(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__isPalListKind(V), V)
a__U12(tt, V) → a__U13(a__isNeList(V))
a__U13(tt) → tt
a__U21(tt, V1, V2) → a__U22(a__isPalListKind(V1), V1, V2)
a__U22(tt, V1, V2) → a__U23(a__isPalListKind(V2), V1, V2)
a__U23(tt, V1, V2) → a__U24(a__isPalListKind(V2), V1, V2)
a__U24(tt, V1, V2) → a__U25(a__isList(V1), V2)
a__U25(tt, V2) → a__U26(a__isList(V2))
a__U26(tt) → tt
a__U31(tt, V) → a__U32(a__isPalListKind(V), V)
a__U32(tt, V) → a__U33(a__isQid(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isPalListKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isPalListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isPalListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isList(V1), V2)
a__U45(tt, V2) → a__U46(a__isNeList(V2))
a__U46(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isPalListKind(V1), V1, V2)
a__U52(tt, V1, V2) → a__U53(a__isPalListKind(V2), V1, V2)
a__U53(tt, V1, V2) → a__U54(a__isPalListKind(V2), V1, V2)
a__U54(tt, V1, V2) → a__U55(a__isNeList(V1), V2)
a__U55(tt, V2) → a__U56(a__isList(V2))
a__U56(tt) → tt
a__U61(tt, V) → a__U62(a__isPalListKind(V), V)
a__U62(tt, V) → a__U63(a__isQid(V))
a__U63(tt) → tt
a__U71(tt, I, P) → a__U72(a__isPalListKind(I), P)
a__U72(tt, P) → a__U73(a__isPal(P), P)
a__U73(tt, P) → a__U74(a__isPalListKind(P))
a__U74(tt) → tt
a__U81(tt, V) → a__U82(a__isPalListKind(V), V)
a__U82(tt, V) → a__U83(a__isNePal(V))
a__U83(tt) → tt
a__U91(tt, V2) → a__U92(a__isPalListKind(V2))
a__U92(tt) → tt
a__isList(V) → a__U11(a__isPalListKind(V), V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isPalListKind(V1), V1, V2)
a__isNeList(V) → a__U31(a__isPalListKind(V), V)
a__isNeList(__(V1, V2)) → a__U41(a__isPalListKind(V1), V1, V2)
a__isNeList(__(V1, V2)) → a__U51(a__isPalListKind(V1), V1, V2)
a__isNePal(V) → a__U61(a__isPalListKind(V), V)
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), I, P)
a__isPal(V) → a__U81(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__U91(a__isPalListKind(V1), 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(X1, X2)) → a__U12(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(U23(X1, X2, X3)) → a__U23(mark(X1), X2, X3)
mark(U24(X1, X2, X3)) → a__U24(mark(X1), X2, X3)
mark(U25(X1, X2)) → a__U25(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(U26(X)) → a__U26(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(U53(X1, X2, X3)) → a__U53(mark(X1), X2, X3)
mark(U54(X1, X2, X3)) → a__U54(mark(X1), X2, X3)
mark(U55(X1, X2)) → a__U55(mark(X1), X2)
mark(U56(X)) → a__U56(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(U73(X1, X2)) → a__U73(mark(X1), X2)
mark(isPal(X)) → a__isPal(X)
mark(U74(X)) → a__U74(mark(X))
mark(U81(X1, X2)) → a__U81(mark(X1), X2)
mark(U82(X1, X2)) → a__U82(mark(X1), X2)
mark(U83(X)) → a__U83(mark(X))
mark(isNePal(X)) → a__isNePal(X)
mark(U91(X1, X2)) → a__U91(mark(X1), X2)
mark(U92(X)) → a__U92(mark(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(X1, X2) → U12(X1, X2)
a__isPalListKind(X) → isPalListKind(X)
a__U13(X) → U13(X)
a__isNeList(X) → isNeList(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__U23(X1, X2, X3) → U23(X1, X2, X3)
a__U24(X1, X2, X3) → U24(X1, X2, X3)
a__U25(X1, X2) → U25(X1, X2)
a__isList(X) → isList(X)
a__U26(X) → U26(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__isQid(X) → isQid(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__U53(X1, X2, X3) → U53(X1, X2, X3)
a__U54(X1, X2, X3) → U54(X1, X2, X3)
a__U55(X1, X2) → U55(X1, X2)
a__U56(X) → U56(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__U73(X1, X2) → U73(X1, X2)
a__isPal(X) → isPal(X)
a__U74(X) → U74(X)
a__U81(X1, X2) → U81(X1, X2)
a__U82(X1, X2) → U82(X1, X2)
a__U83(X) → U83(X)
a__isNePal(X) → isNePal(X)
a__U91(X1, X2) → U91(X1, X2)
a__U92(X) → U92(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__isPalListKind(V), V)
A__U11(tt, V) → A__ISPALLISTKIND(V)
A__U12(tt, V) → A__U13(a__isNeList(V))
A__U12(tt, V) → A__ISNELIST(V)
A__U21(tt, V1, V2) → A__U22(a__isPalListKind(V1), V1, V2)
A__U21(tt, V1, V2) → A__ISPALLISTKIND(V1)
A__U22(tt, V1, V2) → A__U23(a__isPalListKind(V2), V1, V2)
A__U22(tt, V1, V2) → A__ISPALLISTKIND(V2)
A__U23(tt, V1, V2) → A__U24(a__isPalListKind(V2), V1, V2)
A__U23(tt, V1, V2) → A__ISPALLISTKIND(V2)
A__U24(tt, V1, V2) → A__U25(a__isList(V1), V2)
A__U24(tt, V1, V2) → A__ISLIST(V1)
A__U25(tt, V2) → A__U26(a__isList(V2))
A__U25(tt, V2) → A__ISLIST(V2)
A__U31(tt, V) → A__U32(a__isPalListKind(V), V)
A__U31(tt, V) → A__ISPALLISTKIND(V)
A__U32(tt, V) → A__U33(a__isQid(V))
A__U32(tt, V) → A__ISQID(V)
A__U41(tt, V1, V2) → A__U42(a__isPalListKind(V1), V1, V2)
A__U41(tt, V1, V2) → A__ISPALLISTKIND(V1)
A__U42(tt, V1, V2) → A__U43(a__isPalListKind(V2), V1, V2)
A__U42(tt, V1, V2) → A__ISPALLISTKIND(V2)
A__U43(tt, V1, V2) → A__U44(a__isPalListKind(V2), V1, V2)
A__U43(tt, V1, V2) → A__ISPALLISTKIND(V2)
A__U44(tt, V1, V2) → A__U45(a__isList(V1), V2)
A__U44(tt, V1, V2) → A__ISLIST(V1)
A__U45(tt, V2) → A__U46(a__isNeList(V2))
A__U45(tt, V2) → A__ISNELIST(V2)
A__U51(tt, V1, V2) → A__U52(a__isPalListKind(V1), V1, V2)
A__U51(tt, V1, V2) → A__ISPALLISTKIND(V1)
A__U52(tt, V1, V2) → A__U53(a__isPalListKind(V2), V1, V2)
A__U52(tt, V1, V2) → A__ISPALLISTKIND(V2)
A__U53(tt, V1, V2) → A__U54(a__isPalListKind(V2), V1, V2)
A__U53(tt, V1, V2) → A__ISPALLISTKIND(V2)
A__U54(tt, V1, V2) → A__U55(a__isNeList(V1), V2)
A__U54(tt, V1, V2) → A__ISNELIST(V1)
A__U55(tt, V2) → A__U56(a__isList(V2))
A__U55(tt, V2) → A__ISLIST(V2)
A__U61(tt, V) → A__U62(a__isPalListKind(V), V)
A__U61(tt, V) → A__ISPALLISTKIND(V)
A__U62(tt, V) → A__U63(a__isQid(V))
A__U62(tt, V) → A__ISQID(V)
A__U71(tt, I, P) → A__U72(a__isPalListKind(I), P)
A__U71(tt, I, P) → A__ISPALLISTKIND(I)
A__U72(tt, P) → A__U73(a__isPal(P), P)
A__U72(tt, P) → A__ISPAL(P)
A__U73(tt, P) → A__U74(a__isPalListKind(P))
A__U73(tt, P) → A__ISPALLISTKIND(P)
A__U81(tt, V) → A__U82(a__isPalListKind(V), V)
A__U81(tt, V) → A__ISPALLISTKIND(V)
A__U82(tt, V) → A__U83(a__isNePal(V))
A__U82(tt, V) → A__ISNEPAL(V)
A__U91(tt, V2) → A__U92(a__isPalListKind(V2))
A__U91(tt, V2) → A__ISPALLISTKIND(V2)
A__ISLIST(V) → A__U11(a__isPalListKind(V), V)
A__ISLIST(V) → A__ISPALLISTKIND(V)
A__ISLIST(__(V1, V2)) → A__U21(a__isPalListKind(V1), V1, 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__isPalListKind(V1), V1, V2)
A__ISNELIST(__(V1, V2)) → A__ISPALLISTKIND(V1)
A__ISNELIST(__(V1, V2)) → A__U51(a__isPalListKind(V1), V1, V2)
A__ISNEPAL(V) → A__U61(a__isPalListKind(V), V)
A__ISNEPAL(V) → A__ISPALLISTKIND(V)
A__ISNEPAL(__(I, __(P, I))) → A__U71(a__isQid(I), I, P)
A__ISNEPAL(__(I, __(P, I))) → A__ISQID(I)
A__ISPAL(V) → A__U81(a__isPalListKind(V), V)
A__ISPAL(V) → A__ISPALLISTKIND(V)
A__ISPALLISTKIND(__(V1, V2)) → A__U91(a__isPalListKind(V1), 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(X1, X2)) → A__U12(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(isPalListKind(X)) → A__ISPALLISTKIND(X)
MARK(U13(X)) → A__U13(mark(X))
MARK(U13(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, X3)) → A__U22(mark(X1), X2, X3)
MARK(U22(X1, X2, X3)) → MARK(X1)
MARK(U23(X1, X2, X3)) → A__U23(mark(X1), X2, X3)
MARK(U23(X1, X2, X3)) → MARK(X1)
MARK(U24(X1, X2, X3)) → A__U24(mark(X1), X2, X3)
MARK(U24(X1, X2, X3)) → MARK(X1)
MARK(U25(X1, X2)) → A__U25(mark(X1), X2)
MARK(U25(X1, X2)) → MARK(X1)
MARK(isList(X)) → A__ISLIST(X)
MARK(U26(X)) → A__U26(mark(X))
MARK(U26(X)) → MARK(X)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X1, X2)) → A__U32(mark(X1), X2)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U33(X)) → A__U33(mark(X))
MARK(U33(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, X3)) → A__U42(mark(X1), X2, X3)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(U43(X1, X2, X3)) → A__U43(mark(X1), X2, X3)
MARK(U43(X1, X2, X3)) → MARK(X1)
MARK(U44(X1, X2, X3)) → A__U44(mark(X1), X2, X3)
MARK(U44(X1, X2, X3)) → MARK(X1)
MARK(U45(X1, X2)) → A__U45(mark(X1), X2)
MARK(U45(X1, X2)) → MARK(X1)
MARK(U46(X)) → A__U46(mark(X))
MARK(U46(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, X3)) → A__U52(mark(X1), X2, X3)
MARK(U52(X1, X2, X3)) → MARK(X1)
MARK(U53(X1, X2, X3)) → A__U53(mark(X1), X2, X3)
MARK(U53(X1, X2, X3)) → MARK(X1)
MARK(U54(X1, X2, X3)) → A__U54(mark(X1), X2, X3)
MARK(U54(X1, X2, X3)) → MARK(X1)
MARK(U55(X1, X2)) → A__U55(mark(X1), X2)
MARK(U55(X1, X2)) → MARK(X1)
MARK(U56(X)) → A__U56(mark(X))
MARK(U56(X)) → MARK(X)
MARK(U61(X1, X2)) → A__U61(mark(X1), X2)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U62(X1, X2)) → A__U62(mark(X1), X2)
MARK(U62(X1, X2)) → MARK(X1)
MARK(U63(X)) → A__U63(mark(X))
MARK(U63(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U72(X1, X2)) → A__U72(mark(X1), X2)
MARK(U72(X1, X2)) → MARK(X1)
MARK(U73(X1, X2)) → A__U73(mark(X1), X2)
MARK(U73(X1, X2)) → MARK(X1)
MARK(isPal(X)) → A__ISPAL(X)
MARK(U74(X)) → A__U74(mark(X))
MARK(U74(X)) → MARK(X)
MARK(U81(X1, X2)) → A__U81(mark(X1), X2)
MARK(U81(X1, X2)) → MARK(X1)
MARK(U82(X1, X2)) → A__U82(mark(X1), X2)
MARK(U82(X1, X2)) → MARK(X1)
MARK(U83(X)) → A__U83(mark(X))
MARK(U83(X)) → MARK(X)
MARK(isNePal(X)) → A__ISNEPAL(X)
MARK(U91(X1, X2)) → A__U91(mark(X1), X2)
MARK(U91(X1, X2)) → MARK(X1)
MARK(U92(X)) → A__U92(mark(X))
MARK(U92(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__isPalListKind(V), V)
a__U12(tt, V) → a__U13(a__isNeList(V))
a__U13(tt) → tt
a__U21(tt, V1, V2) → a__U22(a__isPalListKind(V1), V1, V2)
a__U22(tt, V1, V2) → a__U23(a__isPalListKind(V2), V1, V2)
a__U23(tt, V1, V2) → a__U24(a__isPalListKind(V2), V1, V2)
a__U24(tt, V1, V2) → a__U25(a__isList(V1), V2)
a__U25(tt, V2) → a__U26(a__isList(V2))
a__U26(tt) → tt
a__U31(tt, V) → a__U32(a__isPalListKind(V), V)
a__U32(tt, V) → a__U33(a__isQid(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isPalListKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isPalListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isPalListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isList(V1), V2)
a__U45(tt, V2) → a__U46(a__isNeList(V2))
a__U46(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isPalListKind(V1), V1, V2)
a__U52(tt, V1, V2) → a__U53(a__isPalListKind(V2), V1, V2)
a__U53(tt, V1, V2) → a__U54(a__isPalListKind(V2), V1, V2)
a__U54(tt, V1, V2) → a__U55(a__isNeList(V1), V2)
a__U55(tt, V2) → a__U56(a__isList(V2))
a__U56(tt) → tt
a__U61(tt, V) → a__U62(a__isPalListKind(V), V)
a__U62(tt, V) → a__U63(a__isQid(V))
a__U63(tt) → tt
a__U71(tt, I, P) → a__U72(a__isPalListKind(I), P)
a__U72(tt, P) → a__U73(a__isPal(P), P)
a__U73(tt, P) → a__U74(a__isPalListKind(P))
a__U74(tt) → tt
a__U81(tt, V) → a__U82(a__isPalListKind(V), V)
a__U82(tt, V) → a__U83(a__isNePal(V))
a__U83(tt) → tt
a__U91(tt, V2) → a__U92(a__isPalListKind(V2))
a__U92(tt) → tt
a__isList(V) → a__U11(a__isPalListKind(V), V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isPalListKind(V1), V1, V2)
a__isNeList(V) → a__U31(a__isPalListKind(V), V)
a__isNeList(__(V1, V2)) → a__U41(a__isPalListKind(V1), V1, V2)
a__isNeList(__(V1, V2)) → a__U51(a__isPalListKind(V1), V1, V2)
a__isNePal(V) → a__U61(a__isPalListKind(V), V)
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), I, P)
a__isPal(V) → a__U81(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__U91(a__isPalListKind(V1), 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(X1, X2)) → a__U12(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(U23(X1, X2, X3)) → a__U23(mark(X1), X2, X3)
mark(U24(X1, X2, X3)) → a__U24(mark(X1), X2, X3)
mark(U25(X1, X2)) → a__U25(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(U26(X)) → a__U26(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(U53(X1, X2, X3)) → a__U53(mark(X1), X2, X3)
mark(U54(X1, X2, X3)) → a__U54(mark(X1), X2, X3)
mark(U55(X1, X2)) → a__U55(mark(X1), X2)
mark(U56(X)) → a__U56(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(U73(X1, X2)) → a__U73(mark(X1), X2)
mark(isPal(X)) → a__isPal(X)
mark(U74(X)) → a__U74(mark(X))
mark(U81(X1, X2)) → a__U81(mark(X1), X2)
mark(U82(X1, X2)) → a__U82(mark(X1), X2)
mark(U83(X)) → a__U83(mark(X))
mark(isNePal(X)) → a__isNePal(X)
mark(U91(X1, X2)) → a__U91(mark(X1), X2)
mark(U92(X)) → a__U92(mark(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(X1, X2) → U12(X1, X2)
a__isPalListKind(X) → isPalListKind(X)
a__U13(X) → U13(X)
a__isNeList(X) → isNeList(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__U23(X1, X2, X3) → U23(X1, X2, X3)
a__U24(X1, X2, X3) → U24(X1, X2, X3)
a__U25(X1, X2) → U25(X1, X2)
a__isList(X) → isList(X)
a__U26(X) → U26(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__isQid(X) → isQid(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__U53(X1, X2, X3) → U53(X1, X2, X3)
a__U54(X1, X2, X3) → U54(X1, X2, X3)
a__U55(X1, X2) → U55(X1, X2)
a__U56(X) → U56(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__U73(X1, X2) → U73(X1, X2)
a__isPal(X) → isPal(X)
a__U74(X) → U74(X)
a__U81(X1, X2) → U81(X1, X2)
a__U82(X1, X2) → U82(X1, X2)
a__U83(X) → U83(X)
a__isNePal(X) → isNePal(X)
a__U91(X1, X2) → U91(X1, X2)
a__U92(X) → U92(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 4 SCCs with 80 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

A__U91(tt, V2) → A__ISPALLISTKIND(V2)
A__ISPALLISTKIND(__(V1, V2)) → A__U91(a__isPalListKind(V1), V2)
A__ISPALLISTKIND(__(V1, V2)) → A__ISPALLISTKIND(V1)

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__isPalListKind(V), V)
a__U12(tt, V) → a__U13(a__isNeList(V))
a__U13(tt) → tt
a__U21(tt, V1, V2) → a__U22(a__isPalListKind(V1), V1, V2)
a__U22(tt, V1, V2) → a__U23(a__isPalListKind(V2), V1, V2)
a__U23(tt, V1, V2) → a__U24(a__isPalListKind(V2), V1, V2)
a__U24(tt, V1, V2) → a__U25(a__isList(V1), V2)
a__U25(tt, V2) → a__U26(a__isList(V2))
a__U26(tt) → tt
a__U31(tt, V) → a__U32(a__isPalListKind(V), V)
a__U32(tt, V) → a__U33(a__isQid(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isPalListKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isPalListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isPalListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isList(V1), V2)
a__U45(tt, V2) → a__U46(a__isNeList(V2))
a__U46(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isPalListKind(V1), V1, V2)
a__U52(tt, V1, V2) → a__U53(a__isPalListKind(V2), V1, V2)
a__U53(tt, V1, V2) → a__U54(a__isPalListKind(V2), V1, V2)
a__U54(tt, V1, V2) → a__U55(a__isNeList(V1), V2)
a__U55(tt, V2) → a__U56(a__isList(V2))
a__U56(tt) → tt
a__U61(tt, V) → a__U62(a__isPalListKind(V), V)
a__U62(tt, V) → a__U63(a__isQid(V))
a__U63(tt) → tt
a__U71(tt, I, P) → a__U72(a__isPalListKind(I), P)
a__U72(tt, P) → a__U73(a__isPal(P), P)
a__U73(tt, P) → a__U74(a__isPalListKind(P))
a__U74(tt) → tt
a__U81(tt, V) → a__U82(a__isPalListKind(V), V)
a__U82(tt, V) → a__U83(a__isNePal(V))
a__U83(tt) → tt
a__U91(tt, V2) → a__U92(a__isPalListKind(V2))
a__U92(tt) → tt
a__isList(V) → a__U11(a__isPalListKind(V), V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isPalListKind(V1), V1, V2)
a__isNeList(V) → a__U31(a__isPalListKind(V), V)
a__isNeList(__(V1, V2)) → a__U41(a__isPalListKind(V1), V1, V2)
a__isNeList(__(V1, V2)) → a__U51(a__isPalListKind(V1), V1, V2)
a__isNePal(V) → a__U61(a__isPalListKind(V), V)
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), I, P)
a__isPal(V) → a__U81(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__U91(a__isPalListKind(V1), 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(X1, X2)) → a__U12(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(U23(X1, X2, X3)) → a__U23(mark(X1), X2, X3)
mark(U24(X1, X2, X3)) → a__U24(mark(X1), X2, X3)
mark(U25(X1, X2)) → a__U25(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(U26(X)) → a__U26(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(U53(X1, X2, X3)) → a__U53(mark(X1), X2, X3)
mark(U54(X1, X2, X3)) → a__U54(mark(X1), X2, X3)
mark(U55(X1, X2)) → a__U55(mark(X1), X2)
mark(U56(X)) → a__U56(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(U73(X1, X2)) → a__U73(mark(X1), X2)
mark(isPal(X)) → a__isPal(X)
mark(U74(X)) → a__U74(mark(X))
mark(U81(X1, X2)) → a__U81(mark(X1), X2)
mark(U82(X1, X2)) → a__U82(mark(X1), X2)
mark(U83(X)) → a__U83(mark(X))
mark(isNePal(X)) → a__isNePal(X)
mark(U91(X1, X2)) → a__U91(mark(X1), X2)
mark(U92(X)) → a__U92(mark(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(X1, X2) → U12(X1, X2)
a__isPalListKind(X) → isPalListKind(X)
a__U13(X) → U13(X)
a__isNeList(X) → isNeList(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__U23(X1, X2, X3) → U23(X1, X2, X3)
a__U24(X1, X2, X3) → U24(X1, X2, X3)
a__U25(X1, X2) → U25(X1, X2)
a__isList(X) → isList(X)
a__U26(X) → U26(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__isQid(X) → isQid(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__U53(X1, X2, X3) → U53(X1, X2, X3)
a__U54(X1, X2, X3) → U54(X1, X2, X3)
a__U55(X1, X2) → U55(X1, X2)
a__U56(X) → U56(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__U73(X1, X2) → U73(X1, X2)
a__isPal(X) → isPal(X)
a__U74(X) → U74(X)
a__U81(X1, X2) → U81(X1, X2)
a__U82(X1, X2) → U82(X1, X2)
a__U83(X) → U83(X)
a__isNePal(X) → isNePal(X)
a__U91(X1, X2) → U91(X1, X2)
a__U92(X) → U92(X)

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


A__ISPALLISTKIND(__(V1, V2)) → A__U91(a__isPalListKind(V1), V2)
A__ISPALLISTKIND(__(V1, V2)) → A__ISPALLISTKIND(V1)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__U91(x1, x2)  =  A__U91(x2)
tt  =  tt
A__ISPALLISTKIND(x1)  =  A__ISPALLISTKIND(x1)
__(x1, x2)  =  __(x1, x2)
a__isPalListKind(x1)  =  a__isPalListKind(x1)
i  =  i
e  =  e
o  =  o
a__U91(x1, x2)  =  x1
a__U92(x1)  =  a__U92
nil  =  nil
u  =  u
U91(x1, x2)  =  U91(x1, x2)
U92(x1)  =  U92(x1)
isPalListKind(x1)  =  isPalListKind(x1)
a  =  a

Recursive Path Order [RPO].
Precedence:
_2 > [AU911, AISPALLISTKIND1] > U921
i > U921
e > U921
o > U921
nil > tt > [AU911, AISPALLISTKIND1] > U921
nil > tt > aisPalListKind1 > U921
nil > tt > aU92 > U921
u > U921
U912 > U921
isPalListKind1 > U921
a > U921


The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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

A__U91(tt, V2) → A__ISPALLISTKIND(V2)

The TRS R consists of the following rules:

a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__U11(tt, V) → a__U12(a__isPalListKind(V), V)
a__U12(tt, V) → a__U13(a__isNeList(V))
a__U13(tt) → tt
a__U21(tt, V1, V2) → a__U22(a__isPalListKind(V1), V1, V2)
a__U22(tt, V1, V2) → a__U23(a__isPalListKind(V2), V1, V2)
a__U23(tt, V1, V2) → a__U24(a__isPalListKind(V2), V1, V2)
a__U24(tt, V1, V2) → a__U25(a__isList(V1), V2)
a__U25(tt, V2) → a__U26(a__isList(V2))
a__U26(tt) → tt
a__U31(tt, V) → a__U32(a__isPalListKind(V), V)
a__U32(tt, V) → a__U33(a__isQid(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isPalListKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isPalListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isPalListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isList(V1), V2)
a__U45(tt, V2) → a__U46(a__isNeList(V2))
a__U46(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isPalListKind(V1), V1, V2)
a__U52(tt, V1, V2) → a__U53(a__isPalListKind(V2), V1, V2)
a__U53(tt, V1, V2) → a__U54(a__isPalListKind(V2), V1, V2)
a__U54(tt, V1, V2) → a__U55(a__isNeList(V1), V2)
a__U55(tt, V2) → a__U56(a__isList(V2))
a__U56(tt) → tt
a__U61(tt, V) → a__U62(a__isPalListKind(V), V)
a__U62(tt, V) → a__U63(a__isQid(V))
a__U63(tt) → tt
a__U71(tt, I, P) → a__U72(a__isPalListKind(I), P)
a__U72(tt, P) → a__U73(a__isPal(P), P)
a__U73(tt, P) → a__U74(a__isPalListKind(P))
a__U74(tt) → tt
a__U81(tt, V) → a__U82(a__isPalListKind(V), V)
a__U82(tt, V) → a__U83(a__isNePal(V))
a__U83(tt) → tt
a__U91(tt, V2) → a__U92(a__isPalListKind(V2))
a__U92(tt) → tt
a__isList(V) → a__U11(a__isPalListKind(V), V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isPalListKind(V1), V1, V2)
a__isNeList(V) → a__U31(a__isPalListKind(V), V)
a__isNeList(__(V1, V2)) → a__U41(a__isPalListKind(V1), V1, V2)
a__isNeList(__(V1, V2)) → a__U51(a__isPalListKind(V1), V1, V2)
a__isNePal(V) → a__U61(a__isPalListKind(V), V)
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), I, P)
a__isPal(V) → a__U81(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__U91(a__isPalListKind(V1), 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(X1, X2)) → a__U12(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(U23(X1, X2, X3)) → a__U23(mark(X1), X2, X3)
mark(U24(X1, X2, X3)) → a__U24(mark(X1), X2, X3)
mark(U25(X1, X2)) → a__U25(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(U26(X)) → a__U26(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(U53(X1, X2, X3)) → a__U53(mark(X1), X2, X3)
mark(U54(X1, X2, X3)) → a__U54(mark(X1), X2, X3)
mark(U55(X1, X2)) → a__U55(mark(X1), X2)
mark(U56(X)) → a__U56(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(U73(X1, X2)) → a__U73(mark(X1), X2)
mark(isPal(X)) → a__isPal(X)
mark(U74(X)) → a__U74(mark(X))
mark(U81(X1, X2)) → a__U81(mark(X1), X2)
mark(U82(X1, X2)) → a__U82(mark(X1), X2)
mark(U83(X)) → a__U83(mark(X))
mark(isNePal(X)) → a__isNePal(X)
mark(U91(X1, X2)) → a__U91(mark(X1), X2)
mark(U92(X)) → a__U92(mark(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(X1, X2) → U12(X1, X2)
a__isPalListKind(X) → isPalListKind(X)
a__U13(X) → U13(X)
a__isNeList(X) → isNeList(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__U23(X1, X2, X3) → U23(X1, X2, X3)
a__U24(X1, X2, X3) → U24(X1, X2, X3)
a__U25(X1, X2) → U25(X1, X2)
a__isList(X) → isList(X)
a__U26(X) → U26(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__isQid(X) → isQid(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__U53(X1, X2, X3) → U53(X1, X2, X3)
a__U54(X1, X2, X3) → U54(X1, X2, X3)
a__U55(X1, X2) → U55(X1, X2)
a__U56(X) → U56(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__U73(X1, X2) → U73(X1, X2)
a__isPal(X) → isPal(X)
a__U74(X) → U74(X)
a__U81(X1, X2) → U81(X1, X2)
a__U82(X1, X2) → U82(X1, X2)
a__U83(X) → U83(X)
a__isNePal(X) → isNePal(X)
a__U91(X1, X2) → U91(X1, X2)
a__U92(X) → U92(X)

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

(8) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(9) TRUE

(10) Obligation:

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

A__ISNEPAL(__(I, __(P, I))) → A__U71(a__isQid(I), I, P)
A__U71(tt, I, P) → A__U72(a__isPalListKind(I), P)
A__U72(tt, P) → A__ISPAL(P)
A__ISPAL(V) → A__U81(a__isPalListKind(V), V)
A__U81(tt, V) → A__U82(a__isPalListKind(V), V)
A__U82(tt, V) → A__ISNEPAL(V)

The TRS R consists of the following rules:

a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__U11(tt, V) → a__U12(a__isPalListKind(V), V)
a__U12(tt, V) → a__U13(a__isNeList(V))
a__U13(tt) → tt
a__U21(tt, V1, V2) → a__U22(a__isPalListKind(V1), V1, V2)
a__U22(tt, V1, V2) → a__U23(a__isPalListKind(V2), V1, V2)
a__U23(tt, V1, V2) → a__U24(a__isPalListKind(V2), V1, V2)
a__U24(tt, V1, V2) → a__U25(a__isList(V1), V2)
a__U25(tt, V2) → a__U26(a__isList(V2))
a__U26(tt) → tt
a__U31(tt, V) → a__U32(a__isPalListKind(V), V)
a__U32(tt, V) → a__U33(a__isQid(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isPalListKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isPalListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isPalListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isList(V1), V2)
a__U45(tt, V2) → a__U46(a__isNeList(V2))
a__U46(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isPalListKind(V1), V1, V2)
a__U52(tt, V1, V2) → a__U53(a__isPalListKind(V2), V1, V2)
a__U53(tt, V1, V2) → a__U54(a__isPalListKind(V2), V1, V2)
a__U54(tt, V1, V2) → a__U55(a__isNeList(V1), V2)
a__U55(tt, V2) → a__U56(a__isList(V2))
a__U56(tt) → tt
a__U61(tt, V) → a__U62(a__isPalListKind(V), V)
a__U62(tt, V) → a__U63(a__isQid(V))
a__U63(tt) → tt
a__U71(tt, I, P) → a__U72(a__isPalListKind(I), P)
a__U72(tt, P) → a__U73(a__isPal(P), P)
a__U73(tt, P) → a__U74(a__isPalListKind(P))
a__U74(tt) → tt
a__U81(tt, V) → a__U82(a__isPalListKind(V), V)
a__U82(tt, V) → a__U83(a__isNePal(V))
a__U83(tt) → tt
a__U91(tt, V2) → a__U92(a__isPalListKind(V2))
a__U92(tt) → tt
a__isList(V) → a__U11(a__isPalListKind(V), V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isPalListKind(V1), V1, V2)
a__isNeList(V) → a__U31(a__isPalListKind(V), V)
a__isNeList(__(V1, V2)) → a__U41(a__isPalListKind(V1), V1, V2)
a__isNeList(__(V1, V2)) → a__U51(a__isPalListKind(V1), V1, V2)
a__isNePal(V) → a__U61(a__isPalListKind(V), V)
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), I, P)
a__isPal(V) → a__U81(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__U91(a__isPalListKind(V1), 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(X1, X2)) → a__U12(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(U23(X1, X2, X3)) → a__U23(mark(X1), X2, X3)
mark(U24(X1, X2, X3)) → a__U24(mark(X1), X2, X3)
mark(U25(X1, X2)) → a__U25(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(U26(X)) → a__U26(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(U53(X1, X2, X3)) → a__U53(mark(X1), X2, X3)
mark(U54(X1, X2, X3)) → a__U54(mark(X1), X2, X3)
mark(U55(X1, X2)) → a__U55(mark(X1), X2)
mark(U56(X)) → a__U56(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(U73(X1, X2)) → a__U73(mark(X1), X2)
mark(isPal(X)) → a__isPal(X)
mark(U74(X)) → a__U74(mark(X))
mark(U81(X1, X2)) → a__U81(mark(X1), X2)
mark(U82(X1, X2)) → a__U82(mark(X1), X2)
mark(U83(X)) → a__U83(mark(X))
mark(isNePal(X)) → a__isNePal(X)
mark(U91(X1, X2)) → a__U91(mark(X1), X2)
mark(U92(X)) → a__U92(mark(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(X1, X2) → U12(X1, X2)
a__isPalListKind(X) → isPalListKind(X)
a__U13(X) → U13(X)
a__isNeList(X) → isNeList(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__U23(X1, X2, X3) → U23(X1, X2, X3)
a__U24(X1, X2, X3) → U24(X1, X2, X3)
a__U25(X1, X2) → U25(X1, X2)
a__isList(X) → isList(X)
a__U26(X) → U26(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__isQid(X) → isQid(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__U53(X1, X2, X3) → U53(X1, X2, X3)
a__U54(X1, X2, X3) → U54(X1, X2, X3)
a__U55(X1, X2) → U55(X1, X2)
a__U56(X) → U56(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__U73(X1, X2) → U73(X1, X2)
a__isPal(X) → isPal(X)
a__U74(X) → U74(X)
a__U81(X1, X2) → U81(X1, X2)
a__U82(X1, X2) → U82(X1, X2)
a__U83(X) → U83(X)
a__isNePal(X) → isNePal(X)
a__U91(X1, X2) → U91(X1, X2)
a__U92(X) → U92(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.


A__ISNEPAL(__(I, __(P, I))) → A__U71(a__isQid(I), I, P)
A__U72(tt, P) → A__ISPAL(P)
A__U81(tt, V) → A__U82(a__isPalListKind(V), V)
A__U82(tt, V) → A__ISNEPAL(V)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__ISNEPAL(x1)  =  A__ISNEPAL(x1)
__(x1, x2)  =  __(x1, x2)
A__U71(x1, x2, x3)  =  A__U71(x1, x3)
a__isQid(x1)  =  a__isQid(x1)
tt  =  tt
A__U72(x1, x2)  =  A__U72(x1, x2)
a__isPalListKind(x1)  =  a__isPalListKind
A__ISPAL(x1)  =  A__ISPAL(x1)
A__U81(x1, x2)  =  A__U81(x2)
A__U82(x1, x2)  =  A__U82(x1, x2)
i  =  i
e  =  e
o  =  o
nil  =  nil
a__U91(x1, x2)  =  x1
u  =  u
a  =  a
U91(x1, x2)  =  x1
a__U92(x1)  =  x1
U92(x1)  =  x1
isPalListKind(x1)  =  isPalListKind
isQid(x1)  =  x1

Recursive Path Order [RPO].
Precedence:
[2, aisQid1] > [AU712, tt, AU722, aisPalListKind, AISPAL1, AU811] > AU822 > AISNEPAL1
[2, aisQid1] > [AU712, tt, AU722, aisPalListKind, AISPAL1, AU811] > isPalListKind
o > [AU712, tt, AU722, aisPalListKind, AISPAL1, AU811] > AU822 > AISNEPAL1
o > [AU712, tt, AU722, aisPalListKind, AISPAL1, AU811] > isPalListKind
nil > [AU712, tt, AU722, aisPalListKind, AISPAL1, AU811] > AU822 > AISNEPAL1
nil > [AU712, tt, AU722, aisPalListKind, AISPAL1, AU811] > isPalListKind


The following usable rules [FROCOS05] were oriented:

a__isPalListKind(i) → tt
a__isPalListKind(e) → tt
a__isPalListKind(o) → tt
a__isPalListKind(nil) → tt
a__isPalListKind(__(V1, V2)) → a__U91(a__isPalListKind(V1), V2)
a__isPalListKind(u) → tt
a__isQid(e) → tt
a__isQid(a) → tt
a__U91(X1, X2) → U91(X1, X2)
a__U92(X) → U92(X)
a__isPalListKind(X) → isPalListKind(X)
a__isPalListKind(a) → tt
a__isQid(X) → isQid(X)
a__U91(tt, V2) → a__U92(a__isPalListKind(V2))
a__U92(tt) → tt
a__isQid(u) → tt
a__isQid(i) → tt
a__isQid(o) → tt

(12) Obligation:

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

A__U71(tt, I, P) → A__U72(a__isPalListKind(I), P)
A__ISPAL(V) → A__U81(a__isPalListKind(V), V)

The TRS R consists of the following rules:

a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__U11(tt, V) → a__U12(a__isPalListKind(V), V)
a__U12(tt, V) → a__U13(a__isNeList(V))
a__U13(tt) → tt
a__U21(tt, V1, V2) → a__U22(a__isPalListKind(V1), V1, V2)
a__U22(tt, V1, V2) → a__U23(a__isPalListKind(V2), V1, V2)
a__U23(tt, V1, V2) → a__U24(a__isPalListKind(V2), V1, V2)
a__U24(tt, V1, V2) → a__U25(a__isList(V1), V2)
a__U25(tt, V2) → a__U26(a__isList(V2))
a__U26(tt) → tt
a__U31(tt, V) → a__U32(a__isPalListKind(V), V)
a__U32(tt, V) → a__U33(a__isQid(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isPalListKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isPalListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isPalListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isList(V1), V2)
a__U45(tt, V2) → a__U46(a__isNeList(V2))
a__U46(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isPalListKind(V1), V1, V2)
a__U52(tt, V1, V2) → a__U53(a__isPalListKind(V2), V1, V2)
a__U53(tt, V1, V2) → a__U54(a__isPalListKind(V2), V1, V2)
a__U54(tt, V1, V2) → a__U55(a__isNeList(V1), V2)
a__U55(tt, V2) → a__U56(a__isList(V2))
a__U56(tt) → tt
a__U61(tt, V) → a__U62(a__isPalListKind(V), V)
a__U62(tt, V) → a__U63(a__isQid(V))
a__U63(tt) → tt
a__U71(tt, I, P) → a__U72(a__isPalListKind(I), P)
a__U72(tt, P) → a__U73(a__isPal(P), P)
a__U73(tt, P) → a__U74(a__isPalListKind(P))
a__U74(tt) → tt
a__U81(tt, V) → a__U82(a__isPalListKind(V), V)
a__U82(tt, V) → a__U83(a__isNePal(V))
a__U83(tt) → tt
a__U91(tt, V2) → a__U92(a__isPalListKind(V2))
a__U92(tt) → tt
a__isList(V) → a__U11(a__isPalListKind(V), V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isPalListKind(V1), V1, V2)
a__isNeList(V) → a__U31(a__isPalListKind(V), V)
a__isNeList(__(V1, V2)) → a__U41(a__isPalListKind(V1), V1, V2)
a__isNeList(__(V1, V2)) → a__U51(a__isPalListKind(V1), V1, V2)
a__isNePal(V) → a__U61(a__isPalListKind(V), V)
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), I, P)
a__isPal(V) → a__U81(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__U91(a__isPalListKind(V1), 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(X1, X2)) → a__U12(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(U23(X1, X2, X3)) → a__U23(mark(X1), X2, X3)
mark(U24(X1, X2, X3)) → a__U24(mark(X1), X2, X3)
mark(U25(X1, X2)) → a__U25(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(U26(X)) → a__U26(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(U53(X1, X2, X3)) → a__U53(mark(X1), X2, X3)
mark(U54(X1, X2, X3)) → a__U54(mark(X1), X2, X3)
mark(U55(X1, X2)) → a__U55(mark(X1), X2)
mark(U56(X)) → a__U56(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(U73(X1, X2)) → a__U73(mark(X1), X2)
mark(isPal(X)) → a__isPal(X)
mark(U74(X)) → a__U74(mark(X))
mark(U81(X1, X2)) → a__U81(mark(X1), X2)
mark(U82(X1, X2)) → a__U82(mark(X1), X2)
mark(U83(X)) → a__U83(mark(X))
mark(isNePal(X)) → a__isNePal(X)
mark(U91(X1, X2)) → a__U91(mark(X1), X2)
mark(U92(X)) → a__U92(mark(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(X1, X2) → U12(X1, X2)
a__isPalListKind(X) → isPalListKind(X)
a__U13(X) → U13(X)
a__isNeList(X) → isNeList(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__U23(X1, X2, X3) → U23(X1, X2, X3)
a__U24(X1, X2, X3) → U24(X1, X2, X3)
a__U25(X1, X2) → U25(X1, X2)
a__isList(X) → isList(X)
a__U26(X) → U26(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__isQid(X) → isQid(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__U53(X1, X2, X3) → U53(X1, X2, X3)
a__U54(X1, X2, X3) → U54(X1, X2, X3)
a__U55(X1, X2) → U55(X1, X2)
a__U56(X) → U56(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__U73(X1, X2) → U73(X1, X2)
a__isPal(X) → isPal(X)
a__U74(X) → U74(X)
a__U81(X1, X2) → U81(X1, X2)
a__U82(X1, X2) → U82(X1, X2)
a__U83(X) → U83(X)
a__isNePal(X) → isNePal(X)
a__U91(X1, X2) → U91(X1, X2)
a__U92(X) → U92(X)

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

(13) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(14) TRUE

(15) Obligation:

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

A__U12(tt, V) → A__ISNELIST(V)
A__ISNELIST(__(V1, V2)) → A__U41(a__isPalListKind(V1), V1, V2)
A__U41(tt, V1, V2) → A__U42(a__isPalListKind(V1), V1, V2)
A__U42(tt, V1, V2) → A__U43(a__isPalListKind(V2), V1, V2)
A__U43(tt, V1, V2) → A__U44(a__isPalListKind(V2), V1, V2)
A__U44(tt, V1, V2) → A__U45(a__isList(V1), V2)
A__U45(tt, V2) → A__ISNELIST(V2)
A__ISNELIST(__(V1, V2)) → A__U51(a__isPalListKind(V1), V1, V2)
A__U51(tt, V1, V2) → A__U52(a__isPalListKind(V1), V1, V2)
A__U52(tt, V1, V2) → A__U53(a__isPalListKind(V2), V1, V2)
A__U53(tt, V1, V2) → A__U54(a__isPalListKind(V2), V1, V2)
A__U54(tt, V1, V2) → A__U55(a__isNeList(V1), V2)
A__U55(tt, V2) → A__ISLIST(V2)
A__ISLIST(V) → A__U11(a__isPalListKind(V), V)
A__U11(tt, V) → A__U12(a__isPalListKind(V), V)
A__ISLIST(__(V1, V2)) → A__U21(a__isPalListKind(V1), V1, V2)
A__U21(tt, V1, V2) → A__U22(a__isPalListKind(V1), V1, V2)
A__U22(tt, V1, V2) → A__U23(a__isPalListKind(V2), V1, V2)
A__U23(tt, V1, V2) → A__U24(a__isPalListKind(V2), V1, V2)
A__U24(tt, V1, V2) → A__U25(a__isList(V1), V2)
A__U25(tt, V2) → A__ISLIST(V2)
A__U24(tt, V1, V2) → A__ISLIST(V1)
A__U54(tt, V1, V2) → A__ISNELIST(V1)
A__U44(tt, V1, V2) → A__ISLIST(V1)

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__isPalListKind(V), V)
a__U12(tt, V) → a__U13(a__isNeList(V))
a__U13(tt) → tt
a__U21(tt, V1, V2) → a__U22(a__isPalListKind(V1), V1, V2)
a__U22(tt, V1, V2) → a__U23(a__isPalListKind(V2), V1, V2)
a__U23(tt, V1, V2) → a__U24(a__isPalListKind(V2), V1, V2)
a__U24(tt, V1, V2) → a__U25(a__isList(V1), V2)
a__U25(tt, V2) → a__U26(a__isList(V2))
a__U26(tt) → tt
a__U31(tt, V) → a__U32(a__isPalListKind(V), V)
a__U32(tt, V) → a__U33(a__isQid(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isPalListKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isPalListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isPalListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isList(V1), V2)
a__U45(tt, V2) → a__U46(a__isNeList(V2))
a__U46(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isPalListKind(V1), V1, V2)
a__U52(tt, V1, V2) → a__U53(a__isPalListKind(V2), V1, V2)
a__U53(tt, V1, V2) → a__U54(a__isPalListKind(V2), V1, V2)
a__U54(tt, V1, V2) → a__U55(a__isNeList(V1), V2)
a__U55(tt, V2) → a__U56(a__isList(V2))
a__U56(tt) → tt
a__U61(tt, V) → a__U62(a__isPalListKind(V), V)
a__U62(tt, V) → a__U63(a__isQid(V))
a__U63(tt) → tt
a__U71(tt, I, P) → a__U72(a__isPalListKind(I), P)
a__U72(tt, P) → a__U73(a__isPal(P), P)
a__U73(tt, P) → a__U74(a__isPalListKind(P))
a__U74(tt) → tt
a__U81(tt, V) → a__U82(a__isPalListKind(V), V)
a__U82(tt, V) → a__U83(a__isNePal(V))
a__U83(tt) → tt
a__U91(tt, V2) → a__U92(a__isPalListKind(V2))
a__U92(tt) → tt
a__isList(V) → a__U11(a__isPalListKind(V), V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isPalListKind(V1), V1, V2)
a__isNeList(V) → a__U31(a__isPalListKind(V), V)
a__isNeList(__(V1, V2)) → a__U41(a__isPalListKind(V1), V1, V2)
a__isNeList(__(V1, V2)) → a__U51(a__isPalListKind(V1), V1, V2)
a__isNePal(V) → a__U61(a__isPalListKind(V), V)
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), I, P)
a__isPal(V) → a__U81(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__U91(a__isPalListKind(V1), 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(X1, X2)) → a__U12(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(U23(X1, X2, X3)) → a__U23(mark(X1), X2, X3)
mark(U24(X1, X2, X3)) → a__U24(mark(X1), X2, X3)
mark(U25(X1, X2)) → a__U25(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(U26(X)) → a__U26(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(U53(X1, X2, X3)) → a__U53(mark(X1), X2, X3)
mark(U54(X1, X2, X3)) → a__U54(mark(X1), X2, X3)
mark(U55(X1, X2)) → a__U55(mark(X1), X2)
mark(U56(X)) → a__U56(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(U73(X1, X2)) → a__U73(mark(X1), X2)
mark(isPal(X)) → a__isPal(X)
mark(U74(X)) → a__U74(mark(X))
mark(U81(X1, X2)) → a__U81(mark(X1), X2)
mark(U82(X1, X2)) → a__U82(mark(X1), X2)
mark(U83(X)) → a__U83(mark(X))
mark(isNePal(X)) → a__isNePal(X)
mark(U91(X1, X2)) → a__U91(mark(X1), X2)
mark(U92(X)) → a__U92(mark(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(X1, X2) → U12(X1, X2)
a__isPalListKind(X) → isPalListKind(X)
a__U13(X) → U13(X)
a__isNeList(X) → isNeList(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__U23(X1, X2, X3) → U23(X1, X2, X3)
a__U24(X1, X2, X3) → U24(X1, X2, X3)
a__U25(X1, X2) → U25(X1, X2)
a__isList(X) → isList(X)
a__U26(X) → U26(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__isQid(X) → isQid(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__U53(X1, X2, X3) → U53(X1, X2, X3)
a__U54(X1, X2, X3) → U54(X1, X2, X3)
a__U55(X1, X2) → U55(X1, X2)
a__U56(X) → U56(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__U73(X1, X2) → U73(X1, X2)
a__isPal(X) → isPal(X)
a__U74(X) → U74(X)
a__U81(X1, X2) → U81(X1, X2)
a__U82(X1, X2) → U82(X1, X2)
a__U83(X) → U83(X)
a__isNePal(X) → isNePal(X)
a__U91(X1, X2) → U91(X1, X2)
a__U92(X) → U92(X)

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

(16) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


A__ISNELIST(__(V1, V2)) → A__U41(a__isPalListKind(V1), V1, V2)
A__U44(tt, V1, V2) → A__U45(a__isList(V1), V2)
A__U45(tt, V2) → A__ISNELIST(V2)
A__ISNELIST(__(V1, V2)) → A__U51(a__isPalListKind(V1), V1, V2)
A__U54(tt, V1, V2) → A__U55(a__isNeList(V1), V2)
A__ISLIST(__(V1, V2)) → A__U21(a__isPalListKind(V1), V1, V2)
A__U22(tt, V1, V2) → A__U23(a__isPalListKind(V2), V1, V2)
A__U23(tt, V1, V2) → A__U24(a__isPalListKind(V2), V1, V2)
A__U24(tt, V1, V2) → A__U25(a__isList(V1), V2)
A__U24(tt, V1, V2) → A__ISLIST(V1)
A__U54(tt, V1, V2) → A__ISNELIST(V1)
A__U44(tt, V1, V2) → A__ISLIST(V1)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__U12(x1, x2)  =  A__U12(x2)
tt  =  tt
A__ISNELIST(x1)  =  A__ISNELIST(x1)
__(x1, x2)  =  __(x1, x2)
A__U41(x1, x2, x3)  =  A__U41(x2, x3)
a__isPalListKind(x1)  =  a__isPalListKind(x1)
A__U42(x1, x2, x3)  =  A__U42(x2, x3)
A__U43(x1, x2, x3)  =  A__U43(x2, x3)
A__U44(x1, x2, x3)  =  A__U44(x2, x3)
A__U45(x1, x2)  =  A__U45(x1, x2)
a__isList(x1)  =  a__isList(x1)
A__U51(x1, x2, x3)  =  A__U51(x2, x3)
A__U52(x1, x2, x3)  =  A__U52(x2, x3)
A__U53(x1, x2, x3)  =  A__U53(x2, x3)
A__U54(x1, x2, x3)  =  A__U54(x2, x3)
A__U55(x1, x2)  =  A__U55(x2)
a__isNeList(x1)  =  a__isNeList(x1)
A__ISLIST(x1)  =  A__ISLIST(x1)
A__U11(x1, x2)  =  A__U11(x2)
A__U21(x1, x2, x3)  =  A__U21(x2, x3)
A__U22(x1, x2, x3)  =  A__U22(x2, x3)
A__U23(x1, x2, x3)  =  A__U23(x2, x3)
A__U24(x1, x2, x3)  =  A__U24(x2, x3)
A__U25(x1, x2)  =  A__U25(x2)
a__U11(x1, x2)  =  a__U11(x2)
U11(x1, x2)  =  U11(x2)
a__U12(x1, x2)  =  x1
U12(x1, x2)  =  x1
a__U22(x1, x2, x3)  =  a__U22(x2, x3)
U22(x1, x2, x3)  =  U22
a__U23(x1, x2, x3)  =  a__U23(x2, x3)
U23(x1, x2, x3)  =  x3
a__U24(x1, x2, x3)  =  a__U24(x1, x2, x3)
U24(x1, x2, x3)  =  U24(x1, x2, x3)
a__U25(x1, x2)  =  a__U25(x2)
U25(x1, x2)  =  U25
isPalListKind(x1)  =  isPalListKind(x1)
a__U13(x1)  =  a__U13
U13(x1)  =  U13
isNeList(x1)  =  isNeList(x1)
a__U21(x1, x2, x3)  =  a__U21(x2, x3)
U21(x1, x2, x3)  =  U21(x2, x3)
a__U55(x1, x2)  =  x1
a__U56(x1)  =  a__U56
a__U54(x1, x2, x3)  =  a__U54(x2)
a__U53(x1, x2, x3)  =  a__U53(x2, x3)
a__U52(x1, x2, x3)  =  a__U52(x2, x3)
a__U51(x1, x2, x3)  =  a__U51(x2, x3)
a__U46(x1)  =  a__U46
U55(x1, x2)  =  x1
a__U45(x1, x2)  =  a__U45(x2)
U56(x1)  =  U56
a__U44(x1, x2, x3)  =  a__U44(x3)
a__U91(x1, x2)  =  a__U91(x2)
U91(x1, x2)  =  U91(x2)
a__U43(x1, x2, x3)  =  a__U43(x2, x3)
a__U92(x1)  =  x1
U92(x1)  =  x1
a__U42(x1, x2, x3)  =  a__U42(x2, x3)
a__U41(x1, x2, x3)  =  a__U41(x1, x2, x3)
a__U33(x1)  =  a__U33(x1)
a__U32(x1, x2)  =  a__U32
a__isQid(x1)  =  a__isQid
a__U31(x1, x2)  =  a__U31
a__U26(x1)  =  a__U26
U42(x1, x2, x3)  =  U42(x2, x3)
U41(x1, x2, x3)  =  U41(x1, x2, x3)
isQid(x1)  =  isQid
U33(x1)  =  U33(x1)
U32(x1, x2)  =  U32
U31(x1, x2)  =  U31
U26(x1)  =  U26
isList(x1)  =  isList(x1)
U54(x1, x2, x3)  =  U54(x2)
U53(x1, x2, x3)  =  U53(x2, x3)
U52(x1, x2, x3)  =  U52(x2, x3)
U51(x1, x2, x3)  =  U51(x2, x3)
U46(x1)  =  U46
U45(x1, x2)  =  U45(x2)
U44(x1, x2, x3)  =  U44(x3)
U43(x1, x2, x3)  =  U43
i  =  i
e  =  e
o  =  o
nil  =  nil
u  =  u
a  =  a

Recursive Path Order [RPO].
Precedence:
[2, AU412, aisPalListKind1, AU422, AU432, AU442, aisList1, aisNeList1, AU212, AU222, aU111, U111, aU222, aU232, isPalListKind1, aU212, U212, aU541, aU532, aU522, aU512, aU911, U911, isList1, U541, U522, U512] > AU452 > [AU121, AISNELIST1, AU512, AU522, AU532, AU542, AU551, AISLIST1, AU111, AU251] > U413
[2, AU412, aisPalListKind1, AU422, AU432, AU442, aisList1, aisNeList1, AU212, AU222, aU111, U111, aU222, aU232, isPalListKind1, aU212, U212, aU541, aU532, aU522, aU512, aU911, U911, isList1, U541, U522, U512] > AU232 > AU242 > [AU121, AISNELIST1, AU512, AU522, AU532, AU542, AU551, AISLIST1, AU111, AU251] > U413
[2, AU412, aisPalListKind1, AU422, AU432, AU442, aisList1, aisNeList1, AU212, AU222, aU111, U111, aU222, aU232, isPalListKind1, aU212, U212, aU541, aU532, aU522, aU512, aU911, U911, isList1, U541, U522, U512] > U22 > U413
[2, AU412, aisPalListKind1, AU422, AU432, AU442, aisList1, aisNeList1, AU212, AU222, aU111, U111, aU222, aU232, isPalListKind1, aU212, U212, aU541, aU532, aU522, aU512, aU911, U911, isList1, U541, U522, U512] > [aU243, U243] > [aU251, aU26, U26] > [tt, aU13, U13, aU56, aU46, aU451, U56, U451] > U46 > U413
[2, AU412, aisPalListKind1, AU422, AU432, AU442, aisList1, aisNeList1, AU212, AU222, aU111, U111, aU222, aU232, isPalListKind1, aU212, U212, aU541, aU532, aU522, aU512, aU911, U911, isList1, U541, U522, U512] > [aU243, U243] > [aU251, aU26, U26] > U25 > U413
[2, AU412, aisPalListKind1, AU422, AU432, AU442, aisList1, aisNeList1, AU212, AU222, aU111, U111, aU222, aU232, isPalListKind1, aU212, U212, aU541, aU532, aU522, aU512, aU911, U911, isList1, U541, U522, U512] > isNeList1 > U413
[2, AU412, aisPalListKind1, AU422, AU432, AU442, aisList1, aisNeList1, AU212, AU222, aU111, U111, aU222, aU232, isPalListKind1, aU212, U212, aU541, aU532, aU522, aU512, aU911, U911, isList1, U541, U522, U512] > aU413 > [aU441, aU432, aU422, U441] > [tt, aU13, U13, aU56, aU46, aU451, U56, U451] > U46 > U413
[2, AU412, aisPalListKind1, AU422, AU432, AU442, aisList1, aisNeList1, AU212, AU222, aU111, U111, aU222, aU232, isPalListKind1, aU212, U212, aU541, aU532, aU522, aU512, aU911, U911, isList1, U541, U522, U512] > aU413 > [aU441, aU432, aU422, U441] > U422 > U413
[2, AU412, aisPalListKind1, AU422, AU432, AU442, aisList1, aisNeList1, AU212, AU222, aU111, U111, aU222, aU232, isPalListKind1, aU212, U212, aU541, aU532, aU522, aU512, aU911, U911, isList1, U541, U522, U512] > aU413 > [aU441, aU432, aU422, U441] > U43 > U413
[2, AU412, aisPalListKind1, AU422, AU432, AU442, aisList1, aisNeList1, AU212, AU222, aU111, U111, aU222, aU232, isPalListKind1, aU212, U212, aU541, aU532, aU522, aU512, aU911, U911, isList1, U541, U522, U512] > [aU32, aU31, U32] > [aU331, U331] > [tt, aU13, U13, aU56, aU46, aU451, U56, U451] > U46 > U413
[2, AU412, aisPalListKind1, AU422, AU432, AU442, aisList1, aisNeList1, AU212, AU222, aU111, U111, aU222, aU232, isPalListKind1, aU212, U212, aU541, aU532, aU522, aU512, aU911, U911, isList1, U541, U522, U512] > [aU32, aU31, U32] > aisQid > [tt, aU13, U13, aU56, aU46, aU451, U56, U451] > U46 > U413
[2, AU412, aisPalListKind1, AU422, AU432, AU442, aisList1, aisNeList1, AU212, AU222, aU111, U111, aU222, aU232, isPalListKind1, aU212, U212, aU541, aU532, aU522, aU512, aU911, U911, isList1, U541, U522, U512] > [aU32, aU31, U32] > aisQid > isQid > U413
[2, AU412, aisPalListKind1, AU422, AU432, AU442, aisList1, aisNeList1, AU212, AU222, aU111, U111, aU222, aU232, isPalListKind1, aU212, U212, aU541, aU532, aU522, aU512, aU911, U911, isList1, U541, U522, U512] > [aU32, aU31, U32] > U31 > U413
[2, AU412, aisPalListKind1, AU422, AU432, AU442, aisList1, aisNeList1, AU212, AU222, aU111, U111, aU222, aU232, isPalListKind1, aU212, U212, aU541, aU532, aU522, aU512, aU911, U911, isList1, U541, U522, U512] > U532 > U413
i > [tt, aU13, U13, aU56, aU46, aU451, U56, U451] > U46 > U413
e > U413
o > U413
nil > U413
u > U413
a > [tt, aU13, U13, aU56, aU46, aU451, U56, U451] > U46 > U413


The following usable rules [FROCOS05] were oriented:

a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__U23(X1, X2, X3) → U23(X1, X2, X3)
a__U24(X1, X2, X3) → U24(X1, X2, X3)
a__U25(X1, X2) → U25(X1, X2)
a__isPalListKind(X) → isPalListKind(X)
a__U13(X) → U13(X)
a__isNeList(X) → isNeList(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U55(tt, V2) → a__U56(a__isList(V2))
a__U54(tt, V1, V2) → a__U55(a__isNeList(V1), V2)
a__U53(tt, V1, V2) → a__U54(a__isPalListKind(V2), V1, V2)
a__U52(tt, V1, V2) → a__U53(a__isPalListKind(V2), V1, V2)
a__U51(tt, V1, V2) → a__U52(a__isPalListKind(V1), V1, V2)
a__U46(tt) → tt
a__U55(X1, X2) → U55(X1, X2)
a__U45(tt, V2) → a__U46(a__isNeList(V2))
a__U56(X) → U56(X)
a__U44(tt, V1, V2) → a__U45(a__isList(V1), V2)
a__U91(X1, X2) → U91(X1, X2)
a__U43(tt, V1, V2) → a__U44(a__isPalListKind(V2), V1, V2)
a__U92(X) → U92(X)
a__U42(tt, V1, V2) → a__U43(a__isPalListKind(V2), V1, V2)
a__U41(tt, V1, V2) → a__U42(a__isPalListKind(V1), V1, V2)
a__U33(tt) → tt
a__U32(tt, V) → a__U33(a__isQid(V))
a__U31(tt, V) → a__U32(a__isPalListKind(V), V)
a__U26(tt) → tt
a__U25(tt, V2) → a__U26(a__isList(V2))
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U23(tt, V1, V2) → a__U24(a__isPalListKind(V2), V1, V2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U24(tt, V1, V2) → a__U25(a__isList(V1), V2)
a__isQid(X) → isQid(X)
a__U21(tt, V1, V2) → a__U22(a__isPalListKind(V1), V1, V2)
a__U33(X) → U33(X)
a__U22(tt, V1, V2) → a__U23(a__isPalListKind(V2), V1, V2)
a__U32(X1, X2) → U32(X1, X2)
a__U12(tt, V) → a__U13(a__isNeList(V))
a__U31(X1, X2) → U31(X1, X2)
a__U13(tt) → tt
a__U26(X) → U26(X)
a__isList(X) → isList(X)
a__U11(tt, V) → a__U12(a__isPalListKind(V), V)
a__U54(X1, X2, X3) → U54(X1, X2, X3)
a__U53(X1, X2, X3) → U53(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U46(X) → U46(X)
a__U45(X1, X2) → U45(X1, X2)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__isPalListKind(i) → tt
a__isPalListKind(e) → tt
a__isPalListKind(o) → tt
a__isPalListKind(nil) → tt
a__isPalListKind(__(V1, V2)) → a__U91(a__isPalListKind(V1), V2)
a__isPalListKind(u) → tt
a__isQid(e) → tt
a__isQid(a) → tt
a__isNeList(__(V1, V2)) → a__U41(a__isPalListKind(V1), V1, V2)
a__isNeList(V) → a__U31(a__isPalListKind(V), V)
a__isNeList(__(V1, V2)) → a__U51(a__isPalListKind(V1), V1, V2)
a__isPalListKind(a) → tt
a__U91(tt, V2) → a__U92(a__isPalListKind(V2))
a__U92(tt) → tt
a__isList(V) → a__U11(a__isPalListKind(V), V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isPalListKind(V1), V1, V2)
a__U56(tt) → tt
a__isQid(u) → tt
a__isQid(i) → tt
a__isQid(o) → tt

(17) Obligation:

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

A__U12(tt, V) → A__ISNELIST(V)
A__U41(tt, V1, V2) → A__U42(a__isPalListKind(V1), V1, V2)
A__U42(tt, V1, V2) → A__U43(a__isPalListKind(V2), V1, V2)
A__U43(tt, V1, V2) → A__U44(a__isPalListKind(V2), V1, V2)
A__U51(tt, V1, V2) → A__U52(a__isPalListKind(V1), V1, V2)
A__U52(tt, V1, V2) → A__U53(a__isPalListKind(V2), V1, V2)
A__U53(tt, V1, V2) → A__U54(a__isPalListKind(V2), V1, V2)
A__U55(tt, V2) → A__ISLIST(V2)
A__ISLIST(V) → A__U11(a__isPalListKind(V), V)
A__U11(tt, V) → A__U12(a__isPalListKind(V), V)
A__U21(tt, V1, V2) → A__U22(a__isPalListKind(V1), V1, V2)
A__U25(tt, V2) → A__ISLIST(V2)

The TRS R consists of the following rules:

a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__U11(tt, V) → a__U12(a__isPalListKind(V), V)
a__U12(tt, V) → a__U13(a__isNeList(V))
a__U13(tt) → tt
a__U21(tt, V1, V2) → a__U22(a__isPalListKind(V1), V1, V2)
a__U22(tt, V1, V2) → a__U23(a__isPalListKind(V2), V1, V2)
a__U23(tt, V1, V2) → a__U24(a__isPalListKind(V2), V1, V2)
a__U24(tt, V1, V2) → a__U25(a__isList(V1), V2)
a__U25(tt, V2) → a__U26(a__isList(V2))
a__U26(tt) → tt
a__U31(tt, V) → a__U32(a__isPalListKind(V), V)
a__U32(tt, V) → a__U33(a__isQid(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isPalListKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isPalListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isPalListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isList(V1), V2)
a__U45(tt, V2) → a__U46(a__isNeList(V2))
a__U46(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isPalListKind(V1), V1, V2)
a__U52(tt, V1, V2) → a__U53(a__isPalListKind(V2), V1, V2)
a__U53(tt, V1, V2) → a__U54(a__isPalListKind(V2), V1, V2)
a__U54(tt, V1, V2) → a__U55(a__isNeList(V1), V2)
a__U55(tt, V2) → a__U56(a__isList(V2))
a__U56(tt) → tt
a__U61(tt, V) → a__U62(a__isPalListKind(V), V)
a__U62(tt, V) → a__U63(a__isQid(V))
a__U63(tt) → tt
a__U71(tt, I, P) → a__U72(a__isPalListKind(I), P)
a__U72(tt, P) → a__U73(a__isPal(P), P)
a__U73(tt, P) → a__U74(a__isPalListKind(P))
a__U74(tt) → tt
a__U81(tt, V) → a__U82(a__isPalListKind(V), V)
a__U82(tt, V) → a__U83(a__isNePal(V))
a__U83(tt) → tt
a__U91(tt, V2) → a__U92(a__isPalListKind(V2))
a__U92(tt) → tt
a__isList(V) → a__U11(a__isPalListKind(V), V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isPalListKind(V1), V1, V2)
a__isNeList(V) → a__U31(a__isPalListKind(V), V)
a__isNeList(__(V1, V2)) → a__U41(a__isPalListKind(V1), V1, V2)
a__isNeList(__(V1, V2)) → a__U51(a__isPalListKind(V1), V1, V2)
a__isNePal(V) → a__U61(a__isPalListKind(V), V)
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), I, P)
a__isPal(V) → a__U81(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__U91(a__isPalListKind(V1), 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(X1, X2)) → a__U12(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(U23(X1, X2, X3)) → a__U23(mark(X1), X2, X3)
mark(U24(X1, X2, X3)) → a__U24(mark(X1), X2, X3)
mark(U25(X1, X2)) → a__U25(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(U26(X)) → a__U26(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(U53(X1, X2, X3)) → a__U53(mark(X1), X2, X3)
mark(U54(X1, X2, X3)) → a__U54(mark(X1), X2, X3)
mark(U55(X1, X2)) → a__U55(mark(X1), X2)
mark(U56(X)) → a__U56(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(U73(X1, X2)) → a__U73(mark(X1), X2)
mark(isPal(X)) → a__isPal(X)
mark(U74(X)) → a__U74(mark(X))
mark(U81(X1, X2)) → a__U81(mark(X1), X2)
mark(U82(X1, X2)) → a__U82(mark(X1), X2)
mark(U83(X)) → a__U83(mark(X))
mark(isNePal(X)) → a__isNePal(X)
mark(U91(X1, X2)) → a__U91(mark(X1), X2)
mark(U92(X)) → a__U92(mark(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(X1, X2) → U12(X1, X2)
a__isPalListKind(X) → isPalListKind(X)
a__U13(X) → U13(X)
a__isNeList(X) → isNeList(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__U23(X1, X2, X3) → U23(X1, X2, X3)
a__U24(X1, X2, X3) → U24(X1, X2, X3)
a__U25(X1, X2) → U25(X1, X2)
a__isList(X) → isList(X)
a__U26(X) → U26(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__isQid(X) → isQid(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__U53(X1, X2, X3) → U53(X1, X2, X3)
a__U54(X1, X2, X3) → U54(X1, X2, X3)
a__U55(X1, X2) → U55(X1, X2)
a__U56(X) → U56(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__U73(X1, X2) → U73(X1, X2)
a__isPal(X) → isPal(X)
a__U74(X) → U74(X)
a__U81(X1, X2) → U81(X1, X2)
a__U82(X1, X2) → U82(X1, X2)
a__U83(X) → U83(X)
a__isNePal(X) → isNePal(X)
a__U91(X1, X2) → U91(X1, X2)
a__U92(X) → U92(X)

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

(18) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 12 less nodes.

(19) TRUE

(20) 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)) → MARK(X1)
MARK(U12(X1, X2)) → MARK(X1)
MARK(U13(X)) → MARK(X)
MARK(U21(X1, X2, X3)) → MARK(X1)
MARK(U22(X1, X2, X3)) → MARK(X1)
MARK(U23(X1, X2, X3)) → MARK(X1)
MARK(U24(X1, X2, X3)) → MARK(X1)
MARK(U25(X1, X2)) → MARK(X1)
MARK(U26(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U33(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(U43(X1, X2, X3)) → MARK(X1)
MARK(U44(X1, X2, X3)) → MARK(X1)
MARK(U45(X1, X2)) → MARK(X1)
MARK(U46(X)) → MARK(X)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2, X3)) → MARK(X1)
MARK(U53(X1, X2, X3)) → MARK(X1)
MARK(U54(X1, X2, X3)) → MARK(X1)
MARK(U55(X1, X2)) → MARK(X1)
MARK(U56(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U62(X1, X2)) → MARK(X1)
MARK(U63(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U72(X1, X2)) → MARK(X1)
MARK(U73(X1, X2)) → MARK(X1)
MARK(U74(X)) → MARK(X)
MARK(U81(X1, X2)) → MARK(X1)
MARK(U82(X1, X2)) → MARK(X1)
MARK(U83(X)) → MARK(X)
MARK(U91(X1, X2)) → MARK(X1)
MARK(U92(X)) → MARK(X)
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__isPalListKind(V), V)
a__U12(tt, V) → a__U13(a__isNeList(V))
a__U13(tt) → tt
a__U21(tt, V1, V2) → a__U22(a__isPalListKind(V1), V1, V2)
a__U22(tt, V1, V2) → a__U23(a__isPalListKind(V2), V1, V2)
a__U23(tt, V1, V2) → a__U24(a__isPalListKind(V2), V1, V2)
a__U24(tt, V1, V2) → a__U25(a__isList(V1), V2)
a__U25(tt, V2) → a__U26(a__isList(V2))
a__U26(tt) → tt
a__U31(tt, V) → a__U32(a__isPalListKind(V), V)
a__U32(tt, V) → a__U33(a__isQid(V))
a__U33(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isPalListKind(V1), V1, V2)
a__U42(tt, V1, V2) → a__U43(a__isPalListKind(V2), V1, V2)
a__U43(tt, V1, V2) → a__U44(a__isPalListKind(V2), V1, V2)
a__U44(tt, V1, V2) → a__U45(a__isList(V1), V2)
a__U45(tt, V2) → a__U46(a__isNeList(V2))
a__U46(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isPalListKind(V1), V1, V2)
a__U52(tt, V1, V2) → a__U53(a__isPalListKind(V2), V1, V2)
a__U53(tt, V1, V2) → a__U54(a__isPalListKind(V2), V1, V2)
a__U54(tt, V1, V2) → a__U55(a__isNeList(V1), V2)
a__U55(tt, V2) → a__U56(a__isList(V2))
a__U56(tt) → tt
a__U61(tt, V) → a__U62(a__isPalListKind(V), V)
a__U62(tt, V) → a__U63(a__isQid(V))
a__U63(tt) → tt
a__U71(tt, I, P) → a__U72(a__isPalListKind(I), P)
a__U72(tt, P) → a__U73(a__isPal(P), P)
a__U73(tt, P) → a__U74(a__isPalListKind(P))
a__U74(tt) → tt
a__U81(tt, V) → a__U82(a__isPalListKind(V), V)
a__U82(tt, V) → a__U83(a__isNePal(V))
a__U83(tt) → tt
a__U91(tt, V2) → a__U92(a__isPalListKind(V2))
a__U92(tt) → tt
a__isList(V) → a__U11(a__isPalListKind(V), V)
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isPalListKind(V1), V1, V2)
a__isNeList(V) → a__U31(a__isPalListKind(V), V)
a__isNeList(__(V1, V2)) → a__U41(a__isPalListKind(V1), V1, V2)
a__isNeList(__(V1, V2)) → a__U51(a__isPalListKind(V1), V1, V2)
a__isNePal(V) → a__U61(a__isPalListKind(V), V)
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), I, P)
a__isPal(V) → a__U81(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__U91(a__isPalListKind(V1), 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(X1, X2)) → a__U12(mark(X1), X2)
mark(isPalListKind(X)) → a__isPalListKind(X)
mark(U13(X)) → a__U13(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(U22(X1, X2, X3)) → a__U22(mark(X1), X2, X3)
mark(U23(X1, X2, X3)) → a__U23(mark(X1), X2, X3)
mark(U24(X1, X2, X3)) → a__U24(mark(X1), X2, X3)
mark(U25(X1, X2)) → a__U25(mark(X1), X2)
mark(isList(X)) → a__isList(X)
mark(U26(X)) → a__U26(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X1, X2)) → a__U32(mark(X1), X2)
mark(U33(X)) → a__U33(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(U43(X1, X2, X3)) → a__U43(mark(X1), X2, X3)
mark(U44(X1, X2, X3)) → a__U44(mark(X1), X2, X3)
mark(U45(X1, X2)) → a__U45(mark(X1), X2)
mark(U46(X)) → a__U46(mark(X))
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(U53(X1, X2, X3)) → a__U53(mark(X1), X2, X3)
mark(U54(X1, X2, X3)) → a__U54(mark(X1), X2, X3)
mark(U55(X1, X2)) → a__U55(mark(X1), X2)
mark(U56(X)) → a__U56(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(U63(X)) → a__U63(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(U73(X1, X2)) → a__U73(mark(X1), X2)
mark(isPal(X)) → a__isPal(X)
mark(U74(X)) → a__U74(mark(X))
mark(U81(X1, X2)) → a__U81(mark(X1), X2)
mark(U82(X1, X2)) → a__U82(mark(X1), X2)
mark(U83(X)) → a__U83(mark(X))
mark(isNePal(X)) → a__isNePal(X)
mark(U91(X1, X2)) → a__U91(mark(X1), X2)
mark(U92(X)) → a__U92(mark(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(X1, X2) → U12(X1, X2)
a__isPalListKind(X) → isPalListKind(X)
a__U13(X) → U13(X)
a__isNeList(X) → isNeList(X)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__U22(X1, X2, X3) → U22(X1, X2, X3)
a__U23(X1, X2, X3) → U23(X1, X2, X3)
a__U24(X1, X2, X3) → U24(X1, X2, X3)
a__U25(X1, X2) → U25(X1, X2)
a__isList(X) → isList(X)
a__U26(X) → U26(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X1, X2) → U32(X1, X2)
a__U33(X) → U33(X)
a__isQid(X) → isQid(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__U43(X1, X2, X3) → U43(X1, X2, X3)
a__U44(X1, X2, X3) → U44(X1, X2, X3)
a__U45(X1, X2) → U45(X1, X2)
a__U46(X) → U46(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__U53(X1, X2, X3) → U53(X1, X2, X3)
a__U54(X1, X2, X3) → U54(X1, X2, X3)
a__U55(X1, X2) → U55(X1, X2)
a__U56(X) → U56(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X1, X2) → U62(X1, X2)
a__U63(X) → U63(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__U73(X1, X2) → U73(X1, X2)
a__isPal(X) → isPal(X)
a__U74(X) → U74(X)
a__U81(X1, X2) → U81(X1, X2)
a__U82(X1, X2) → U82(X1, X2)
a__U83(X) → U83(X)
a__isNePal(X) → isNePal(X)
a__U91(X1, X2) → U91(X1, X2)
a__U92(X) → U92(X)

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