(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__ISPALLISTKIND(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
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(__(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(U31(X1, X2)) → MARK(X1)
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(__(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: SCNP Order with the following components:
Level mapping:
Top level AFS:
A____(x0, x1, x2)  =  A____(x0, x2)
MARK(x0, x1)  =  MARK(x0, x1)
A__U11(x0, x1, x2)  =  A__U11(x0, x2)
A__ISNELIST(x0, x1)  =  A__ISNELIST(x0, x1)
A__ISPALLISTKIND(x0, x1)  =  A__ISPALLISTKIND(x0, x1)
A__AND(x0, x1, x2)  =  A__AND(x1, x2)
A__U41(x0, x1, x2, x3)  =  A__U41(x0, x2)
A__U42(x0, x1, x2)  =  A__U42(x0, x2)
A__U51(x0, x1, x2, x3)  =  A__U51(x0, x1, x2, x3)
A__U52(x0, x1, x2)  =  A__U52(x0, x1)
A__ISLIST(x0, x1)  =  A__ISLIST(x0, x1)
A__U21(x0, x1, x2, x3)  =  A__U21(x0, x2, x3)
A__U22(x0, x1, x2)  =  A__U22(x0, x2)
A__U71(x0, x1, x2)  =  A__U71(x0, x1)
A__ISNEPAL(x0, x1)  =  A__ISNEPAL(x0, x1)
A__ISPAL(x0, x1)  =  A__ISPAL(x0, x1)

Tags:
A____ has argument tags [0,0,16] and root tag 8
MARK has argument tags [32,43] and root tag 0
A__U11 has argument tags [36,6,8] and root tag 10
A__ISNELIST has argument tags [9,8] and root tag 10
A__ISPALLISTKIND has argument tags [0,8] and root tag 10
A__AND has argument tags [15,44,43] and root tag 0
A__U41 has argument tags [20,0,6,15] and root tag 4
A__U42 has argument tags [48,54,7] and root tag 9
A__U51 has argument tags [20,0,7,1] and root tag 0
A__U52 has argument tags [52,27,4] and root tag 6
A__ISLIST has argument tags [36,8] and root tag 10
A__U21 has argument tags [1,0,8,35] and root tag 13
A__U22 has argument tags [1,24,4] and root tag 10
A__U71 has argument tags [35,40,7] and root tag 15
A__ISNEPAL has argument tags [0,8] and root tag 10
A__ISPAL has argument tags [0,0] and root tag 8

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
A____(x1, x2)  =  A____(x1, x2)
__(x1, x2)  =  __(x1, x2)
MARK(x1)  =  MARK
mark(x1)  =  x1
a____(x1, x2)  =  a____(x1, x2)
U11(x1, x2)  =  U11(x1, x2)
A__U11(x1, x2)  =  A__U11
tt  =  tt
A__ISNELIST(x1)  =  A__ISNELIST
A__ISPALLISTKIND(x1)  =  A__ISPALLISTKIND
A__AND(x1, x2)  =  A__AND(x1, x2)
a__isPalListKind(x1)  =  a__isPalListKind(x1)
isPalListKind(x1)  =  isPalListKind(x1)
U12(x1)  =  x1
isNeList(x1)  =  isNeList(x1)
A__U41(x1, x2, x3)  =  A__U41(x1, 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(x1, x2, x3)
A__U52(x1, x2)  =  A__U52(x1, x2)
a__isNeList(x1)  =  a__isNeList(x1)
A__ISLIST(x1)  =  A__ISLIST
A__U21(x1, x2, x3)  =  A__U21(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)  =  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)  =  x1
U61(x1, x2)  =  U61(x1, x2)
U62(x1)  =  x1
U71(x1, x2)  =  U71(x1, x2)
A__U71(x1, x2)  =  A__U71(x1, x2)
A__ISNEPAL(x1)  =  A__ISNEPAL
a__isQid(x1)  =  a__isQid(x1)
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)  =  x1
a__U31(x1, x2)  =  a__U31(x1, x2)
a__U32(x1)  =  a__U32(x1)
isQid(x1)  =  isQid(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, 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:
[A2, 2, a2] > [AU513, U513, aU513] > [U522, aU522] > [AU413, aisList1, AU211, AU221, isList1] > [U112, isNeList1, aisNeList1, aU112] > [aisPalListKind1, isPalListKind1] > [AU11, AISNELIST, AISPALLISTKIND, AISLIST, AISNEPAL] > [MARK, aand2, and2]
[A2, 2, a2] > [AU513, U513, aU513] > [U522, aU522] > [AU413, aisList1, AU211, AU221, isList1] > [U112, isNeList1, aisNeList1, aU112] > [aisPalListKind1, isPalListKind1] > tt > [MARK, aand2, and2]
[A2, 2, a2] > [AU513, U513, aU513] > [U522, aU522] > [AU413, aisList1, AU211, AU221, isList1] > [U112, isNeList1, aisNeList1, aU112] > [aisPalListKind1, isPalListKind1] > tt > [U321, aU321]
[A2, 2, a2] > [AU513, U513, aU513] > [U522, aU522] > [AU413, aisList1, AU211, AU221, isList1] > [U112, isNeList1, aisNeList1, aU112] > [U312, aU312] > [aisQid1, isQid1] > tt > [MARK, aand2, and2]
[A2, 2, a2] > [AU513, U513, aU513] > [U522, aU522] > [AU413, aisList1, AU211, AU221, isList1] > [U112, isNeList1, aisNeList1, aU112] > [U312, aU312] > [aisQid1, isQid1] > tt > [U321, aU321]
[A2, 2, a2] > [AU513, U513, aU513] > [U522, aU522] > [AU413, aisList1, AU211, AU221, isList1] > AU421 > [AU11, AISNELIST, AISPALLISTKIND, AISLIST, AISNEPAL] > [MARK, aand2, and2]
[A2, 2, a2] > [AU513, U513, aU513] > [U522, aU522] > AU522 > [AU11, AISNELIST, AISPALLISTKIND, AISLIST, AISNEPAL] > [MARK, aand2, and2]
[A2, 2, a2] > [U213, aU213] > [U222, aU222] > [AU413, aisList1, AU211, AU221, isList1] > [U112, isNeList1, aisNeList1, aU112] > [aisPalListKind1, isPalListKind1] > [AU11, AISNELIST, AISPALLISTKIND, AISLIST, AISNEPAL] > [MARK, aand2, and2]
[A2, 2, a2] > [U213, aU213] > [U222, aU222] > [AU413, aisList1, AU211, AU221, isList1] > [U112, isNeList1, aisNeList1, aU112] > [aisPalListKind1, isPalListKind1] > tt > [MARK, aand2, and2]
[A2, 2, a2] > [U213, aU213] > [U222, aU222] > [AU413, aisList1, AU211, AU221, isList1] > [U112, isNeList1, aisNeList1, aU112] > [aisPalListKind1, isPalListKind1] > tt > [U321, aU321]
[A2, 2, a2] > [U213, aU213] > [U222, aU222] > [AU413, aisList1, AU211, AU221, isList1] > [U112, isNeList1, aisNeList1, aU112] > [U312, aU312] > [aisQid1, isQid1] > tt > [MARK, aand2, and2]
[A2, 2, a2] > [U213, aU213] > [U222, aU222] > [AU413, aisList1, AU211, AU221, isList1] > [U112, isNeList1, aisNeList1, aU112] > [U312, aU312] > [aisQid1, isQid1] > tt > [U321, aU321]
[A2, 2, a2] > [U213, aU213] > [U222, aU222] > [AU413, aisList1, AU211, AU221, isList1] > AU421 > [AU11, AISNELIST, AISPALLISTKIND, AISLIST, AISNEPAL] > [MARK, aand2, and2]
[A2, 2, a2] > [U413, aU413] > [AU413, aisList1, AU211, AU221, isList1] > [U112, isNeList1, aisNeList1, aU112] > [aisPalListKind1, isPalListKind1] > [AU11, AISNELIST, AISPALLISTKIND, AISLIST, AISNEPAL] > [MARK, aand2, and2]
[A2, 2, a2] > [U413, aU413] > [AU413, aisList1, AU211, AU221, isList1] > [U112, isNeList1, aisNeList1, aU112] > [aisPalListKind1, isPalListKind1] > tt > [MARK, aand2, and2]
[A2, 2, a2] > [U413, aU413] > [AU413, aisList1, AU211, AU221, isList1] > [U112, isNeList1, aisNeList1, aU112] > [aisPalListKind1, isPalListKind1] > tt > [U321, aU321]
[A2, 2, a2] > [U413, aU413] > [AU413, aisList1, AU211, AU221, isList1] > [U112, isNeList1, aisNeList1, aU112] > [U312, aU312] > [aisQid1, isQid1] > tt > [MARK, aand2, and2]
[A2, 2, a2] > [U413, aU413] > [AU413, aisList1, AU211, AU221, isList1] > [U112, isNeList1, aisNeList1, aU112] > [U312, aU312] > [aisQid1, isQid1] > tt > [U321, aU321]
[A2, 2, a2] > [U413, aU413] > [AU413, aisList1, AU211, AU221, isList1] > AU421 > [AU11, AISNELIST, AISPALLISTKIND, AISLIST, AISNEPAL] > [MARK, aand2, and2]
[A2, 2, a2] > [U413, aU413] > [U422, aU422] > [U112, isNeList1, aisNeList1, aU112] > [aisPalListKind1, isPalListKind1] > [AU11, AISNELIST, AISPALLISTKIND, AISLIST, AISNEPAL] > [MARK, aand2, and2]
[A2, 2, a2] > [U413, aU413] > [U422, aU422] > [U112, isNeList1, aisNeList1, aU112] > [aisPalListKind1, isPalListKind1] > tt > [MARK, aand2, and2]
[A2, 2, a2] > [U413, aU413] > [U422, aU422] > [U112, isNeList1, aisNeList1, aU112] > [aisPalListKind1, isPalListKind1] > tt > [U321, aU321]
[A2, 2, a2] > [U413, aU413] > [U422, aU422] > [U112, isNeList1, aisNeList1, aU112] > [U312, aU312] > [aisQid1, isQid1] > tt > [MARK, aand2, and2]
[A2, 2, a2] > [U413, aU413] > [U422, aU422] > [U112, isNeList1, aisNeList1, aU112] > [U312, aU312] > [aisQid1, isQid1] > tt > [U321, aU321]
[A2, 2, a2] > [U413, aU413] > [U422, aU422] > AU421 > [AU11, AISNELIST, AISPALLISTKIND, AISLIST, AISNEPAL] > [MARK, aand2, and2]
[A2, 2, a2] > [isPal1, AISPAL1, aisPal1] > [U712, aU712] > AU712 > [AU11, AISNELIST, AISPALLISTKIND, AISLIST, AISNEPAL] > [MARK, aand2, and2]
[A2, 2, a2] > [isPal1, AISPAL1, aisPal1] > [U712, aU712] > [isNePal1, aisNePal1] > [aisPalListKind1, isPalListKind1] > [AU11, AISNELIST, AISPALLISTKIND, AISLIST, AISNEPAL] > [MARK, aand2, and2]
[A2, 2, a2] > [isPal1, AISPAL1, aisPal1] > [U712, aU712] > [isNePal1, aisNePal1] > [aisPalListKind1, isPalListKind1] > tt > [MARK, aand2, and2]
[A2, 2, a2] > [isPal1, AISPAL1, aisPal1] > [U712, aU712] > [isNePal1, aisNePal1] > [aisPalListKind1, isPalListKind1] > tt > [U321, aU321]
[A2, 2, a2] > [isPal1, AISPAL1, aisPal1] > [U712, aU712] > [isNePal1, aisNePal1] > [U612, aU612] > [aisQid1, isQid1] > tt > [MARK, aand2, and2]
[A2, 2, a2] > [isPal1, AISPAL1, aisPal1] > [U712, aU712] > [isNePal1, aisNePal1] > [U612, aU612] > [aisQid1, isQid1] > tt > [U321, aU321]
AAND2 > [MARK, aand2, and2]
i > tt > [MARK, aand2, and2]
i > tt > [U321, aU321]
o > tt > [MARK, aand2, and2]
o > tt > [U321, aU321]
u > tt > [MARK, aand2, and2]
u > tt > [U321, aU321]

Status:
A2: [1,2]
_2: [1,2]
MARK: []
a2: [1,2]
U112: multiset
AU11: multiset
tt: multiset
AISNELIST: multiset
AISPALLISTKIND: multiset
AAND2: multiset
aisPalListKind1: [1]
isPalListKind1: [1]
isNeList1: multiset
AU413: multiset
aand2: multiset
AU421: multiset
aisList1: [1]
AU513: multiset
AU522: multiset
aisNeList1: multiset
AISLIST: multiset
AU211: [1]
AU221: [1]
U213: [2,3,1]
U222: multiset
isList1: [1]
U312: [1,2]
U321: [1]
U413: multiset
U422: multiset
U513: multiset
U522: multiset
U612: multiset
U712: multiset
AU712: multiset
AISNEPAL: multiset
aisQid1: multiset
and2: multiset
isPal1: multiset
isNePal1: multiset
AISPAL1: multiset
nil: multiset
aisNePal1: multiset
aU112: multiset
aU213: [2,3,1]
aU222: multiset
aU312: [1,2]
aU321: [1]
isQid1: multiset
aU413: multiset
aU422: multiset
aU513: multiset
aU522: multiset
aU612: multiset
aU712: multiset
aisPal1: 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(tt, V) → a__U12(a__isNeList(V))
a__isNePal(V) → a__U61(a__isPalListKind(V), V)
a__U21(tt, V1, V2) → a__U22(a__isList(V1), V2)
a__U22(tt, V2) → a__U23(a__isList(V2))
a__U41(tt, V1, V2) → a__U42(a__isList(V1), V2)
a__U51(tt, V1, V2) → a__U52(a__isNeList(V1), V2)
a__U42(tt, V2) → a__U43(a__isNeList(V2))
a__U52(tt, V2) → a__U53(a__isList(V2))
a__U71(tt, V) → a__U72(a__isNePal(V))
a__isPal(V) → a__U71(a__isPalListKind(V), V)
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__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U23(tt) → tt
a__U23(X) → U23(X)
a__U43(tt) → tt
a__U43(X) → U43(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
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)

(6) Obligation:

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

A__U11(tt, V) → A__ISNELIST(V)
A__ISNELIST(V) → A__ISPALLISTKIND(V)
A__AND(tt, X) → MARK(X)
MARK(U12(X)) → MARK(X)
A__ISLIST(V) → A__U11(a__isPalListKind(V), V)
A__ISLIST(V) → A__ISPALLISTKIND(V)
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)
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 6 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(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(U23(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1)  =  MARK(x0, x1)

Tags:
MARK has argument tags [0,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
MARK(x1)  =  MARK
U23(x1)  =  U23(x1)
U12(x1)  =  x1
U32(x1)  =  x1
U43(x1)  =  x1
U53(x1)  =  x1
U62(x1)  =  x1
U72(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
U231 > MARK

Status:
MARK: multiset
U231: [1]


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(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(U72(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1)  =  MARK(x0)

Tags:
MARK has argument tags [0,1] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
MARK(x1)  =  MARK(x1)
U12(x1)  =  x1
U32(x1)  =  x1
U43(x1)  =  x1
U53(x1)  =  x1
U62(x1)  =  x1
U72(x1)  =  U72(x1)

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

Status:
MARK1: [1]
U721: multiset


The following usable rules [FROCOS05] were oriented: none

(12) Obligation:

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

MARK(U12(X)) → MARK(X)
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
MARK(U53(X)) → MARK(X)
MARK(U62(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(U32(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1)  =  MARK(x0)

Tags:
MARK has argument tags [0,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
MARK(x1)  =  x1
U12(x1)  =  x1
U32(x1)  =  U32(x1)
U43(x1)  =  x1
U53(x1)  =  x1
U62(x1)  =  x1

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

Status:
U321: [1]


The following usable rules [FROCOS05] were oriented: none

(14) Obligation:

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

MARK(U12(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
MARK(U53(X)) → MARK(X)
MARK(U62(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(U43(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1)  =  MARK(x0)

Tags:
MARK has argument tags [0,1] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
MARK(x1)  =  x1
U12(x1)  =  x1
U43(x1)  =  U43(x1)
U53(x1)  =  x1
U62(x1)  =  x1

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

Status:
U431: multiset


The following usable rules [FROCOS05] were oriented: none

(16) Obligation:

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

MARK(U12(X)) → MARK(X)
MARK(U53(X)) → MARK(X)
MARK(U62(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(U12(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1)  =  MARK(x0, x1)

Tags:
MARK has argument tags [0,1] and root tag 0

Comparison: MS
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
MARK(x1)  =  MARK
U12(x1)  =  U12(x1)
U53(x1)  =  x1
U62(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[MARK, U121]

Status:
MARK: []
U121: multiset


The following usable rules [FROCOS05] were oriented: none

(18) Obligation:

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

MARK(U53(X)) → MARK(X)
MARK(U62(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: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1)  =  MARK(x0, x1)

Tags:
MARK has argument tags [0,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
MARK(x1)  =  MARK
U53(x1)  =  U53(x1)
U62(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
U531 > MARK

Status:
MARK: multiset
U531: multiset


The following usable rules [FROCOS05] were oriented: none

(20) Obligation:

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

MARK(U62(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(U62(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
MARK(x0, x1)  =  MARK(x0, x1)

Tags:
MARK has argument tags [0,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
MARK(x1)  =  MARK
U62(x1)  =  U62(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[MARK, U621]

Status:
MARK: multiset
U621: multiset


The following usable rules [FROCOS05] were oriented: none

(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