Termination w.r.t. Q proof of /tmp/tmphrFA7g/PALINDROME_complete

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

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 QDP

↳5 QDPOrderProof (⇔)

↳6 QDP

↳7 DependencyGraphProof (⇔)

↳8 QDP

↳9 QDPOrderProof (⇔)

↳10 QDP

↳11 QDPOrderProof (⇔)

↳12 QDP

↳13 QDPOrderProof (⇔)

↳14 QDP

↳15 PisEmptyProof (⇔)

↳16 TRUE

(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__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(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__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(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(U71(X1, X2)) → A__U71(mark(X1), X2)
A__U71(tt, V) → A__ISNEPAL(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(x2)
tt = tt
A__ISNELIST(x1) = A__ISNELIST(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) = isNeList(x1)
A__U41(x1, x2, x3) = A__U41(x2, x3)
a__and(x1, x2) = a__and(x1, x2)
A__U42(x1, x2) = A__U42(x2)
a__isList(x1) = a__isList(x1)
A__U51(x1, x2, x3) = A__U51(x2, x3)
A__U52(x1, x2) = A__U52(x2)
a__isNeList(x1) = a__isNeList(x1)
A__ISLIST(x1) = A__ISLIST(x1)
A__U21(x1, x2, x3) = A__U21(x2, x3)
A__U22(x1, x2) = A__U22(x2)
U21(x1, x2, x3) = U21(x1, x2, x3)
U22(x1, x2) = U22(x1, x2)
isList(x1) = isList(x1)
U23(x1) = U23(x1)
U31(x1, x2) = U31(x1, x2)
U32(x1) = U32(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) = U53(x1)
U61(x1, x2) = U61(x1, x2)
U62(x1) = x1
U71(x1, x2) = U71(x1, x2)
A__U71(x1, x2) = A__U71(x2)
A__ISNEPAL(x1) = 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__isNePal(x1) = a__isNePal(x1)
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) = a__U23(x1)
a__U31(x1, x2) = a__U31(x1, x2)
a__U32(x1) = a__U32(x1)
isQid(x1) = isQid
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) = a__U53(x1)
a__U61(x1, x2) = a__U61(x1, x2)
a__U62(x1) = x1
a__U71(x1, x2) = a__U71(x1, x2)
a__U72(x1) = x1
a__isPal(x1) = a__isPal(x1)
a = a
e = e
i = i
o = o
u = u

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

nil > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > [A_{_U41₂}, A_{_U42₁}] > [a_{_isList₁}, isList₁] > [U11₂, isNeList₁, a_{_isNeList₁}, a_{_U11₂}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
nil > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > [A_{_U41₂}, A_{_U42₁}] > [A_{_ISLIST₁}, A_{_U22₁}] > [A_{_U11₁}, A_{_ISNELIST₁}] > [a_{_and₂}, and₂] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
nil > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > A_{_U51₂} > [U11₂, isNeList₁, a_{_isNeList₁}, a_{_U11₂}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
nil > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > A_{_U51₂} > A_{_U52₁} > [A_{_ISLIST₁}, A_{_U22₁}] > [A_{_U11₁}, A_{_ISNELIST₁}] > [a_{_and₂}, and₂] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
nil > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > A_{_U21₂} > [a_{_isList₁}, isList₁] > [U11₂, isNeList₁, a_{_isNeList₁}, a_{_U11₂}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
nil > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > A_{_U21₂} > [A_{_ISLIST₁}, A_{_U22₁}] > [A_{_U11₁}, A_{_ISNELIST₁}] > [a_{_and₂}, and₂] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
nil > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > [U22₂, a_{_U22₂}] > [a_{_isList₁}, isList₁] > [U11₂, isNeList₁, a_{_isNeList₁}, a_{_U11₂}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
nil > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > [U22₂, a_{_U22₂}] > [U23₁, a_{_U23₁}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
nil > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > [U42₂, a_{_U42₂}] > [U11₂, isNeList₁, a_{_isNeList₁}, a_{_U11₂}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
nil > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > [U52₂, a_{_U52₂}] > [a_{_isList₁}, isList₁] > [U11₂, isNeList₁, a_{_isNeList₁}, a_{_U11₂}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
nil > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > [U52₂, a_{_U52₂}] > A_{_U52₁} > [A_{_ISLIST₁}, A_{_U22₁}] > [A_{_U11₁}, A_{_ISNELIST₁}] > [a_{_and₂}, and₂] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
nil > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > A_{_ISPAL₁} > A_{_U71₁} > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
nil > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [isPal₁, a_{_isPal₁}] > [U71₂, isNePal₁, a_{_isNePal₁}, a_{_U71₂}] > [a_{_and₂}, and₂] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
nil > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [isPal₁, a_{_isPal₁}] > [U71₂, isNePal₁, a_{_isNePal₁}, a_{_U71₂}] > [U61₂, a_{_U61₂}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
e > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > [A_{_U41₂}, A_{_U42₁}] > [a_{_isList₁}, isList₁] > [U11₂, isNeList₁, a_{_isNeList₁}, a_{_U11₂}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
e > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > [A_{_U41₂}, A_{_U42₁}] > [A_{_ISLIST₁}, A_{_U22₁}] > [A_{_U11₁}, A_{_ISNELIST₁}] > [a_{_and₂}, and₂] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
e > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > A_{_U51₂} > [U11₂, isNeList₁, a_{_isNeList₁}, a_{_U11₂}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
e > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > A_{_U51₂} > A_{_U52₁} > [A_{_ISLIST₁}, A_{_U22₁}] > [A_{_U11₁}, A_{_ISNELIST₁}] > [a_{_and₂}, and₂] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
e > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > A_{_U21₂} > [a_{_isList₁}, isList₁] > [U11₂, isNeList₁, a_{_isNeList₁}, a_{_U11₂}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
e > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > A_{_U21₂} > [A_{_ISLIST₁}, A_{_U22₁}] > [A_{_U11₁}, A_{_ISNELIST₁}] > [a_{_and₂}, and₂] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
e > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > [U22₂, a_{_U22₂}] > [a_{_isList₁}, isList₁] > [U11₂, isNeList₁, a_{_isNeList₁}, a_{_U11₂}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
e > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > [U22₂, a_{_U22₂}] > [U23₁, a_{_U23₁}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
e > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > [U42₂, a_{_U42₂}] > [U11₂, isNeList₁, a_{_isNeList₁}, a_{_U11₂}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
e > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > [U52₂, a_{_U52₂}] > [a_{_isList₁}, isList₁] > [U11₂, isNeList₁, a_{_isNeList₁}, a_{_U11₂}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
e > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > [U52₂, a_{_U52₂}] > A_{_U52₁} > [A_{_ISLIST₁}, A_{_U22₁}] > [A_{_U11₁}, A_{_ISNELIST₁}] > [a_{_and₂}, and₂] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
e > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > A_{_ISPAL₁} > A_{_U71₁} > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
e > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [isPal₁, a_{_isPal₁}] > [U71₂, isNePal₁, a_{_isNePal₁}, a_{_U71₂}] > [a_{_and₂}, and₂] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
e > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [isPal₁, a_{_isPal₁}] > [U71₂, isNePal₁, a_{_isNePal₁}, a_{_U71₂}] > [U61₂, a_{_U61₂}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
i > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > [A_{_U41₂}, A_{_U42₁}] > [a_{_isList₁}, isList₁] > [U11₂, isNeList₁, a_{_isNeList₁}, a_{_U11₂}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
i > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > [A_{_U41₂}, A_{_U42₁}] > [A_{_ISLIST₁}, A_{_U22₁}] > [A_{_U11₁}, A_{_ISNELIST₁}] > [a_{_and₂}, and₂] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
i > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > A_{_U51₂} > [U11₂, isNeList₁, a_{_isNeList₁}, a_{_U11₂}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
i > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > A_{_U51₂} > A_{_U52₁} > [A_{_ISLIST₁}, A_{_U22₁}] > [A_{_U11₁}, A_{_ISNELIST₁}] > [a_{_and₂}, and₂] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
i > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > A_{_U21₂} > [a_{_isList₁}, isList₁] > [U11₂, isNeList₁, a_{_isNeList₁}, a_{_U11₂}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
i > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > A_{_U21₂} > [A_{_ISLIST₁}, A_{_U22₁}] > [A_{_U11₁}, A_{_ISNELIST₁}] > [a_{_and₂}, and₂] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
i > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > [U22₂, a_{_U22₂}] > [a_{_isList₁}, isList₁] > [U11₂, isNeList₁, a_{_isNeList₁}, a_{_U11₂}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
i > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > [U22₂, a_{_U22₂}] > [U23₁, a_{_U23₁}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
i > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > [U42₂, a_{_U42₂}] > [U11₂, isNeList₁, a_{_isNeList₁}, a_{_U11₂}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
i > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > [U52₂, a_{_U52₂}] > [a_{_isList₁}, isList₁] > [U11₂, isNeList₁, a_{_isNeList₁}, a_{_U11₂}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
i > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > [U52₂, a_{_U52₂}] > A_{_U52₁} > [A_{_ISLIST₁}, A_{_U22₁}] > [A_{_U11₁}, A_{_ISNELIST₁}] > [a_{_and₂}, and₂] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
i > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > A_{_ISPAL₁} > A_{_U71₁} > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
i > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [isPal₁, a_{_isPal₁}] > [U71₂, isNePal₁, a_{_isNePal₁}, a_{_U71₂}] > [a_{_and₂}, and₂] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
i > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [isPal₁, a_{_isPal₁}] > [U71₂, isNePal₁, a_{_isNePal₁}, a_{_U71₂}] > [U61₂, a_{_U61₂}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
o > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > [A_{_U41₂}, A_{_U42₁}] > [a_{_isList₁}, isList₁] > [U11₂, isNeList₁, a_{_isNeList₁}, a_{_U11₂}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
o > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > [A_{_U41₂}, A_{_U42₁}] > [A_{_ISLIST₁}, A_{_U22₁}] > [A_{_U11₁}, A_{_ISNELIST₁}] > [a_{_and₂}, and₂] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
o > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > A_{_U51₂} > [U11₂, isNeList₁, a_{_isNeList₁}, a_{_U11₂}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
o > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > A_{_U51₂} > A_{_U52₁} > [A_{_ISLIST₁}, A_{_U22₁}] > [A_{_U11₁}, A_{_ISNELIST₁}] > [a_{_and₂}, and₂] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
o > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > A_{_U21₂} > [a_{_isList₁}, isList₁] > [U11₂, isNeList₁, a_{_isNeList₁}, a_{_U11₂}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
o > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > A_{_U21₂} > [A_{_ISLIST₁}, A_{_U22₁}] > [A_{_U11₁}, A_{_ISNELIST₁}] > [a_{_and₂}, and₂] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
o > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > [U22₂, a_{_U22₂}] > [a_{_isList₁}, isList₁] > [U11₂, isNeList₁, a_{_isNeList₁}, a_{_U11₂}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
o > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > [U22₂, a_{_U22₂}] > [U23₁, a_{_U23₁}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
o > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > [U42₂, a_{_U42₂}] > [U11₂, isNeList₁, a_{_isNeList₁}, a_{_U11₂}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
o > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > [U52₂, a_{_U52₂}] > [a_{_isList₁}, isList₁] > [U11₂, isNeList₁, a_{_isNeList₁}, a_{_U11₂}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
o > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > [U52₂, a_{_U52₂}] > A_{_U52₁} > [A_{_ISLIST₁}, A_{_U22₁}] > [A_{_U11₁}, A_{_ISNELIST₁}] > [a_{_and₂}, and₂] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
o > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [MARK₁, A_{_AND₂}, U21₃, U41₃, U51₃, a_{_U21₃}, a_{_U41₃}, a_{_U51₃}] > A_{_ISPAL₁} > A_{_U71₁} > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
o > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [isPal₁, a_{_isPal₁}] > [U71₂, isNePal₁, a_{_isNePal₁}, a_{_U71₂}] > [a_{_and₂}, and₂] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]
o > [A_{_{_{_₂}}}, _{_₂}, a_{_{_{_₂}}}, tt, a_{_isQid}, isQid, a, u] > [isPal₁, a_{_isPal₁}] > [U71₂, isNePal₁, a_{_isNePal₁}, a_{_U71₂}] > [U61₂, a_{_U61₂}] > [U31₂, U32₁, U53₁, a_{_U31₂}, a_{_U32₁}, a_{_U53₁}]

Status:

A_{_{_{_₂}}}: [1,2]
__₂: [1,2]
MARK₁: multiset
a_{_{_{_₂}}}: [1,2]
U11₂: multiset
A_{_U11₁}: multiset
tt: multiset
A_{_ISNELIST₁}: multiset
A_{_AND₂}: multiset
isNeList₁: multiset
A_{_U41₂}: [2,1]
a_{_and₂}: [2,1]
A_{_U42₁}: [1]
a_{_isList₁}: multiset
A_{_U51₂}: multiset
A_{_U52₁}: multiset
a_{_isNeList₁}: multiset
A_{_ISLIST₁}: multiset
A_{_U21₂}: [2,1]
A_{_U22₁}: multiset
U21₃: [1,2,3]
U22₂: [1,2]
isList₁: multiset
U23₁: multiset
U31₂: multiset
U32₁: multiset
U41₃: multiset
U42₂: multiset
U51₃: [1,2,3]
U52₂: multiset
U53₁: multiset
U61₂: [2,1]
U71₂: multiset
A_{_U71₁}: [1]
a_{_isQid}: []
and₂: [2,1]
isPal₁: multiset
isNePal₁: multiset
A_{_ISPAL₁}: [1]
nil: multiset
a_{_isNePal₁}: multiset
a_{_U11₂}: multiset
a_{_U21₃}: [1,2,3]
a_{_U22₂}: [1,2]
a_{_U23₁}: multiset
a_{_U31₂}: multiset
a_{_U32₁}: multiset
isQid: []
a_{_U41₃}: multiset
a_{_U42₂}: multiset
a_{_U51₃}: [1,2,3]
a_{_U52₂}: multiset
a_{_U53₁}: multiset
a_{_U61₂}: [2,1]
a_{_U71₂}: multiset
a_{_isPal₁}: multiset
a: multiset
e: multiset
i: multiset
o: multiset
u: multiset

The following usable rules [FROCOS05] were oriented:

a____(X, nil) → mark(X)
mark(__(X1, X2)) → a____(mark(X1), mark(X2))
a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(nil, X) → mark(X)
mark(isNePal(X)) → a__isNePal(X)
a__isNePal(__(I, __(P, I))) → a__and(a__and(a__isQid(I), isPalListKind(I)), and(isPal(P), isPalListKind(P)))
a__and(tt, X) → mark(X)
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(X)
a__isPalListKind(__(V1, V2)) → a__and(a__isPalListKind(V1), isPalListKind(V2))
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(isPal(X)) → a__isPal(X)
mark(nil) → nil
mark(tt) → tt
mark(a) → a
mark(e) → e
mark(i) → i
mark(o) → o
mark(u) → u
a____(X1, X2) → __(X1, X2)
a__isPalListKind(a) → tt
a__isPalListKind(e) → tt
a__isPalListKind(i) → tt
a__isPalListKind(nil) → tt
a__isPalListKind(o) → tt
a__isPalListKind(u) → tt
a__isPalListKind(X) → isPalListKind(X)
a__and(X1, X2) → and(X1, X2)
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__isList(X) → isList(X)
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__isNeList(X) → isNeList(X)
a__isQid(a) → tt
a__isQid(e) → tt
a__isQid(i) → tt
a__isQid(o) → tt
a__isQid(u) → tt
a__isQid(X) → isQid(X)
a__U11(X1, X2) → U11(X1, X2)
a__U12(tt) → tt
a__U12(X) → U12(X)
a__U31(tt, V) → a__U32(a__isQid(V))
a__U31(X1, X2) → U31(X1, X2)
a__isNePal(X) → isNePal(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2) → U22(X1, X2)
a__U11(tt, V) → a__U12(a__isNeList(V))
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U41(tt, V1, V2) → a__U42(a__isList(V1), V2)
a__U42(X1, X2) → U42(X1, X2)
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__U23(X) → U23(X)
a__U42(tt, V2) → a__U43(a__isNeList(V2))
a__U43(tt) → tt
a__U43(X) → U43(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U51(tt, V1, V2) → a__U52(a__isNeList(V1), V2)
a__U52(X1, X2) → U52(X1, X2)
a__U52(tt, V2) → a__U53(a__isList(V2))
a__U53(tt) → tt
a__U53(X) → U53(X)
a__U32(tt) → tt
a__U32(X) → U32(X)
a__U61(tt, V) → a__U62(a__isQid(V))
a__U61(X1, X2) → U61(X1, X2)
a__U62(tt) → tt
a__U62(X) → U62(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(tt) → tt
a__U72(X) → U72(X)
a__isPal(nil) → tt
a__isPal(X) → isPal(X)
a__isPal(V) → a__U71(a__isPalListKind(V), V)
a__U71(tt, V) → a__U72(a__isNePal(V))
a__isNePal(V) → a__U61(a__isPalListKind(V), V)

(6) Obligation:

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

A__U11(tt, V) → A__ISNELIST(V)
MARK(U12(X)) → MARK(X)
A__U22(tt, V2) → A__ISLIST(V2)
MARK(U43(X)) → MARK(X)
MARK(U62(X)) → MARK(X)
A__ISNEPAL(V) → A__ISPALLISTKIND(V)
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 3 less nodes.

(8) Obligation:

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

MARK(U43(X)) → MARK(X)
MARK(U12(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(U43(X)) → MARK(X)
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)
U43(x1) = U43(x1)
U12(x1) = x1
U62(x1) = U62(x1)
U72(x1) = x1

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

[MARK₁, U43₁, U62₁]

Status:

MARK₁: multiset
U43₁: multiset
U62₁: multiset

The following usable rules [FROCOS05] were oriented: none

(10) Obligation:

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

MARK(U12(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(U12(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
U12(x1) = U12(x1)
U72(x1) = x1

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

trivial

Status:

U12₁: multiset

The following usable rules [FROCOS05] were oriented: none

(12) Obligation:

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

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(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
U72(x1) = U72(x1)

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

trivial

Status:

U72₁: multiset

The following usable rules [FROCOS05] were oriented: none

(14) 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.

(15) PisEmptyProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Obligation:

(5) QDPOrderProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Obligation:

(9) QDPOrderProof (EQUIVALENT transformation)

(10) Obligation:

(11) QDPOrderProof (EQUIVALENT transformation)

(12) Obligation:

(13) QDPOrderProof (EQUIVALENT transformation)

(14) Obligation:

(15) PisEmptyProof (EQUIVALENT transformation)

(16) TRUE