Termination w.r.t. Q of the given QTRS could be proven:

(0) Obligation:

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.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

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

A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
A____(__(X, Y), Z) → MARK(X)
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
A____(__(X, Y), Z) → MARK(Y)
A____(__(X, Y), Z) → MARK(Z)
A____(X, nil) → MARK(X)
A____(nil, X) → MARK(X)
A__U11(tt, V) → A__U12(a__isNeList(V))
A__U11(tt, V) → A__ISNELIST(V)
A__U21(tt, V1, V2) → A__U22(a__isList(V1), V2)
A__U21(tt, V1, V2) → A__ISLIST(V1)
A__U22(tt, V2) → A__U23(a__isList(V2))
A__U22(tt, V2) → A__ISLIST(V2)
A__U31(tt, V) → A__U32(a__isQid(V))
A__U31(tt, V) → A__ISQID(V)
A__U41(tt, V1, V2) → A__U42(a__isList(V1), V2)
A__U41(tt, V1, V2) → A__ISLIST(V1)
A__U42(tt, V2) → A__U43(a__isNeList(V2))
A__U42(tt, V2) → A__ISNELIST(V2)
A__U51(tt, V1, V2) → A__U52(a__isNeList(V1), V2)
A__U51(tt, V1, V2) → A__ISNELIST(V1)
A__U52(tt, V2) → A__U53(a__isList(V2))
A__U52(tt, V2) → A__ISLIST(V2)
A__U61(tt, V) → A__U62(a__isQid(V))
A__U61(tt, V) → A__ISQID(V)
A__U71(tt, V) → A__U72(a__isNePal(V))
A__U71(tt, V) → A__ISNEPAL(V)
A__AND(tt, X) → MARK(X)
A__ISLIST(V) → A__U11(a__isPalListKind(V), V)
A__ISLIST(V) → A__ISPALLISTKIND(V)
A__ISLIST(__(V1, V2)) → A__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
A__ISLIST(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__ISLIST(__(V1, V2)) → A__ISPALLISTKIND(V1)
A__ISNELIST(V) → A__U31(a__isPalListKind(V), V)
A__ISNELIST(V) → A__ISPALLISTKIND(V)
A__ISNELIST(__(V1, V2)) → A__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
A__ISNELIST(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__ISNELIST(__(V1, V2)) → A__ISPALLISTKIND(V1)
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(V) → A__ISPALLISTKIND(V)
A__ISNEPAL(__(I, __(P, I))) → A__AND(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))
A__ISNEPAL(__(I, __(P, I))) → A__AND(a__isQid(I), isPalListKind(I))
A__ISNEPAL(__(I, __(P, I))) → A__ISQID(I)
A__ISPAL(V) → A__U71(a__isPalListKind(V), V)
A__ISPAL(V) → A__ISPALLISTKIND(V)
A__ISPALLISTKIND(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__ISPALLISTKIND(__(V1, V2)) → A__ISPALLISTKIND(V1)
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
MARK(__(X1, X2)) → MARK(X1)
MARK(__(X1, X2)) → MARK(X2)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X)) → A__U12(mark(X))
MARK(U12(X)) → MARK(X)
MARK(isNeList(X)) → A__ISNELIST(X)
MARK(U21(X1, X2, X3)) → A__U21(mark(X1), X2, X3)
MARK(U21(X1, X2, X3)) → MARK(X1)
MARK(U22(X1, X2)) → A__U22(mark(X1), X2)
MARK(U22(X1, X2)) → MARK(X1)
MARK(isList(X)) → A__ISLIST(X)
MARK(U23(X)) → A__U23(mark(X))
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → A__U32(mark(X))
MARK(U32(X)) → MARK(X)
MARK(isQid(X)) → A__ISQID(X)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U43(X)) → A__U43(mark(X))
MARK(U43(X)) → MARK(X)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U53(X)) → A__U53(mark(X))
MARK(U53(X)) → MARK(X)
MARK(U61(X1, X2)) → A__U61(mark(X1), X2)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U62(X)) → A__U62(mark(X))
MARK(U62(X)) → MARK(X)
MARK(U71(X1, X2)) → A__U71(mark(X1), X2)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X)) → A__U72(mark(X))
MARK(U72(X)) → MARK(X)
MARK(isNePal(X)) → A__ISNEPAL(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isPalListKind(X)) → A__ISPALLISTKIND(X)
MARK(isPal(X)) → A__ISPAL(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.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 22 less nodes.

(4) Obligation:

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

A____(__(X, Y), Z) → MARK(X)
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
A____(__(X, Y), Z) → MARK(Y)
MARK(__(X1, X2)) → MARK(X1)
MARK(__(X1, X2)) → MARK(X2)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__U11(tt, V) → A__ISNELIST(V)
A__ISNELIST(V) → A__ISPALLISTKIND(V)
A__ISPALLISTKIND(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__AND(tt, X) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(isNeList(X)) → A__ISNELIST(X)
A__ISNELIST(__(V1, V2)) → A__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
A__U41(tt, V1, V2) → A__U42(a__isList(V1), V2)
A__U42(tt, V2) → A__ISNELIST(V2)
A__ISNELIST(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__ISNELIST(__(V1, V2)) → A__ISPALLISTKIND(V1)
A__ISPALLISTKIND(__(V1, V2)) → A__ISPALLISTKIND(V1)
A__ISNELIST(__(V1, V2)) → A__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
A__U51(tt, V1, V2) → A__U52(a__isNeList(V1), V2)
A__U52(tt, V2) → A__ISLIST(V2)
A__ISLIST(V) → A__U11(a__isPalListKind(V), V)
A__ISLIST(V) → A__ISPALLISTKIND(V)
A__ISLIST(__(V1, V2)) → A__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
A__U21(tt, V1, V2) → A__U22(a__isList(V1), V2)
A__U22(tt, V2) → A__ISLIST(V2)
A__ISLIST(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__ISLIST(__(V1, V2)) → A__ISPALLISTKIND(V1)
A__U21(tt, V1, V2) → A__ISLIST(V1)
A__U51(tt, V1, V2) → A__ISNELIST(V1)
A__U41(tt, V1, V2) → A__ISLIST(V1)
MARK(U21(X1, X2, X3)) → A__U21(mark(X1), X2, X3)
MARK(U21(X1, X2, X3)) → MARK(X1)
MARK(U22(X1, X2)) → A__U22(mark(X1), X2)
MARK(U22(X1, X2)) → MARK(X1)
MARK(isList(X)) → A__ISLIST(X)
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U53(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U62(X)) → MARK(X)
MARK(U71(X1, X2)) → A__U71(mark(X1), X2)
A__U71(tt, V) → A__ISNEPAL(V)
A__ISNEPAL(V) → A__ISPALLISTKIND(V)
A__ISNEPAL(__(I, __(P, I))) → A__AND(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))
A__ISNEPAL(__(I, __(P, I))) → A__AND(a__isQid(I), isPalListKind(I))
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X)) → MARK(X)
MARK(isNePal(X)) → A__ISNEPAL(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isPalListKind(X)) → A__ISPALLISTKIND(X)
MARK(isPal(X)) → A__ISPAL(X)
A__ISPAL(V) → A__U71(a__isPalListKind(V), V)
A__ISPAL(V) → A__ISPALLISTKIND(V)
A____(__(X, Y), Z) → MARK(Z)
A____(X, nil) → MARK(X)
A____(nil, 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.

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A____(__(X, Y), Z) → MARK(X)
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
A____(__(X, Y), Z) → MARK(Y)
MARK(__(X1, X2)) → MARK(X1)
MARK(__(X1, X2)) → MARK(X2)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__U11(tt, V) → A__ISNELIST(V)
A__ISPALLISTKIND(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__AND(tt, X) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
MARK(isNeList(X)) → A__ISNELIST(X)
A__ISNELIST(__(V1, V2)) → A__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
A__U41(tt, V1, V2) → A__U42(a__isList(V1), V2)
A__ISNELIST(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__ISNELIST(__(V1, V2)) → A__ISPALLISTKIND(V1)
A__ISPALLISTKIND(__(V1, V2)) → A__ISPALLISTKIND(V1)
A__ISNELIST(__(V1, V2)) → A__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
A__U51(tt, V1, V2) → A__U52(a__isNeList(V1), V2)
A__U52(tt, V2) → A__ISLIST(V2)
A__ISLIST(V) → A__U11(a__isPalListKind(V), V)
A__ISLIST(V) → A__ISPALLISTKIND(V)
A__ISLIST(__(V1, V2)) → A__U21(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
A__U21(tt, V1, V2) → A__U22(a__isList(V1), V2)
A__U22(tt, V2) → A__ISLIST(V2)
A__ISLIST(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__ISLIST(__(V1, V2)) → A__ISPALLISTKIND(V1)
A__U21(tt, V1, V2) → A__ISLIST(V1)
A__U51(tt, V1, V2) → A__ISNELIST(V1)
A__U41(tt, V1, V2) → A__ISLIST(V1)
MARK(U21(X1, X2, X3)) → A__U21(mark(X1), X2, X3)
MARK(U21(X1, X2, X3)) → MARK(X1)
MARK(U22(X1, X2)) → A__U22(mark(X1), X2)
MARK(U22(X1, X2)) → MARK(X1)
MARK(isList(X)) → A__ISLIST(X)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → A__U71(mark(X1), X2)
A__U71(tt, V) → A__ISNEPAL(V)
A__ISNEPAL(V) → A__ISPALLISTKIND(V)
A__ISNEPAL(__(I, __(P, I))) → A__AND(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))
A__ISNEPAL(__(I, __(P, I))) → A__AND(a__isQid(I), isPalListKind(I))
MARK(U71(X1, X2)) → MARK(X1)
MARK(isNePal(X)) → A__ISNEPAL(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isPalListKind(X)) → A__ISPALLISTKIND(X)
MARK(isPal(X)) → A__ISPAL(X)
A__ISPAL(V) → A__U71(a__isPalListKind(V), V)
A__ISPAL(V) → A__ISPALLISTKIND(V)
A____(__(X, Y), Z) → MARK(Z)
A____(X, nil) → MARK(X)
A____(nil, X) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A____(x1, x2)  =  A____(x1, x2)
__(x1, x2)  =  __(x1, x2)
MARK(x1)  =  MARK(x1)
mark(x1)  =  x1
a____(x1, x2)  =  a____(x1, x2)
U11(x1, x2)  =  U11(x1, x2)
A__U11(x1, x2)  =  A__U11(x1, x2)
tt  =  tt
A__ISNELIST(x1)  =  x1
A__ISPALLISTKIND(x1)  =  x1
A__AND(x1, x2)  =  A__AND(x1, x2)
a__isPalListKind(x1)  =  x1
isPalListKind(x1)  =  x1
U12(x1)  =  x1
isNeList(x1)  =  x1
A__U41(x1, x2, x3)  =  A__U41(x1, x2, x3)
a__and(x1, x2)  =  a__and(x1, x2)
A__U42(x1, x2)  =  x2
a__isList(x1)  =  a__isList(x1)
A__U51(x1, x2, x3)  =  A__U51(x2, x3)
A__U52(x1, x2)  =  A__U52(x1, x2)
a__isNeList(x1)  =  x1
A__ISLIST(x1)  =  A__ISLIST(x1)
A__U21(x1, x2, x3)  =  A__U21(x2, x3)
A__U22(x1, x2)  =  A__U22(x1, x2)
U21(x1, x2, x3)  =  U21(x1, x2, x3)
U22(x1, x2)  =  U22(x1, x2)
isList(x1)  =  isList(x1)
U23(x1)  =  x1
U31(x1, x2)  =  x1
U32(x1)  =  x1
U41(x1, x2, x3)  =  U41(x1, x2, x3)
U42(x1, x2)  =  U42(x1, x2)
U43(x1)  =  x1
U51(x1, x2, x3)  =  U51(x1, x2, x3)
U52(x1, x2)  =  U52(x1, x2)
U53(x1)  =  x1
U61(x1, x2)  =  U61(x1)
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
A__U71(x1, x2)  =  A__U71(x1, x2)
A__ISNEPAL(x1)  =  A__ISNEPAL(x1)
a__isQid(x1)  =  a__isQid
and(x1, x2)  =  and(x1, x2)
isPal(x1)  =  isPal(x1)
U72(x1)  =  x1
isNePal(x1)  =  isNePal(x1)
A__ISPAL(x1)  =  A__ISPAL(x1)
nil  =  nil
a__U11(x1, x2)  =  a__U11(x1, x2)
a__U12(x1)  =  x1
a__U21(x1, x2, x3)  =  a__U21(x1, x2, x3)
a__U22(x1, x2)  =  a__U22(x1, x2)
a__U23(x1)  =  x1
a__U31(x1, x2)  =  x1
a__U32(x1)  =  x1
a__U41(x1, x2, x3)  =  a__U41(x1, x2, x3)
a__U42(x1, x2)  =  a__U42(x1, x2)
a__U43(x1)  =  x1
a__U51(x1, x2, x3)  =  a__U51(x1, x2, x3)
a__U52(x1, x2)  =  a__U52(x1, x2)
a__U53(x1)  =  x1
a__U61(x1, x2)  =  a__U61(x1)
a__U62(x1)  =  x1
a__U71(x1, x2)  =  a__U71(x1, x2)
a__U72(x1)  =  x1
a__isNePal(x1)  =  a__isNePal(x1)
a__isPal(x1)  =  a__isPal(x1)
a  =  a
e  =  e
i  =  i
o  =  o
u  =  u
isQid(x1)  =  isQid

Recursive path order with status [RPO].
Quasi-Precedence:

[A₂, ₂, a₂, isPal₁, a_isPal1] > [A_AND2, a_and2, and₂] > [MARK₁, A_U112, U61₁, A_ISNEPAL1, a_U611] > [tt, a_isQid, isQid]
[A₂, ₂, a₂, isPal₁, a_isPal1] > [U21₃, a_U213] > A_U212 > A_U222 > [a_isList1, A_U522, A_ISLIST1, isList₁, U52₂, a_U522] > [MARK₁, A_U112, U61₁, A_ISNEPAL1, a_U611] > [tt, a_isQid, isQid]
[A₂, ₂, a₂, isPal₁, a_isPal1] > [U21₃, a_U213] > A_U212 > A_U222 > [a_isList1, A_U522, A_ISLIST1, isList₁, U52₂, a_U522] > [U11₂, a_U112]
[A₂, ₂, a₂, isPal₁, a_isPal1] > [U21₃, a_U213] > [U22₂, a_U222] > A_U222 > [a_isList1, A_U522, A_ISLIST1, isList₁, U52₂, a_U522] > [MARK₁, A_U112, U61₁, A_ISNEPAL1, a_U611] > [tt, a_isQid, isQid]
[A₂, ₂, a₂, isPal₁, a_isPal1] > [U21₃, a_U213] > [U22₂, a_U222] > A_U222 > [a_isList1, A_U522, A_ISLIST1, isList₁, U52₂, a_U522] > [U11₂, a_U112]
[A₂, ₂, a₂, isPal₁, a_isPal1] > [U41₃, a_U413] > A_U413 > [a_isList1, A_U522, A_ISLIST1, isList₁, U52₂, a_U522] > [MARK₁, A_U112, U61₁, A_ISNEPAL1, a_U611] > [tt, a_isQid, isQid]
[A₂, ₂, a₂, isPal₁, a_isPal1] > [U41₃, a_U413] > A_U413 > [a_isList1, A_U522, A_ISLIST1, isList₁, U52₂, a_U522] > [U11₂, a_U112]
[A₂, ₂, a₂, isPal₁, a_isPal1] > [U41₃, a_U413] > [U42₂, a_U422] > [MARK₁, A_U112, U61₁, A_ISNEPAL1, a_U611] > [tt, a_isQid, isQid]
[A₂, ₂, a₂, isPal₁, a_isPal1] > [U51₃, a_U513] > A_U512 > [a_isList1, A_U522, A_ISLIST1, isList₁, U52₂, a_U522] > [MARK₁, A_U112, U61₁, A_ISNEPAL1, a_U611] > [tt, a_isQid, isQid]
[A₂, ₂, a₂, isPal₁, a_isPal1] > [U51₃, a_U513] > A_U512 > [a_isList1, A_U522, A_ISLIST1, isList₁, U52₂, a_U522] > [U11₂, a_U112]
[A₂, ₂, a₂, isPal₁, a_isPal1] > [U71₂, a_U712] > A_U712 > [MARK₁, A_U112, U61₁, A_ISNEPAL1, a_U611] > [tt, a_isQid, isQid]
[A₂, ₂, a₂, isPal₁, a_isPal1] > [U71₂, a_U712] > [isNePal₁, a_isNePal1] > [MARK₁, A_U112, U61₁, A_ISNEPAL1, a_U611] > [tt, a_isQid, isQid]
[A₂, ₂, a₂, isPal₁, a_isPal1] > A_ISPAL1 > A_U712 > [MARK₁, A_U112, U61₁, A_ISNEPAL1, a_U611] > [tt, a_isQid, isQid]
nil > [MARK₁, A_U112, U61₁, A_ISNEPAL1, a_U611] > [tt, a_isQid, isQid]
a > [tt, a_isQid, isQid]
e > [tt, a_isQid, isQid]
i > [tt, a_isQid, isQid]
o > [tt, a_isQid, isQid]
u > [tt, a_isQid, isQid]

Status:

A₂: [1,2]
_₂: [1,2]
MARK₁: [1]
a₂: [1,2]
U11₂: multiset
A_U112: [1,2]
tt: multiset
A_AND2: multiset
A_U413: [2,3,1]
a_and2: multiset
a_isList1: multiset
A_U512: multiset
A_U522: multiset
A_ISLIST1: multiset
A_U212: multiset
A_U222: multiset
U21₃: multiset
U22₂: multiset
isList₁: multiset
U41₃: multiset
U42₂: multiset
U51₃: multiset
U52₂: multiset
U61₁: multiset
U71₂: multiset
A_U712: multiset
A_ISNEPAL1: multiset
a_isQid: multiset
and₂: multiset
isPal₁: [1]
isNePal₁: multiset
A_ISPAL1: [1]
nil: multiset
a_U112: multiset
a_U213: multiset
a_U222: multiset
a_U413: multiset
a_U422: multiset
a_U513: multiset
a_U522: multiset
a_U611: multiset
a_U712: multiset
a_isNePal1: multiset
a_isPal1: [1]
a: multiset
e: multiset
i: multiset
o: multiset
u: multiset
isQid: multiset

The following usable rules [FROCOS05] were oriented:

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)

(6) Obligation:

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

A__ISNELIST(V) → A__ISPALLISTKIND(V)
MARK(U12(X)) → MARK(X)
A__U42(tt, V2) → A__ISNELIST(V2)
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
MARK(U53(X)) → MARK(X)
MARK(U62(X)) → MARK(X)
MARK(U72(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.

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(8) Obligation:

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

MARK(U23(X)) → MARK(X)
MARK(U12(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
MARK(U53(X)) → MARK(X)
MARK(U62(X)) → MARK(X)
MARK(U72(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.

(9) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U12(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  MARK(x1)
U23(x1)  =  x1
U12(x1)  =  U12(x1)
U31(x1, x2)  =  U31(x1, x2)
U32(x1)  =  x1
U43(x1)  =  x1
U53(x1)  =  x1
U62(x1)  =  x1
U72(x1)  =  x1
a____(x1, x2)  =  a____(x1, x2)
__(x1, x2)  =  __(x1, x2)
mark(x1)  =  x1
nil  =  nil
a__U11(x1, x2)  =  a__U11(x1, x2)
tt  =  tt
a__U12(x1)  =  a__U12(x1)
a__isNeList(x1)  =  a__isNeList(x1)
a__U21(x1, x2, x3)  =  a__U21(x3)
a__U22(x1, x2)  =  a__U22(x2)
a__isList(x1)  =  a__isList(x1)
a__U23(x1)  =  x1
a__U31(x1, x2)  =  a__U31(x1, x2)
a__U32(x1)  =  x1
a__isQid(x1)  =  a__isQid
a__U41(x1, x2, x3)  =  a__U41(x3)
a__U42(x1, x2)  =  a__U42(x2)
a__U43(x1)  =  x1
a__U51(x1, x2, x3)  =  a__U51(x1, x2, x3)
a__U52(x1, x2)  =  a__U52(x1, x2)
a__U53(x1)  =  x1
a__U61(x1, x2)  =  x1
a__U62(x1)  =  x1
a__U71(x1, x2)  =  a__U71(x1, x2)
a__U72(x1)  =  x1
a__isNePal(x1)  =  a__isNePal(x1)
a__and(x1, x2)  =  x2
a__isPalListKind(x1)  =  a__isPalListKind(x1)
isPalListKind(x1)  =  isPalListKind(x1)
and(x1, x2)  =  x2
isPal(x1)  =  isPal(x1)
a__isPal(x1)  =  a__isPal(x1)
a  =  a
e  =  e
i  =  i
o  =  o
u  =  u
U11(x1, x2)  =  U11(x1, x2)
isNeList(x1)  =  isNeList(x1)
U21(x1, x2, x3)  =  U21(x3)
U22(x1, x2)  =  U22(x2)
isList(x1)  =  isList(x1)
isQid(x1)  =  isQid
U41(x1, x2, x3)  =  U41(x3)
U42(x1, x2)  =  U42(x2)
U51(x1, x2, x3)  =  U51(x1, x2, x3)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2)  =  x1
U71(x1, x2)  =  U71(x1, x2)
isNePal(x1)  =  isNePal(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[a₂, ₂] > [tt, a_isQid, a, isQid] > [a_U211, a_U221, U21₁, U22₁] > [a_isList1, isList₁] > [a_U112, U11₂] > [U12₁, a_U121]
[a₂, ₂] > [tt, a_isQid, a, isQid] > [a_U211, a_U221, U21₁, U22₁] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [U31₂, a_U312]
[a₂, ₂] > [tt, a_isQid, a, isQid] > [a_U211, a_U221, U21₁, U22₁] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [a_{isPalListKind1}, isPalListKind₁]
[a₂, ₂] > [tt, a_isQid, a, isQid] > [a_U522, U52₂] > [a_isList1, isList₁] > [a_U112, U11₂] > [U12₁, a_U121]
[a₂, ₂] > [tt, a_isQid, a, isQid] > [a_U522, U52₂] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [U31₂, a_U312]
[a₂, ₂] > [tt, a_isQid, a, isQid] > [a_U522, U52₂] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [a_{isPalListKind1}, isPalListKind₁]
[a₂, ₂] > [tt, a_isQid, a, isQid] > [a_isNePal1, isNePal₁] > [a_{isPalListKind1}, isPalListKind₁]
[a₂, ₂] > [a_U411, a_U421, U41₁, U42₁] > [a_isList1, isList₁] > [a_U112, U11₂] > [U12₁, a_U121]
[a₂, ₂] > [a_U411, a_U421, U41₁, U42₁] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [U31₂, a_U312]
[a₂, ₂] > [a_U411, a_U421, U41₁, U42₁] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [a_{isPalListKind1}, isPalListKind₁]
[a₂, ₂] > [a_U513, U51₃] > [a_U522, U52₂] > [a_isList1, isList₁] > [a_U112, U11₂] > [U12₁, a_U121]
[a₂, ₂] > [a_U513, U51₃] > [a_U522, U52₂] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [U31₂, a_U312]
[a₂, ₂] > [a_U513, U51₃] > [a_U522, U52₂] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [a_{isPalListKind1}, isPalListKind₁]
nil > [tt, a_isQid, a, isQid] > [a_U211, a_U221, U21₁, U22₁] > [a_isList1, isList₁] > [a_U112, U11₂] > [U12₁, a_U121]
nil > [tt, a_isQid, a, isQid] > [a_U211, a_U221, U21₁, U22₁] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [U31₂, a_U312]
nil > [tt, a_isQid, a, isQid] > [a_U211, a_U221, U21₁, U22₁] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [a_{isPalListKind1}, isPalListKind₁]
nil > [tt, a_isQid, a, isQid] > [a_U522, U52₂] > [a_isList1, isList₁] > [a_U112, U11₂] > [U12₁, a_U121]
nil > [tt, a_isQid, a, isQid] > [a_U522, U52₂] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [U31₂, a_U312]
nil > [tt, a_isQid, a, isQid] > [a_U522, U52₂] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [a_{isPalListKind1}, isPalListKind₁]
nil > [tt, a_isQid, a, isQid] > [a_isNePal1, isNePal₁] > [a_{isPalListKind1}, isPalListKind₁]
[isPal₁, a_isPal1] > [tt, a_isQid, a, isQid] > [a_U211, a_U221, U21₁, U22₁] > [a_isList1, isList₁] > [a_U112, U11₂] > [U12₁, a_U121]
[isPal₁, a_isPal1] > [tt, a_isQid, a, isQid] > [a_U211, a_U221, U21₁, U22₁] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [U31₂, a_U312]
[isPal₁, a_isPal1] > [tt, a_isQid, a, isQid] > [a_U211, a_U221, U21₁, U22₁] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [a_{isPalListKind1}, isPalListKind₁]
[isPal₁, a_isPal1] > [tt, a_isQid, a, isQid] > [a_U522, U52₂] > [a_isList1, isList₁] > [a_U112, U11₂] > [U12₁, a_U121]
[isPal₁, a_isPal1] > [tt, a_isQid, a, isQid] > [a_U522, U52₂] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [U31₂, a_U312]
[isPal₁, a_isPal1] > [tt, a_isQid, a, isQid] > [a_U522, U52₂] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [a_{isPalListKind1}, isPalListKind₁]
[isPal₁, a_isPal1] > [tt, a_isQid, a, isQid] > [a_isNePal1, isNePal₁] > [a_{isPalListKind1}, isPalListKind₁]
[isPal₁, a_isPal1] > [a_U712, U71₂] > [a_isNePal1, isNePal₁] > [a_{isPalListKind1}, isPalListKind₁]
e > [tt, a_isQid, a, isQid] > [a_U211, a_U221, U21₁, U22₁] > [a_isList1, isList₁] > [a_U112, U11₂] > [U12₁, a_U121]
e > [tt, a_isQid, a, isQid] > [a_U211, a_U221, U21₁, U22₁] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [U31₂, a_U312]
e > [tt, a_isQid, a, isQid] > [a_U211, a_U221, U21₁, U22₁] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [a_{isPalListKind1}, isPalListKind₁]
e > [tt, a_isQid, a, isQid] > [a_U522, U52₂] > [a_isList1, isList₁] > [a_U112, U11₂] > [U12₁, a_U121]
e > [tt, a_isQid, a, isQid] > [a_U522, U52₂] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [U31₂, a_U312]
e > [tt, a_isQid, a, isQid] > [a_U522, U52₂] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [a_{isPalListKind1}, isPalListKind₁]
e > [tt, a_isQid, a, isQid] > [a_isNePal1, isNePal₁] > [a_{isPalListKind1}, isPalListKind₁]
i > [tt, a_isQid, a, isQid] > [a_U211, a_U221, U21₁, U22₁] > [a_isList1, isList₁] > [a_U112, U11₂] > [U12₁, a_U121]
i > [tt, a_isQid, a, isQid] > [a_U211, a_U221, U21₁, U22₁] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [U31₂, a_U312]
i > [tt, a_isQid, a, isQid] > [a_U211, a_U221, U21₁, U22₁] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [a_{isPalListKind1}, isPalListKind₁]
i > [tt, a_isQid, a, isQid] > [a_U522, U52₂] > [a_isList1, isList₁] > [a_U112, U11₂] > [U12₁, a_U121]
i > [tt, a_isQid, a, isQid] > [a_U522, U52₂] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [U31₂, a_U312]
i > [tt, a_isQid, a, isQid] > [a_U522, U52₂] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [a_{isPalListKind1}, isPalListKind₁]
i > [tt, a_isQid, a, isQid] > [a_isNePal1, isNePal₁] > [a_{isPalListKind1}, isPalListKind₁]
o > [tt, a_isQid, a, isQid] > [a_U211, a_U221, U21₁, U22₁] > [a_isList1, isList₁] > [a_U112, U11₂] > [U12₁, a_U121]
o > [tt, a_isQid, a, isQid] > [a_U211, a_U221, U21₁, U22₁] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [U31₂, a_U312]
o > [tt, a_isQid, a, isQid] > [a_U211, a_U221, U21₁, U22₁] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [a_{isPalListKind1}, isPalListKind₁]
o > [tt, a_isQid, a, isQid] > [a_U522, U52₂] > [a_isList1, isList₁] > [a_U112, U11₂] > [U12₁, a_U121]
o > [tt, a_isQid, a, isQid] > [a_U522, U52₂] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [U31₂, a_U312]
o > [tt, a_isQid, a, isQid] > [a_U522, U52₂] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [a_{isPalListKind1}, isPalListKind₁]
o > [tt, a_isQid, a, isQid] > [a_isNePal1, isNePal₁] > [a_{isPalListKind1}, isPalListKind₁]
u > [tt, a_isQid, a, isQid] > [a_U211, a_U221, U21₁, U22₁] > [a_isList1, isList₁] > [a_U112, U11₂] > [U12₁, a_U121]
u > [tt, a_isQid, a, isQid] > [a_U211, a_U221, U21₁, U22₁] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [U31₂, a_U312]
u > [tt, a_isQid, a, isQid] > [a_U211, a_U221, U21₁, U22₁] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [a_{isPalListKind1}, isPalListKind₁]
u > [tt, a_isQid, a, isQid] > [a_U522, U52₂] > [a_isList1, isList₁] > [a_U112, U11₂] > [U12₁, a_U121]
u > [tt, a_isQid, a, isQid] > [a_U522, U52₂] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [U31₂, a_U312]
u > [tt, a_isQid, a, isQid] > [a_U522, U52₂] > [a_isList1, isList₁] > [a_U112, U11₂] > [a_isNeList1, isNeList₁] > [a_{isPalListKind1}, isPalListKind₁]
u > [tt, a_isQid, a, isQid] > [a_isNePal1, isNePal₁] > [a_{isPalListKind1}, isPalListKind₁]

Status:

MARK₁: [1]
U12₁: [1]
U31₂: multiset
a₂: [1,2]
_₂: [1,2]
nil: multiset
a_U112: multiset
tt: multiset
a_U121: [1]
a_isNeList1: multiset
a_U211: multiset
a_U221: multiset
a_isList1: [1]
a_U312: multiset
a_isQid: multiset
a_U411: [1]
a_U421: [1]
a_U513: [2,1,3]
a_U522: [1,2]
a_U712: [1,2]
a_isNePal1: multiset
a_{isPalListKind1}: [1]
isPalListKind₁: [1]
isPal₁: [1]
a_isPal1: [1]
a: multiset
e: multiset
i: multiset
o: multiset
u: multiset
U11₂: multiset
isNeList₁: multiset
U21₁: multiset
U22₁: multiset
isList₁: [1]
isQid: multiset
U41₁: [1]
U42₁: [1]
U51₃: [2,1,3]
U52₂: [1,2]
U71₂: [1,2]
isNePal₁: multiset

The following usable rules [FROCOS05] were oriented:

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)

(10) Obligation:

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

MARK(U23(X)) → MARK(X)
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
MARK(U53(X)) → MARK(X)
MARK(U62(X)) → MARK(X)
MARK(U72(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.

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U62(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  MARK(x1)
U23(x1)  =  x1
U32(x1)  =  x1
U43(x1)  =  x1
U53(x1)  =  x1
U62(x1)  =  U62(x1)
U72(x1)  =  x1
a____(x1, x2)  =  a____(x1, x2)
__(x1, x2)  =  __(x1, x2)
mark(x1)  =  x1
nil  =  nil
a__U11(x1, x2)  =  a__U11(x1, x2)
tt  =  tt
a__U12(x1)  =  a__U12(x1)
a__isNeList(x1)  =  x1
a__U21(x1, x2, x3)  =  a__U21(x2, x3)
a__U22(x1, x2)  =  a__U22(x1, x2)
a__isList(x1)  =  a__isList(x1)
a__U23(x1)  =  x1
a__U31(x1, x2)  =  a__U31
a__U32(x1)  =  x1
a__isQid(x1)  =  a__isQid
a__U41(x1, x2, x3)  =  a__U41(x1, x2, x3)
a__U42(x1, x2)  =  a__U42(x2)
a__U43(x1)  =  x1
a__U51(x1, x2, x3)  =  a__U51(x1, x2, x3)
a__U52(x1, x2)  =  a__U52(x2)
a__U53(x1)  =  x1
a__U61(x1, x2)  =  a__U61(x2)
a__U62(x1)  =  a__U62(x1)
a__U71(x1, x2)  =  a__U71(x1, x2)
a__U72(x1)  =  x1
a__isNePal(x1)  =  a__isNePal(x1)
a__and(x1, x2)  =  x2
a__isPalListKind(x1)  =  a__isPalListKind(x1)
isPalListKind(x1)  =  isPalListKind(x1)
and(x1, x2)  =  x2
isPal(x1)  =  isPal(x1)
a__isPal(x1)  =  a__isPal(x1)
a  =  a
e  =  e
i  =  i
o  =  o
u  =  u
U11(x1, x2)  =  U11(x1, x2)
U12(x1)  =  U12(x1)
isNeList(x1)  =  x1
U21(x1, x2, x3)  =  U21(x2, x3)
U22(x1, x2)  =  U22(x1, x2)
isList(x1)  =  isList(x1)
U31(x1, x2)  =  U31
isQid(x1)  =  isQid
U41(x1, x2, x3)  =  U41(x1, x2, x3)
U42(x1, x2)  =  U42(x2)
U51(x1, x2, x3)  =  U51(x1, x2, x3)
U52(x1, x2)  =  U52(x2)
U61(x1, x2)  =  U61(x2)
U71(x1, x2)  =  U71(x1, x2)
isNePal(x1)  =  isNePal(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

MARK₁ > [nil, a_U112, tt, a_U121, a_U31, a_isQid, a_{isPalListKind1}, isPalListKind₁, a, o, U11₂, U12₁, U31, isQid]
[a₂, ₂] > [a_U212, U21₂] > [a_U222, a_isList1, a_U521, U22₂, isList₁, U52₁] > [nil, a_U112, tt, a_U121, a_U31, a_isQid, a_{isPalListKind1}, isPalListKind₁, a, o, U11₂, U12₁, U31, isQid]
[a₂, ₂] > [a_U413, U41₃] > [a_U222, a_isList1, a_U521, U22₂, isList₁, U52₁] > [nil, a_U112, tt, a_U121, a_U31, a_isQid, a_{isPalListKind1}, isPalListKind₁, a, o, U11₂, U12₁, U31, isQid]
[a₂, ₂] > [a_U413, U41₃] > [a_U421, U42₁] > [nil, a_U112, tt, a_U121, a_U31, a_isQid, a_{isPalListKind1}, isPalListKind₁, a, o, U11₂, U12₁, U31, isQid]
[a₂, ₂] > [a_U513, U51₃] > [a_U222, a_isList1, a_U521, U22₂, isList₁, U52₁] > [nil, a_U112, tt, a_U121, a_U31, a_isQid, a_{isPalListKind1}, isPalListKind₁, a, o, U11₂, U12₁, U31, isQid]
[isPal₁, a_isPal1] > [a_U712, U71₂] > [a_U611, a_isNePal1, U61₁, isNePal₁] > [U62₁, a_U621] > [nil, a_U112, tt, a_U121, a_U31, a_isQid, a_{isPalListKind1}, isPalListKind₁, a, o, U11₂, U12₁, U31, isQid]
e > [nil, a_U112, tt, a_U121, a_U31, a_isQid, a_{isPalListKind1}, isPalListKind₁, a, o, U11₂, U12₁, U31, isQid]
i > [nil, a_U112, tt, a_U121, a_U31, a_isQid, a_{isPalListKind1}, isPalListKind₁, a, o, U11₂, U12₁, U31, isQid]
u > [nil, a_U112, tt, a_U121, a_U31, a_isQid, a_{isPalListKind1}, isPalListKind₁, a, o, U11₂, U12₁, U31, isQid]

Status:

MARK₁: [1]
U62₁: multiset
a₂: [1,2]
_₂: [1,2]
nil: multiset
a_U112: multiset
tt: multiset
a_U121: multiset
a_U212: multiset
a_U222: multiset
a_isList1: multiset
a_U31: []
a_isQid: []
a_U413: multiset
a_U421: [1]
a_U513: multiset
a_U521: multiset
a_U611: [1]
a_U621: multiset
a_U712: [1,2]
a_isNePal1: [1]
a_{isPalListKind1}: [1]
isPalListKind₁: [1]
isPal₁: multiset
a_isPal1: multiset
a: multiset
e: multiset
i: multiset
o: multiset
u: multiset
U11₂: multiset
U12₁: multiset
U21₂: multiset
U22₂: multiset
isList₁: multiset
U31: []
isQid: []
U41₃: multiset
U42₁: [1]
U51₃: multiset
U52₁: multiset
U61₁: [1]
U71₂: [1,2]
isNePal₁: [1]

The following usable rules [FROCOS05] were oriented:

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)

(12) Obligation:

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

MARK(U23(X)) → MARK(X)
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
MARK(U53(X)) → MARK(X)
MARK(U72(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.

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U43(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  MARK(x1)
U23(x1)  =  x1
U32(x1)  =  x1
U43(x1)  =  U43(x1)
U53(x1)  =  x1
U72(x1)  =  x1
a____(x1, x2)  =  a____(x1, x2)
__(x1, x2)  =  __(x1, x2)
mark(x1)  =  x1
nil  =  nil
a__U11(x1, x2)  =  x1
tt  =  tt
a__U12(x1)  =  a__U12
a__isNeList(x1)  =  x1
a__U21(x1, x2, x3)  =  a__U21(x1, x2, x3)
a__U22(x1, x2)  =  a__U22(x1, x2)
a__isList(x1)  =  a__isList(x1)
a__U23(x1)  =  x1
a__U31(x1, x2)  =  x2
a__U32(x1)  =  x1
a__isQid(x1)  =  x1
a__U41(x1, x2, x3)  =  a__U41(x2, x3)
a__U42(x1, x2)  =  a__U42(x1, x2)
a__U43(x1)  =  a__U43(x1)
a__U51(x1, x2, x3)  =  a__U51(x1, x2, x3)
a__U52(x1, x2)  =  a__U52(x1, x2)
a__U53(x1)  =  x1
a__U61(x1, x2)  =  x2
a__U62(x1)  =  x1
a__U71(x1, x2)  =  x2
a__U72(x1)  =  x1
a__isNePal(x1)  =  x1
a__and(x1, x2)  =  x2
a__isPalListKind(x1)  =  a__isPalListKind
isPalListKind(x1)  =  isPalListKind
and(x1, x2)  =  x2
isPal(x1)  =  isPal(x1)
a__isPal(x1)  =  a__isPal(x1)
a  =  a
e  =  e
i  =  i
o  =  o
u  =  u
U11(x1, x2)  =  x1
U12(x1)  =  U12
isNeList(x1)  =  x1
U21(x1, x2, x3)  =  U21(x1, x2, x3)
U22(x1, x2)  =  U22(x1, x2)
isList(x1)  =  isList(x1)
U31(x1, x2)  =  x2
isQid(x1)  =  x1
U41(x1, x2, x3)  =  U41(x2, x3)
U42(x1, x2)  =  U42(x1, x2)
U51(x1, x2, x3)  =  U51(x1, x2, x3)
U52(x1, x2)  =  U52(x1, x2)
U61(x1, x2)  =  x2
U62(x1)  =  x1
U71(x1, x2)  =  x2
isNePal(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:

[a₂, ₂, a_U412, isPal₁, a_isPal1, U41₂] > [a_U213, U21₃] > [a_U222, U22₂] > [a_isList1, isList₁] > [a_{isPalListKind}, isPalListKind] > [tt, a_U12, U12]
[a₂, ₂, a_U412, isPal₁, a_isPal1, U41₂] > [a_U422, U42₂] > [MARK₁, U43₁, a_U431] > [tt, a_U12, U12]
[a₂, ₂, a_U412, isPal₁, a_isPal1, U41₂] > [a_U513, a_U522, U51₃, U52₂] > [a_isList1, isList₁] > [a_{isPalListKind}, isPalListKind] > [tt, a_U12, U12]
a > [tt, a_U12, U12]
e > [tt, a_U12, U12]
i > [tt, a_U12, U12]
o > [tt, a_U12, U12]
u > [tt, a_U12, U12]

Status:

MARK₁: [1]
U43₁: multiset
a₂: [1,2]
_₂: [1,2]
nil: multiset
tt: multiset
a_U12: []
a_U213: multiset
a_U222: multiset
a_isList1: [1]
a_U412: [1,2]
a_U422: multiset
a_U431: multiset
a_U513: multiset
a_U522: multiset
a_{isPalListKind}: multiset
isPalListKind: multiset
isPal₁: multiset
a_isPal1: multiset
a: multiset
e: multiset
i: multiset
o: multiset
u: multiset
U12: []
U21₃: multiset
U22₂: multiset
isList₁: [1]
U41₂: [1,2]
U42₂: multiset
U51₃: multiset
U52₂: multiset

The following usable rules [FROCOS05] were oriented:

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)

(14) Obligation:

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

MARK(U23(X)) → MARK(X)
MARK(U32(X)) → MARK(X)
MARK(U53(X)) → MARK(X)
MARK(U72(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.

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U72(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  x1
U23(x1)  =  x1
U32(x1)  =  x1
U53(x1)  =  x1
U72(x1)  =  U72(x1)
a____(x1, x2)  =  a____(x1, x2)
__(x1, x2)  =  __(x1, x2)
mark(x1)  =  x1
nil  =  nil
a__U11(x1, x2)  =  x1
tt  =  tt
a__U12(x1)  =  a__U12
a__isNeList(x1)  =  a__isNeList
a__U21(x1, x2, x3)  =  a__U21
a__U22(x1, x2)  =  a__U22
a__isList(x1)  =  a__isList
a__U23(x1)  =  x1
a__U31(x1, x2)  =  a__U31
a__U32(x1)  =  x1
a__isQid(x1)  =  a__isQid
a__U41(x1, x2, x3)  =  a__U41
a__U42(x1, x2)  =  a__U42
a__U43(x1)  =  a__U43
a__U51(x1, x2, x3)  =  a__U51
a__U52(x1, x2)  =  a__U52
a__U53(x1)  =  x1
a__U61(x1, x2)  =  x1
a__U62(x1)  =  a__U62
a__U71(x1, x2)  =  a__U71(x1, x2)
a__U72(x1)  =  a__U72(x1)
a__isNePal(x1)  =  a__isNePal(x1)
a__and(x1, x2)  =  x2
a__isPalListKind(x1)  =  a__isPalListKind
isPalListKind(x1)  =  isPalListKind
and(x1, x2)  =  x2
isPal(x1)  =  isPal(x1)
a__isPal(x1)  =  a__isPal(x1)
a  =  a
e  =  e
i  =  i
o  =  o
u  =  u
U11(x1, x2)  =  x1
U12(x1)  =  U12
isNeList(x1)  =  isNeList
U21(x1, x2, x3)  =  U21
U22(x1, x2)  =  U22
isList(x1)  =  isList
U31(x1, x2)  =  U31
isQid(x1)  =  isQid
U41(x1, x2, x3)  =  U41
U42(x1, x2)  =  U42
U43(x1)  =  U43
U51(x1, x2, x3)  =  U51
U52(x1, x2)  =  U52
U61(x1, x2)  =  x1
U62(x1)  =  U62
U71(x1, x2)  =  U71(x1, x2)
isNePal(x1)  =  isNePal(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[a₂, ₂]
nil > [tt, a_U12, a_U42, a_U43, a_U62, a, u, U12, U42, U43, U62]
[isPal₁, a_isPal1] > [a_U712, a_isNePal1, U71₂, isNePal₁] > [U72₁, a_U721] > [tt, a_U12, a_U42, a_U43, a_U62, a, u, U12, U42, U43, U62]
[isPal₁, a_isPal1] > [a_U712, a_isNePal1, U71₂, isNePal₁] > [a_isNeList, a_U21, a_U22, a_isList, a_U31, a_isQid, a_U51, a_U52, a_{isPalListKind}, isPalListKind, isNeList, U21, U22, isList, U31, isQid, U51, U52] > [a_U41, U41] > [tt, a_U12, a_U42, a_U43, a_U62, a, u, U12, U42, U43, U62]
e > [tt, a_U12, a_U42, a_U43, a_U62, a, u, U12, U42, U43, U62]
i > [tt, a_U12, a_U42, a_U43, a_U62, a, u, U12, U42, U43, U62]

Status:

U72₁: multiset
a₂: [1,2]
_₂: [1,2]
nil: multiset
tt: multiset
a_U12: []
a_isNeList: multiset
a_U21: multiset
a_U22: multiset
a_isList: multiset
a_U31: multiset
a_isQid: multiset
a_U41: multiset
a_U42: []
a_U43: []
a_U51: multiset
a_U52: multiset
a_U62: []
a_U712: multiset
a_U721: multiset
a_isNePal1: multiset
a_{isPalListKind}: multiset
isPalListKind: multiset
isPal₁: multiset
a_isPal1: multiset
a: multiset
e: multiset
i: multiset
o: multiset
u: multiset
U12: []
isNeList: multiset
U21: multiset
U22: multiset
isList: multiset
U31: multiset
isQid: multiset
U41: multiset
U42: []
U43: []
U51: multiset
U52: multiset
U62: []
U71₂: multiset
isNePal₁: multiset

The following usable rules [FROCOS05] were oriented:

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)

(16) Obligation:

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

MARK(U23(X)) → MARK(X)
MARK(U32(X)) → MARK(X)
MARK(U53(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.

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U32(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  MARK(x1)
U23(x1)  =  x1
U32(x1)  =  U32(x1)
U53(x1)  =  x1
a____(x1, x2)  =  a____(x1, x2)
__(x1, x2)  =  __(x1, x2)
mark(x1)  =  x1
nil  =  nil
a__U11(x1, x2)  =  a__U11
tt  =  tt
a__U12(x1)  =  a__U12
a__isNeList(x1)  =  a__isNeList
a__U21(x1, x2, x3)  =  a__U21
a__U22(x1, x2)  =  x1
a__isList(x1)  =  a__isList
a__U23(x1)  =  x1
a__U31(x1, x2)  =  a__U31
a__U32(x1)  =  a__U32(x1)
a__isQid(x1)  =  a__isQid
a__U41(x1, x2, x3)  =  a__U41
a__U42(x1, x2)  =  a__U42
a__U43(x1)  =  x1
a__U51(x1, x2, x3)  =  x1
a__U52(x1, x2)  =  a__U52
a__U53(x1)  =  x1
a__U61(x1, x2)  =  a__U61
a__U62(x1)  =  x1
a__U71(x1, x2)  =  x1
a__U72(x1)  =  a__U72
a__isNePal(x1)  =  a__isNePal(x1)
a__and(x1, x2)  =  x2
a__isPalListKind(x1)  =  a__isPalListKind
isPalListKind(x1)  =  isPalListKind
and(x1, x2)  =  x2
isPal(x1)  =  isPal
a__isPal(x1)  =  a__isPal
a  =  a
e  =  e
i  =  i
o  =  o
u  =  u
U11(x1, x2)  =  U11
U12(x1)  =  U12
isNeList(x1)  =  isNeList
U21(x1, x2, x3)  =  U21
U22(x1, x2)  =  x1
isList(x1)  =  isList
U31(x1, x2)  =  U31
isQid(x1)  =  isQid
U41(x1, x2, x3)  =  U41
U42(x1, x2)  =  U42
U43(x1)  =  x1
U51(x1, x2, x3)  =  x1
U52(x1, x2)  =  U52
U61(x1, x2)  =  U61
U62(x1)  =  x1
U71(x1, x2)  =  x1
U72(x1)  =  U72
isNePal(x1)  =  isNePal(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[a_isNePal1, isNePal₁] > [a₂, ₂, a_isNeList, a_U31, a_U41, a_U42, a_{isPalListKind}, isPalListKind, isNeList, U31, U41, U42] > [a_U11, tt, a_U12, a_U21, a_isList, a_isQid, a_U52, a_U72, a, U11, U12, U21, isList, isQid, U52, U72] > [MARK₁, U32₁, a_U321]
[a_isNePal1, isNePal₁] > [a_U61, U61] > [a_U11, tt, a_U12, a_U21, a_isList, a_isQid, a_U52, a_U72, a, U11, U12, U21, isList, isQid, U52, U72] > [MARK₁, U32₁, a_U321]
[isPal, a_isPal] > [a₂, ₂, a_isNeList, a_U31, a_U41, a_U42, a_{isPalListKind}, isPalListKind, isNeList, U31, U41, U42] > [a_U11, tt, a_U12, a_U21, a_isList, a_isQid, a_U52, a_U72, a, U11, U12, U21, isList, isQid, U52, U72] > [MARK₁, U32₁, a_U321]
e > [a_U11, tt, a_U12, a_U21, a_isList, a_isQid, a_U52, a_U72, a, U11, U12, U21, isList, isQid, U52, U72] > [MARK₁, U32₁, a_U321]

Status:

MARK₁: multiset
U32₁: multiset
a₂: [1,2]
_₂: [1,2]
nil: multiset
a_U11: multiset
tt: multiset
a_U12: multiset
a_isNeList: multiset
a_U21: multiset
a_isList: multiset
a_U31: multiset
a_U321: multiset
a_isQid: multiset
a_U41: multiset
a_U42: multiset
a_U52: multiset
a_U61: multiset
a_U72: multiset
a_isNePal1: [1]
a_{isPalListKind}: multiset
isPalListKind: multiset
isPal: []
a_isPal: []
a: multiset
e: multiset
i: multiset
o: multiset
u: multiset
U11: multiset
U12: multiset
isNeList: multiset
U21: multiset
isList: multiset
U31: multiset
isQid: multiset
U41: multiset
U42: multiset
U52: multiset
U61: multiset
U72: multiset
isNePal₁: [1]

The following usable rules [FROCOS05] were oriented:

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)

(18) Obligation:

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

MARK(U23(X)) → MARK(X)
MARK(U53(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.

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U53(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  MARK(x1)
U23(x1)  =  x1
U53(x1)  =  U53(x1)
a____(x1, x2)  =  a____(x1, x2)
__(x1, x2)  =  __(x1, x2)
mark(x1)  =  x1
nil  =  nil
a__U11(x1, x2)  =  a__U11
tt  =  tt
a__U12(x1)  =  a__U12
a__isNeList(x1)  =  a__isNeList
a__U21(x1, x2, x3)  =  a__U21
a__U22(x1, x2)  =  a__U22
a__isList(x1)  =  a__isList
a__U23(x1)  =  x1
a__U31(x1, x2)  =  x1
a__U32(x1)  =  x1
a__isQid(x1)  =  a__isQid
a__U41(x1, x2, x3)  =  a__U41
a__U42(x1, x2)  =  a__U42(x1)
a__U43(x1)  =  a__U43
a__U51(x1, x2, x3)  =  a__U51
a__U52(x1, x2)  =  a__U52
a__U53(x1)  =  a__U53(x1)
a__U61(x1, x2)  =  x1
a__U62(x1)  =  x1
a__U71(x1, x2)  =  a__U71(x2)
a__U72(x1)  =  x1
a__isNePal(x1)  =  a__isNePal(x1)
a__and(x1, x2)  =  x2
a__isPalListKind(x1)  =  a__isPalListKind
isPalListKind(x1)  =  isPalListKind
and(x1, x2)  =  x2
isPal(x1)  =  isPal(x1)
a__isPal(x1)  =  a__isPal(x1)
a  =  a
e  =  e
i  =  i
o  =  o
u  =  u
U11(x1, x2)  =  U11
U12(x1)  =  U12
isNeList(x1)  =  isNeList
U21(x1, x2, x3)  =  U21
U22(x1, x2)  =  U22
isList(x1)  =  isList
U31(x1, x2)  =  x1
U32(x1)  =  x1
isQid(x1)  =  isQid
U41(x1, x2, x3)  =  U41
U42(x1, x2)  =  U42(x1)
U43(x1)  =  U43
U51(x1, x2, x3)  =  U51
U52(x1, x2)  =  U52
U61(x1, x2)  =  x1
U62(x1)  =  x1
U71(x1, x2)  =  U71(x2)
U72(x1)  =  x1
isNePal(x1)  =  isNePal(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[a₂, ₂] > [isPal₁, a_isPal1] > [a_U711, U71₁] > [a_isNePal1, a_{isPalListKind}, isPalListKind, isNePal₁] > [tt, a_isQid, e, i, isQid]
[a_isNeList, a_U51, isNeList, U51] > [a_U41, U41] > [a_U21, a_U22, a_isList, U21, U22, isList] > [a_U11, U11] > [a_U12, U12] > [tt, a_isQid, e, i, isQid]
[a_isNeList, a_U51, isNeList, U51] > [a_U41, U41] > [a_U21, a_U22, a_isList, U21, U22, isList] > [a_isNePal1, a_{isPalListKind}, isPalListKind, isNePal₁] > [tt, a_isQid, e, i, isQid]
[a_isNeList, a_U51, isNeList, U51] > [a_U41, U41] > [a_U421, a_U43, U42₁, U43] > [tt, a_isQid, e, i, isQid]
[a_isNeList, a_U51, isNeList, U51] > [a_U52, U52] > [U53₁, a_U531] > MARK₁
[a_isNeList, a_U51, isNeList, U51] > [a_U52, U52] > [U53₁, a_U531] > [tt, a_isQid, e, i, isQid]
[a_isNeList, a_U51, isNeList, U51] > [a_U52, U52] > [a_U21, a_U22, a_isList, U21, U22, isList] > [a_U11, U11] > [a_U12, U12] > [tt, a_isQid, e, i, isQid]
[a_isNeList, a_U51, isNeList, U51] > [a_U52, U52] > [a_U21, a_U22, a_isList, U21, U22, isList] > [a_isNePal1, a_{isPalListKind}, isPalListKind, isNePal₁] > [tt, a_isQid, e, i, isQid]
o > [tt, a_isQid, e, i, isQid]

Status:

MARK₁: [1]
U53₁: [1]
a₂: [1,2]
_₂: [1,2]
nil: multiset
a_U11: []
tt: multiset
a_U12: []
a_isNeList: multiset
a_U21: []
a_U22: []
a_isList: []
a_isQid: multiset
a_U41: multiset
a_U421: [1]
a_U43: []
a_U51: multiset
a_U52: []
a_U531: [1]
a_U711: multiset
a_isNePal1: multiset
a_{isPalListKind}: multiset
isPalListKind: multiset
isPal₁: [1]
a_isPal1: [1]
a: multiset
e: multiset
i: multiset
o: multiset
u: multiset
U11: []
U12: []
isNeList: multiset
U21: []
U22: []
isList: []
isQid: multiset
U41: multiset
U42₁: [1]
U43: []
U51: multiset
U52: []
U71₁: multiset
isNePal₁: multiset

The following usable rules [FROCOS05] were oriented:

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)

(20) Obligation:

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

MARK(U23(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.

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U23(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1)  =  x1
U23(x1)  =  U23(x1)
a____(x1, x2)  =  a____(x1, x2)
__(x1, x2)  =  __(x1, x2)
mark(x1)  =  x1
nil  =  nil
a__U11(x1, x2)  =  x1
tt  =  tt
a__U12(x1)  =  a__U12
a__isNeList(x1)  =  a__isNeList(x1)
a__U21(x1, x2, x3)  =  a__U21(x1, x2, x3)
a__U22(x1, x2)  =  a__U22(x2)
a__isList(x1)  =  x1
a__U23(x1)  =  a__U23(x1)
a__U31(x1, x2)  =  a__U31(x1)
a__U32(x1)  =  x1
a__isQid(x1)  =  a__isQid
a__U41(x1, x2, x3)  =  a__U41(x2, x3)
a__U42(x1, x2)  =  a__U42(x1, x2)
a__U43(x1)  =  a__U43(x1)
a__U51(x1, x2, x3)  =  a__U51(x1, x2, x3)
a__U52(x1, x2)  =  a__U52
a__U53(x1)  =  a__U53
a__U61(x1, x2)  =  a__U61(x1)
a__U62(x1)  =  a__U62(x1)
a__U71(x1, x2)  =  a__U71(x2)
a__U72(x1)  =  a__U72
a__isNePal(x1)  =  a__isNePal(x1)
a__and(x1, x2)  =  a__and(x2)
a__isPalListKind(x1)  =  x1
isPalListKind(x1)  =  x1
and(x1, x2)  =  and(x2)
isPal(x1)  =  isPal(x1)
a__isPal(x1)  =  a__isPal(x1)
a  =  a
e  =  e
i  =  i
o  =  o
u  =  u
U11(x1, x2)  =  x1
U12(x1)  =  U12
isNeList(x1)  =  isNeList(x1)
U21(x1, x2, x3)  =  U21(x1, x2, x3)
U22(x1, x2)  =  U22(x2)
isList(x1)  =  x1
U31(x1, x2)  =  U31(x1)
U32(x1)  =  x1
isQid(x1)  =  isQid
U41(x1, x2, x3)  =  U41(x2, x3)
U42(x1, x2)  =  U42(x1, x2)
U43(x1)  =  U43(x1)
U51(x1, x2, x3)  =  U51(x1, x2, x3)
U52(x1, x2)  =  U52
U53(x1)  =  U53
U61(x1, x2)  =  U61(x1)
U62(x1)  =  U62(x1)
U71(x1, x2)  =  U71(x2)
U72(x1)  =  U72
isNePal(x1)  =  isNePal(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

[a₂, ₂] > [a_U213, U21₃] > [U23₁, a_U221, a_U231, U22₁] > [a_U621, U62₁]
[a₂, ₂] > [a_isQid, isQid] > [nil, tt, a_U12, a, U12] > [U23₁, a_U221, a_U231, U22₁] > [a_U621, U62₁]
[a₂, ₂] > [a_isQid, isQid] > [nil, tt, a_U12, a, U12] > [a_U431, U43₁] > [a_U621, U62₁]
[a₂, ₂] > [a_and1, and₁] > [a_U621, U62₁]
[a_isNeList1, a_U412, a_U422, isNeList₁, U41₂, U42₂] > [a_U311, U31₁] > [a_isQid, isQid] > [nil, tt, a_U12, a, U12] > [U23₁, a_U221, a_U231, U22₁] > [a_U621, U62₁]
[a_isNeList1, a_U412, a_U422, isNeList₁, U41₂, U42₂] > [a_U311, U31₁] > [a_isQid, isQid] > [nil, tt, a_U12, a, U12] > [a_U431, U43₁] > [a_U621, U62₁]
[a_isNeList1, a_U412, a_U422, isNeList₁, U41₂, U42₂] > [a_U513, U51₃] > [a_U52, U52] > [a_U53, U53] > [nil, tt, a_U12, a, U12] > [U23₁, a_U221, a_U231, U22₁] > [a_U621, U62₁]
[a_isNeList1, a_U412, a_U422, isNeList₁, U41₂, U42₂] > [a_U513, U51₃] > [a_U52, U52] > [a_U53, U53] > [nil, tt, a_U12, a, U12] > [a_U431, U43₁] > [a_U621, U62₁]
[a_isNeList1, a_U412, a_U422, isNeList₁, U41₂, U42₂] > [a_and1, and₁] > [a_U621, U62₁]
[isPal₁, a_isPal1] > [a_U711, a_isNePal1, U71₁, isNePal₁] > [a_U611, U61₁] > [a_isQid, isQid] > [nil, tt, a_U12, a, U12] > [U23₁, a_U221, a_U231, U22₁] > [a_U621, U62₁]
[isPal₁, a_isPal1] > [a_U711, a_isNePal1, U71₁, isNePal₁] > [a_U611, U61₁] > [a_isQid, isQid] > [nil, tt, a_U12, a, U12] > [a_U431, U43₁] > [a_U621, U62₁]
[isPal₁, a_isPal1] > [a_U711, a_isNePal1, U71₁, isNePal₁] > [a_U72, U72] > [nil, tt, a_U12, a, U12] > [U23₁, a_U221, a_U231, U22₁] > [a_U621, U62₁]
[isPal₁, a_isPal1] > [a_U711, a_isNePal1, U71₁, isNePal₁] > [a_U72, U72] > [nil, tt, a_U12, a, U12] > [a_U431, U43₁] > [a_U621, U62₁]
[isPal₁, a_isPal1] > [a_U711, a_isNePal1, U71₁, isNePal₁] > [a_and1, and₁] > [a_U621, U62₁]
e > [nil, tt, a_U12, a, U12] > [U23₁, a_U221, a_U231, U22₁] > [a_U621, U62₁]
e > [nil, tt, a_U12, a, U12] > [a_U431, U43₁] > [a_U621, U62₁]
i > [nil, tt, a_U12, a, U12] > [U23₁, a_U221, a_U231, U22₁] > [a_U621, U62₁]
i > [nil, tt, a_U12, a, U12] > [a_U431, U43₁] > [a_U621, U62₁]
o > [nil, tt, a_U12, a, U12] > [U23₁, a_U221, a_U231, U22₁] > [a_U621, U62₁]
o > [nil, tt, a_U12, a, U12] > [a_U431, U43₁] > [a_U621, U62₁]
u > [nil, tt, a_U12, a, U12] > [U23₁, a_U221, a_U231, U22₁] > [a_U621, U62₁]
u > [nil, tt, a_U12, a, U12] > [a_U431, U43₁] > [a_U621, U62₁]

Status:

U23₁: multiset
a₂: [1,2]
_₂: [1,2]
nil: multiset
tt: multiset
a_U12: multiset
a_isNeList1: multiset
a_U213: multiset
a_U221: multiset
a_U231: multiset
a_U311: [1]
a_isQid: multiset
a_U412: multiset
a_U422: multiset
a_U431: [1]
a_U513: multiset
a_U52: []
a_U53: []
a_U611: [1]
a_U621: [1]
a_U711: [1]
a_U72: []
a_isNePal1: multiset
a_and1: [1]
and₁: [1]
isPal₁: multiset
a_isPal1: multiset
a: multiset
e: multiset
i: multiset
o: multiset
u: multiset
U12: multiset
isNeList₁: multiset
U21₃: multiset
U22₁: multiset
U31₁: [1]
isQid: multiset
U41₂: multiset
U42₂: multiset
U43₁: [1]
U51₃: multiset
U52: []
U53: []
U61₁: [1]
U62₁: [1]
U71₁: [1]
U72: []
isNePal₁: multiset

The following usable rules [FROCOS05] were oriented:

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)

(22) Obligation:

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.

(23) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(24) TRUE