(X)) → A__ISPALLISTKIND(X)
A__ISLIST(V) → A__U11(a__isPalListKind(V), 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, V) → a__U12(a__isNeList(V))
a__U12(tt) → tt
a__U21(tt, V1, V2) → a__U22(a__isList(V1), V2)
a__U22(tt, V2) → a__U23(a__isList(V2))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isQid(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isList(V1), V2)
a__U42(tt, V2) → a__U43(a__isNeList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNeList(V1), V2)
a__U52(tt, V2) → a__U53(a__isList(V2))
a__U53(tt) → tt
a__U61(tt, V) → a__U62(a__isQid(V))
a__U62(tt) → tt
a__U71(tt, V) → a__U72(a__isNePal(V))
a__U72(tt) → tt
a__and(tt, X) → mark(X)
a__isList(V) → a__U11(a__isPalListKind(V), V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNeList(V) → a__U31(a__isPalListKind(V), V)
a__isNeList(__(V1, V2)) → a__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNeList(__(V1, V2)) → a__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNePal(V) → a__U61(a__isPalListKind(V), V)
a__isNePal(__(I, __(P, I))) → a__and(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))
a__isPal(V) → a__U71(a__isPalListKind(V), V)
a__isPal(nil) → tt
a__isPalListKind(a) → tt
a__isPalListKind(e) → tt
a__isPalListKind(i) → tt
a__isPalListKind(nil) → tt
a__isPalListKind(o) → tt
a__isPalListKind(u) → tt
a__isPalListKind(__(V1, V2)) → a__and(a__isPalListKind(V1), isPalListKind(V2))
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(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X)) → a__U72(mark(X))
mark(isNePal(X)) → a__isNePal(X)
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(X)
mark(isPal(X)) → a__isPal(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(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNeList(X) → isNeList(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2) → U22(X1, X2)
a__isList(X) → isList(X)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__isQid(X) → isQid(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isNePal(X) → isNePal(X)
a__and(X1, X2) → and(X1, X2)
a__isPalListKind(X) → isPalListKind(X)
a__isPal(X) → isPal(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 11 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                        ↳ QDPOrderProof

                      ↳ QDP

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

A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))

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, V) → a__U12(a__isNeList(V))
a__U12(tt) → tt
a__U21(tt, V1, V2) → a__U22(a__isList(V1), V2)
a__U22(tt, V2) → a__U23(a__isList(V2))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isQid(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isList(V1), V2)
a__U42(tt, V2) → a__U43(a__isNeList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNeList(V1), V2)
a__U52(tt, V2) → a__U53(a__isList(V2))
a__U53(tt) → tt
a__U61(tt, V) → a__U62(a__isQid(V))
a__U62(tt) → tt
a__U71(tt, V) → a__U72(a__isNePal(V))
a__U72(tt) → tt
a__and(tt, X) → mark(X)
a__isList(V) → a__U11(a__isPalListKind(V), V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNeList(V) → a__U31(a__isPalListKind(V), V)
a__isNeList(__(V1, V2)) → a__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNeList(__(V1, V2)) → a__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNePal(V) → a__U61(a__isPalListKind(V), V)
a__isNePal(__(I, __(P, I))) → a__and(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))
a__isPal(V) → a__U71(a__isPalListKind(V), V)
a__isPal(nil) → tt
a__isPalListKind(a) → tt
a__isPalListKind(e) → tt
a__isPalListKind(i) → tt
a__isPalListKind(nil) → tt
a__isPalListKind(o) → tt
a__isPalListKind(u) → tt
a__isPalListKind(__(V1, V2)) → a__and(a__isPalListKind(V1), isPalListKind(V2))
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(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X)) → a__U72(mark(X))
mark(isNePal(X)) → a__isNePal(X)
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(X)
mark(isPal(X)) → a__isPal(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(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNeList(X) → isNeList(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2) → U22(X1, X2)
a__isList(X) → isList(X)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__isQid(X) → isQid(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isNePal(X) → isNePal(X)
a__and(X1, X2) → and(X1, X2)
a__isPalListKind(X) → isPalListKind(X)
a__isPal(X) → isPal(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)))

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation with max and min functions [25]:

POL(A____(x₁, x₂)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂, x₃)) = x₁
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 0
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U72(x₁)) = 0
POL(__(x₁, x₂)) = 1 + x₁ + x₂
POL(a) = 0
POL(a__U11(x₁, x₂)) = 0
POL(a__U12(x₁)) = 0
POL(a__U21(x₁, x₂, x₃)) = x₁
POL(a__U22(x₁, x₂)) = 0
POL(a__U23(x₁)) = 0
POL(a__U31(x₁, x₂)) = 0
POL(a__U32(x₁)) = 0
POL(a__U41(x₁, x₂, x₃)) = 0
POL(a__U42(x₁, x₂)) = 0
POL(a__U43(x₁)) = 0
POL(a__U51(x₁, x₂, x₃)) = 0
POL(a__U52(x₁, x₂)) = 0
POL(a__U53(x₁)) = 0
POL(a__U61(x₁, x₂)) = 0
POL(a__U62(x₁)) = 0
POL(a__U71(x₁, x₂)) = 0
POL(a__U72(x₁)) = 0
POL(a____(x₁, x₂)) = 1 + x₁ + x₂
POL(a__and(x₁, x₂)) = x₂
POL(a__isList(x₁)) = 0
POL(a__isNeList(x₁)) = 0
POL(a__isNePal(x₁)) = 0
POL(a__isPal(x₁)) = 0
POL(a__isPalListKind(x₁)) = 0
POL(a__isQid(x₁)) = x₁
POL(and(x₁, x₂)) = x₂
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 0
POL(isNeList(x₁)) = 0
POL(isNePal(x₁)) = 0
POL(isPal(x₁)) = 0
POL(isPalListKind(x₁)) = 0
POL(isQid(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented:

mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(isNeList(X)) → a__isNeList(X)
mark(U12(X)) → a__U12(mark(X))
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U23(X)) → a__U23(mark(X))
mark(isList(X)) → a__isList(X)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(isQid(X)) → a__isQid(X)
mark(U32(X)) → a__U32(mark(X))
mark(U53(X)) → a__U53(mark(X))
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U43(X)) → a__U43(mark(X))
a__isPal(nil) → tt
a__isPalListKind(a) → tt
a__isPal(V) → a__U71(a__isPalListKind(V), V)
a__isPalListKind(nil) → tt
a__isPalListKind(o) → tt
a__isPalListKind(e) → tt
a__isPalListKind(i) → tt
a__isQid(a) → tt
a__isQid(e) → tt
a__isPalListKind(u) → tt
a__isQid(u) → tt
a__isQid(i) → tt
a__isQid(o) → tt
a__U12(X) → U12(X)
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__isNeList(X) → isNeList(X)
a__isList(X) → isList(X)
a__U22(X1, X2) → U22(X1, X2)
a__U31(X1, X2) → U31(X1, X2)
a__U23(X) → U23(X)
a__isQid(X) → isQid(X)
a__U32(X) → U32(X)
a__U42(X1, X2) → U42(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U43(X) → U43(X)
a__U53(X) → U53(X)
a__U52(X1, X2) → U52(X1, X2)
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(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(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__isPalListKind(X) → isPalListKind(X)
a__isPal(X) → isPal(X)
a__isNePal(X) → isNePal(X)
a__and(X1, X2) → and(X1, X2)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U42(tt, V2) → a__U43(a__isNeList(V2))
a__U41(tt, V1, V2) → a__U42(a__isList(V1), V2)
a__U32(tt) → tt
a__U31(tt, V) → a__U32(a__isQid(V))
a__U23(tt) → tt
a__U22(tt, V2) → a__U23(a__isList(V2))
a__U21(tt, V1, V2) → a__U22(a__isList(V1), V2)
a__U12(tt) → tt
a__U11(tt, V) → a__U12(a__isNeList(V))
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
a__isNePal(__(I, __(P, I))) → a__and(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(X)
a____(nil, X) → mark(X)
a____(X, nil) → mark(X)
mark(isNePal(X)) → a__isNePal(X)
a__and(tt, X) → mark(X)
a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a__isPalListKind(__(V1, V2)) → a__and(a__isPalListKind(V1), isPalListKind(V2))
a__isNeList(__(V1, V2)) → a__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNePal(V) → a__U61(a__isPalListKind(V), V)
a__isNeList(V) → a__U31(a__isPalListKind(V), V)
a__isNeList(__(V1, V2)) → a__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isList(V) → a__U11(a__isPalListKind(V), V)
a__U71(tt, V) → a__U72(a__isNePal(V))
a__U72(tt) → tt
a__U61(tt, V) → a__U62(a__isQid(V))
a__U62(tt) → tt
a__U52(tt, V2) → a__U53(a__isList(V2))
a__U53(tt) → tt
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNeList(V1), V2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ PisEmptyProof

                      ↳ QDP

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, V) → a__U12(a__isNeList(V))
a__U12(tt) → tt
a__U21(tt, V1, V2) → a__U22(a__isList(V1), V2)
a__U22(tt, V2) → a__U23(a__isList(V2))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isQid(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isList(V1), V2)
a__U42(tt, V2) → a__U43(a__isNeList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNeList(V1), V2)
a__U52(tt, V2) → a__U53(a__isList(V2))
a__U53(tt) → tt
a__U61(tt, V) → a__U62(a__isQid(V))
a__U62(tt) → tt
a__U71(tt, V) → a__U72(a__isNePal(V))
a__U72(tt) → tt
a__and(tt, X) → mark(X)
a__isList(V) → a__U11(a__isPalListKind(V), V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNeList(V) → a__U31(a__isPalListKind(V), V)
a__isNeList(__(V1, V2)) → a__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNeList(__(V1, V2)) → a__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNePal(V) → a__U61(a__isPalListKind(V), V)
a__isNePal(__(I, __(P, I))) → a__and(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))
a__isPal(V) → a__U71(a__isPalListKind(V), V)
a__isPal(nil) → tt
a__isPalListKind(a) → tt
a__isPalListKind(e) → tt
a__isPalListKind(i) → tt
a__isPalListKind(nil) → tt
a__isPalListKind(o) → tt
a__isPalListKind(u) → tt
a__isPalListKind(__(V1, V2)) → a__and(a__isPalListKind(V1), isPalListKind(V2))
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(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X)) → a__U72(mark(X))
mark(isNePal(X)) → a__isNePal(X)
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(X)
mark(isPal(X)) → a__isPal(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(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNeList(X) → isNeList(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2) → U22(X1, X2)
a__isList(X) → isList(X)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__isQid(X) → isQid(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isNePal(X) → isNePal(X)
a__and(X1, X2) → and(X1, X2)
a__isPalListKind(X) → isPalListKind(X)
a__isPal(X) → isPal(X)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

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

MARK(U12(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
A__U11(tt, V) → A__ISNELIST(V)
A__ISPALLISTKIND(__(V1, V2)) → A__ISPALLISTKIND(V1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(isNeList(X)) → A__ISNELIST(X)
MARK(U72(X)) → MARK(X)
A__ISNELIST(V) → A__ISPALLISTKIND(V)
MARK(isList(X)) → A__ISLIST(X)
A__AND(tt, X) → MARK(X)
A__ISLIST(V) → A__ISPALLISTKIND(V)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isPalListKind(X)) → A__ISPALLISTKIND(X)
A__ISPALLISTKIND(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__ISLIST(V) → A__U11(a__isPalListKind(V), 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, V) → a__U12(a__isNeList(V))
a__U12(tt) → tt
a__U21(tt, V1, V2) → a__U22(a__isList(V1), V2)
a__U22(tt, V2) → a__U23(a__isList(V2))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isQid(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isList(V1), V2)
a__U42(tt, V2) → a__U43(a__isNeList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNeList(V1), V2)
a__U52(tt, V2) → a__U53(a__isList(V2))
a__U53(tt) → tt
a__U61(tt, V) → a__U62(a__isQid(V))
a__U62(tt) → tt
a__U71(tt, V) → a__U72(a__isNePal(V))
a__U72(tt) → tt
a__and(tt, X) → mark(X)
a__isList(V) → a__U11(a__isPalListKind(V), V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNeList(V) → a__U31(a__isPalListKind(V), V)
a__isNeList(__(V1, V2)) → a__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNeList(__(V1, V2)) → a__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNePal(V) → a__U61(a__isPalListKind(V), V)
a__isNePal(__(I, __(P, I))) → a__and(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))
a__isPal(V) → a__U71(a__isPalListKind(V), V)
a__isPal(nil) → tt
a__isPalListKind(a) → tt
a__isPalListKind(e) → tt
a__isPalListKind(i) → tt
a__isPalListKind(nil) → tt
a__isPalListKind(o) → tt
a__isPalListKind(u) → tt
a__isPalListKind(__(V1, V2)) → a__and(a__isPalListKind(V1), isPalListKind(V2))
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(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X)) → a__U72(mark(X))
mark(isNePal(X)) → a__isNePal(X)
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(X)
mark(isPal(X)) → a__isPal(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(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNeList(X) → isNeList(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2) → U22(X1, X2)
a__isList(X) → isList(X)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__isQid(X) → isQid(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isNePal(X) → isNePal(X)
a__and(X1, X2) → and(X1, X2)
a__isPalListKind(X) → isPalListKind(X)
a__isPal(X) → isPal(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(U12(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
A__ISPALLISTKIND(__(V1, V2)) → A__ISPALLISTKIND(V1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(isNeList(X)) → A__ISNELIST(X)
MARK(U72(X)) → MARK(X)
A__ISLIST(V) → A__ISPALLISTKIND(V)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
A__ISPALLISTKIND(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__ISLIST(V) → A__U11(a__isPalListKind(V), V)

The remaining pairs can at least be oriented weakly.

A__U11(tt, V) → A__ISNELIST(V)
A__ISNELIST(V) → A__ISPALLISTKIND(V)
MARK(isList(X)) → A__ISLIST(X)
A__AND(tt, X) → MARK(X)
MARK(isPalListKind(X)) → A__ISPALLISTKIND(X)

Used ordering: Polynomial interpretation [25]:

POL(A__AND(x₁, x₂)) = x₂
POL(A__ISLIST(x₁)) = 1 + x₁
POL(A__ISNELIST(x₁)) = x₁
POL(A__ISPALLISTKIND(x₁)) = x₁
POL(A__U11(x₁, x₂)) = x₂
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = 1 + x₁ + x₂
POL(U12(x₁)) = 1 + x₁
POL(U21(x₁, x₂, x₃)) = 0
POL(U22(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U31(x₁, x₂)) = 1 + x₁
POL(U32(x₁)) = 1 + x₁
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 0
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂)) = 0
POL(U62(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U72(x₁)) = 1 + x₁
POL(__(x₁, x₂)) = 1 + x₁ + x₂
POL(a) = 0
POL(a__U11(x₁, x₂)) = 0
POL(a__U12(x₁)) = 0
POL(a__U21(x₁, x₂, x₃)) = 0
POL(a__U22(x₁, x₂)) = 0
POL(a__U23(x₁)) = 0
POL(a__U31(x₁, x₂)) = 0
POL(a__U32(x₁)) = 0
POL(a__U41(x₁, x₂, x₃)) = 0
POL(a__U42(x₁, x₂)) = 0
POL(a__U43(x₁)) = 0
POL(a__U51(x₁, x₂, x₃)) = 0
POL(a__U52(x₁, x₂)) = 0
POL(a__U53(x₁)) = 0
POL(a__U61(x₁, x₂)) = 0
POL(a__U62(x₁)) = 0
POL(a__U71(x₁, x₂)) = 0
POL(a__U72(x₁)) = 0
POL(a____(x₁, x₂)) = 0
POL(a__and(x₁, x₂)) = 0
POL(a__isList(x₁)) = 0
POL(a__isNeList(x₁)) = 0
POL(a__isNePal(x₁)) = 0
POL(a__isPal(x₁)) = 0
POL(a__isPalListKind(x₁)) = 0
POL(a__isQid(x₁)) = 0
POL(and(x₁, x₂)) = 1 + x₁ + x₂
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 1 + x₁
POL(isNeList(x₁)) = 1 + x₁
POL(isNePal(x₁)) = 0
POL(isPal(x₁)) = 0
POL(isPalListKind(x₁)) = x₁
POL(isQid(x₁)) = 0
POL(mark(x₁)) = 0
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

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

MARK(isPalListKind(X)) → A__ISPALLISTKIND(X)
A__U11(tt, V) → A__ISNELIST(V)
MARK(isList(X)) → A__ISLIST(X)
A__ISNELIST(V) → A__ISPALLISTKIND(V)
A__AND(tt, 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, V) → a__U12(a__isNeList(V))
a__U12(tt) → tt
a__U21(tt, V1, V2) → a__U22(a__isList(V1), V2)
a__U22(tt, V2) → a__U23(a__isList(V2))
a__U23(tt) → tt
a__U31(tt, V) → a__U32(a__isQid(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isList(V1), V2)
a__U42(tt, V2) → a__U43(a__isNeList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNeList(V1), V2)
a__U52(tt, V2) → a__U53(a__isList(V2))
a__U53(tt) → tt
a__U61(tt, V) → a__U62(a__isQid(V))
a__U62(tt) → tt
a__U71(tt, V) → a__U72(a__isNePal(V))
a__U72(tt) → tt
a__and(tt, X) → mark(X)
a__isList(V) → a__U11(a__isPalListKind(V), V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNeList(V) → a__U31(a__isPalListKind(V), V)
a__isNeList(__(V1, V2)) → a__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNeList(__(V1, V2)) → a__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNePal(V) → a__U61(a__isPalListKind(V), V)
a__isNePal(__(I, __(P, I))) → a__and(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))
a__isPal(V) → a__U71(a__isPalListKind(V), V)
a__isPal(nil) → tt
a__isPalListKind(a) → tt
a__isPalListKind(e) → tt
a__isPalListKind(i) → tt
a__isPalListKind(nil) → tt
a__isPalListKind(o) → tt
a__isPalListKind(u) → tt
a__isPalListKind(__(V1, V2)) → a__and(a__isPalListKind(V1), isPalListKind(V2))
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(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2)) → a__U22(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(U23(X)) → a__U23(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2)) → a__U71(mark(X1), X2)
mark(U72(X)) → a__U72(mark(X))
mark(isNePal(X)) → a__isNePal(X)
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(X)
mark(isPal(X)) → a__isPal(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(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNeList(X) → isNeList(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2) → U22(X1, X2)
a__isList(X) → isList(X)
a__U23(X) → U23(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__isQid(X) → isQid(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isNePal(X) → isNePal(X)
a__and(X1, X2) → and(X1, X2)
a__isPalListKind(X) → isPalListKind(X)
a__isPal(X) → isPal(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 5 less nodes.