Termination w.r.t. Q proof of /tmp/tmp3gg8MS/PALINDROME_complete

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

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 QDP

↳5 QDPOrderProof (⇔)

↳6 QDP

↳7 DependencyGraphProof (⇔)

↳8 AND

    ↳9 QDP

        ↳10 QDPOrderProof (⇔)

        ↳11 QDP

        ↳12 DependencyGraphProof (⇔)

        ↳13 TRUE

    ↳14 QDP

        ↳15 QDPOrderProof (⇔)

        ↳16 QDP

        ↳17 QDPOrderProof (⇔)

        ↳18 QDP

        ↳19 DependencyGraphProof (⇔)

        ↳20 QDP

        ↳21 QDPOrderProof (⇔)

        ↳22 QDP

        ↳23 QDPOrderProof (⇔)

        ↳24 QDP

        ↳25 QDPOrderProof (⇔)

        ↳26 QDP

        ↳27 QDPOrderProof (⇔)

        ↳28 QDP

        ↳29 QDPOrderProof (⇔)

        ↳30 QDP

        ↳31 QDPOrderProof (⇔)

        ↳32 QDP

        ↳33 QDPOrderProof (⇔)

        ↳34 QDP

        ↳35 PisEmptyProof (⇔)

        ↳36 TRUE

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

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

(2) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(3) DependencyGraphProof (EQUIVALENT transformation)

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

(4) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNELIST(V) → A__ISPALLISTKIND(V)
MARK(isNeList(X)) → A__ISNELIST(X)
A__ISNELIST(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__ISNELIST(__(V1, V2)) → A__ISPALLISTKIND(V1)
A__ISLIST(V) → A__ISPALLISTKIND(V)
A__ISLIST(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__ISLIST(__(V1, V2)) → A__ISPALLISTKIND(V1)
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(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
A__ISNEPAL(__(I, __(P, I))) → A__AND(a__isQid(I), isPalListKind(I))

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)
MARK(x0, x1) = MARK(x0, x1)
A__U11(x0, x1, x2) = A__U11(x1)
A__ISNELIST(x0, x1) = A__ISNELIST(x0, x1)
A__ISPALLISTKIND(x0, x1) = A__ISPALLISTKIND(x0, x1)
A__AND(x0, x1, x2) = A__AND(x0, x2)
A__U41(x0, x1, x2, x3) = A__U41(x1)
A__U42(x0, x1, x2) = A__U42(x0, x1)
A__U51(x0, x1, x2, x3) = A__U51(x0, x1)
A__U52(x0, x1, x2) = A__U52(x0, x1)
A__ISLIST(x0, x1) = A__ISLIST(x0)
A__U21(x0, x1, x2, x3) = A__U21(x0, x1)
A__U22(x0, x1, x2) = A__U22(x1)
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 [0,0,27] and root tag 1
MARK has argument tags [0,32] and root tag 1
A__U11 has argument tags [12,0,61] and root tag 0
A__ISNELIST has argument tags [0,57] and root tag 0
A__ISPALLISTKIND has argument tags [32,28] and root tag 1
A__AND has argument tags [0,59,32] and root tag 1
A__U41 has argument tags [32,0,0,28] and root tag 0
A__U42 has argument tags [20,0,32] and root tag 0
A__U51 has argument tags [49,0,6,0] and root tag 0
A__U52 has argument tags [56,0,0] and root tag 0
A__ISLIST has argument tags [0,60] and root tag 0
A__U21 has argument tags [0,16,0,8] and root tag 0
A__U22 has argument tags [56,0,53] and root tag 0
A__U71 has argument tags [28,0,0] and root tag 1
A__ISNEPAL has argument tags [0,28] and root tag 1
A__ISPAL has argument tags [0,28] and root tag 1

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

A_{_U11} > [A, , MARK, mark, a, U11, tt, A_{_ISNELIST}, a_{_{isPalListKind}}, isNeList, a_{_and}, a_{_isList}, a_{_isNeList}, A_{_ISLIST}, A_{_U21}, U21, U22, isList, U41, U42, U51, U52, U61, U71, A_{_ISNEPAL}, a_{_isQid}, and, isPal, isNePal, A_{_ISPAL}, a_{_isNePal}, a_{_U11}, a_{_U21}, a_{_U22}, a_{_U41}, a_{_U42}, a_{_U51}, a_{_U52}, a_{_U61}, a_{_U71}, a_{_isPal}, u] > [A_{_{ISPALLISTKIND}}, isPalListKind]
A_{_U11} > [A, , MARK, mark, a, U11, tt, A_{_ISNELIST}, a_{_{isPalListKind}}, isNeList, a_{_and}, a_{_isList}, a_{_isNeList}, A_{_ISLIST}, A_{_U21}, U21, U22, isList, U41, U42, U51, U52, U61, U71, A_{_ISNEPAL}, a_{_isQid}, and, isPal, isNePal, A_{_ISPAL}, a_{_isNePal}, a_{_U11}, a_{_U21}, a_{_U22}, a_{_U41}, a_{_U42}, a_{_U51}, a_{_U52}, a_{_U61}, a_{_U71}, a_{_isPal}, u] > A_{_U42₁}
A_{_U11} > [A, , MARK, mark, a, U11, tt, A_{_ISNELIST}, a_{_{isPalListKind}}, isNeList, a_{_and}, a_{_isList}, a_{_isNeList}, A_{_ISLIST}, A_{_U21}, U21, U22, isList, U41, U42, U51, U52, U61, U71, A_{_ISNEPAL}, a_{_isQid}, and, isPal, isNePal, A_{_ISPAL}, a_{_isNePal}, a_{_U11}, a_{_U21}, a_{_U22}, a_{_U41}, a_{_U42}, a_{_U51}, a_{_U52}, a_{_U61}, a_{_U71}, a_{_isPal}, u] > A_{_U51₃}
A_{_U11} > [A, , MARK, mark, a, U11, tt, A_{_ISNELIST}, a_{_{isPalListKind}}, isNeList, a_{_and}, a_{_isList}, a_{_isNeList}, A_{_ISLIST}, A_{_U21}, U21, U22, isList, U41, U42, U51, U52, U61, U71, A_{_ISNEPAL}, a_{_isQid}, and, isPal, isNePal, A_{_ISPAL}, a_{_isNePal}, a_{_U11}, a_{_U21}, a_{_U22}, a_{_U41}, a_{_U42}, a_{_U51}, a_{_U52}, a_{_U61}, a_{_U71}, a_{_isPal}, u] > A_{_U22₂}
A_{_U11} > [A, , MARK, mark, a, U11, tt, A_{_ISNELIST}, a_{_{isPalListKind}}, isNeList, a_{_and}, a_{_isList}, a_{_isNeList}, A_{_ISLIST}, A_{_U21}, U21, U22, isList, U41, U42, U51, U52, U61, U71, A_{_ISNEPAL}, a_{_isQid}, and, isPal, isNePal, A_{_ISPAL}, a_{_isNePal}, a_{_U11}, a_{_U21}, a_{_U22}, a_{_U41}, a_{_U42}, a_{_U51}, a_{_U52}, a_{_U61}, a_{_U71}, a_{_isPal}, u] > nil
A_{_U11} > [A, , MARK, mark, a, U11, tt, A_{_ISNELIST}, a_{_{isPalListKind}}, isNeList, a_{_and}, a_{_isList}, a_{_isNeList}, A_{_ISLIST}, A_{_U21}, U21, U22, isList, U41, U42, U51, U52, U61, U71, A_{_ISNEPAL}, a_{_isQid}, and, isPal, isNePal, A_{_ISPAL}, a_{_isNePal}, a_{_U11}, a_{_U21}, a_{_U22}, a_{_U41}, a_{_U42}, a_{_U51}, a_{_U52}, a_{_U61}, a_{_U71}, a_{_isPal}, u] > isQid
A_{_U11} > [A, , MARK, mark, a, U11, tt, A_{_ISNELIST}, a_{_{isPalListKind}}, isNeList, a_{_and}, a_{_isList}, a_{_isNeList}, A_{_ISLIST}, A_{_U21}, U21, U22, isList, U41, U42, U51, U52, U61, U71, A_{_ISNEPAL}, a_{_isQid}, and, isPal, isNePal, A_{_ISPAL}, a_{_isNePal}, a_{_U11}, a_{_U21}, a_{_U22}, a_{_U41}, a_{_U42}, a_{_U51}, a_{_U52}, a_{_U61}, a_{_U71}, a_{_isPal}, u] > a
A_{_U11} > [A, , MARK, mark, a, U11, tt, A_{_ISNELIST}, a_{_{isPalListKind}}, isNeList, a_{_and}, a_{_isList}, a_{_isNeList}, A_{_ISLIST}, A_{_U21}, U21, U22, isList, U41, U42, U51, U52, U61, U71, A_{_ISNEPAL}, a_{_isQid}, and, isPal, isNePal, A_{_ISPAL}, a_{_isNePal}, a_{_U11}, a_{_U21}, a_{_U22}, a_{_U41}, a_{_U42}, a_{_U51}, a_{_U52}, a_{_U61}, a_{_U71}, a_{_isPal}, u] > e
A_{_U11} > [A, , MARK, mark, a, U11, tt, A_{_ISNELIST}, a_{_{isPalListKind}}, isNeList, a_{_and}, a_{_isList}, a_{_isNeList}, A_{_ISLIST}, A_{_U21}, U21, U22, isList, U41, U42, U51, U52, U61, U71, A_{_ISNEPAL}, a_{_isQid}, and, isPal, isNePal, A_{_ISPAL}, a_{_isNePal}, a_{_U11}, a_{_U21}, a_{_U22}, a_{_U41}, a_{_U42}, a_{_U51}, a_{_U52}, a_{_U61}, a_{_U71}, a_{_isPal}, u] > i
A_{_U11} > [A, , MARK, mark, a, U11, tt, A_{_ISNELIST}, a_{_{isPalListKind}}, isNeList, a_{_and}, a_{_isList}, a_{_isNeList}, A_{_ISLIST}, A_{_U21}, U21, U22, isList, U41, U42, U51, U52, U61, U71, A_{_ISNEPAL}, a_{_isQid}, and, isPal, isNePal, A_{_ISPAL}, a_{_isNePal}, a_{_U11}, a_{_U21}, a_{_U22}, a_{_U41}, a_{_U42}, a_{_U51}, a_{_U52}, a_{_U61}, a_{_U71}, a_{_isPal}, u] > o
A_{_U52₂} > [A, , MARK, mark, a, U11, tt, A_{_ISNELIST}, a_{_{isPalListKind}}, isNeList, a_{_and}, a_{_isList}, a_{_isNeList}, A_{_ISLIST}, A_{_U21}, U21, U22, isList, U41, U42, U51, U52, U61, U71, A_{_ISNEPAL}, a_{_isQid}, and, isPal, isNePal, A_{_ISPAL}, a_{_isNePal}, a_{_U11}, a_{_U21}, a_{_U22}, a_{_U41}, a_{_U42}, a_{_U51}, a_{_U52}, a_{_U61}, a_{_U71}, a_{_isPal}, u] > [A_{_{ISPALLISTKIND}}, isPalListKind]
A_{_U52₂} > [A, , MARK, mark, a, U11, tt, A_{_ISNELIST}, a_{_{isPalListKind}}, isNeList, a_{_and}, a_{_isList}, a_{_isNeList}, A_{_ISLIST}, A_{_U21}, U21, U22, isList, U41, U42, U51, U52, U61, U71, A_{_ISNEPAL}, a_{_isQid}, and, isPal, isNePal, A_{_ISPAL}, a_{_isNePal}, a_{_U11}, a_{_U21}, a_{_U22}, a_{_U41}, a_{_U42}, a_{_U51}, a_{_U52}, a_{_U61}, a_{_U71}, a_{_isPal}, u] > A_{_U42₁}
A_{_U52₂} > [A, , MARK, mark, a, U11, tt, A_{_ISNELIST}, a_{_{isPalListKind}}, isNeList, a_{_and}, a_{_isList}, a_{_isNeList}, A_{_ISLIST}, A_{_U21}, U21, U22, isList, U41, U42, U51, U52, U61, U71, A_{_ISNEPAL}, a_{_isQid}, and, isPal, isNePal, A_{_ISPAL}, a_{_isNePal}, a_{_U11}, a_{_U21}, a_{_U22}, a_{_U41}, a_{_U42}, a_{_U51}, a_{_U52}, a_{_U61}, a_{_U71}, a_{_isPal}, u] > A_{_U51₃}
A_{_U52₂} > [A, , MARK, mark, a, U11, tt, A_{_ISNELIST}, a_{_{isPalListKind}}, isNeList, a_{_and}, a_{_isList}, a_{_isNeList}, A_{_ISLIST}, A_{_U21}, U21, U22, isList, U41, U42, U51, U52, U61, U71, A_{_ISNEPAL}, a_{_isQid}, and, isPal, isNePal, A_{_ISPAL}, a_{_isNePal}, a_{_U11}, a_{_U21}, a_{_U22}, a_{_U41}, a_{_U42}, a_{_U51}, a_{_U52}, a_{_U61}, a_{_U71}, a_{_isPal}, u] > A_{_U22₂}
A_{_U52₂} > [A, , MARK, mark, a, U11, tt, A_{_ISNELIST}, a_{_{isPalListKind}}, isNeList, a_{_and}, a_{_isList}, a_{_isNeList}, A_{_ISLIST}, A_{_U21}, U21, U22, isList, U41, U42, U51, U52, U61, U71, A_{_ISNEPAL}, a_{_isQid}, and, isPal, isNePal, A_{_ISPAL}, a_{_isNePal}, a_{_U11}, a_{_U21}, a_{_U22}, a_{_U41}, a_{_U42}, a_{_U51}, a_{_U52}, a_{_U61}, a_{_U71}, a_{_isPal}, u] > nil
A_{_U52₂} > [A, , MARK, mark, a, U11, tt, A_{_ISNELIST}, a_{_{isPalListKind}}, isNeList, a_{_and}, a_{_isList}, a_{_isNeList}, A_{_ISLIST}, A_{_U21}, U21, U22, isList, U41, U42, U51, U52, U61, U71, A_{_ISNEPAL}, a_{_isQid}, and, isPal, isNePal, A_{_ISPAL}, a_{_isNePal}, a_{_U11}, a_{_U21}, a_{_U22}, a_{_U41}, a_{_U42}, a_{_U51}, a_{_U52}, a_{_U61}, a_{_U71}, a_{_isPal}, u] > isQid
A_{_U52₂} > [A, , MARK, mark, a, U11, tt, A_{_ISNELIST}, a_{_{isPalListKind}}, isNeList, a_{_and}, a_{_isList}, a_{_isNeList}, A_{_ISLIST}, A_{_U21}, U21, U22, isList, U41, U42, U51, U52, U61, U71, A_{_ISNEPAL}, a_{_isQid}, and, isPal, isNePal, A_{_ISPAL}, a_{_isNePal}, a_{_U11}, a_{_U21}, a_{_U22}, a_{_U41}, a_{_U42}, a_{_U51}, a_{_U52}, a_{_U61}, a_{_U71}, a_{_isPal}, u] > a
A_{_U52₂} > [A, , MARK, mark, a, U11, tt, A_{_ISNELIST}, a_{_{isPalListKind}}, isNeList, a_{_and}, a_{_isList}, a_{_isNeList}, A_{_ISLIST}, A_{_U21}, U21, U22, isList, U41, U42, U51, U52, U61, U71, A_{_ISNEPAL}, a_{_isQid}, and, isPal, isNePal, A_{_ISPAL}, a_{_isNePal}, a_{_U11}, a_{_U21}, a_{_U22}, a_{_U41}, a_{_U42}, a_{_U51}, a_{_U52}, a_{_U61}, a_{_U71}, a_{_isPal}, u] > e
A_{_U52₂} > [A, , MARK, mark, a, U11, tt, A_{_ISNELIST}, a_{_{isPalListKind}}, isNeList, a_{_and}, a_{_isList}, a_{_isNeList}, A_{_ISLIST}, A_{_U21}, U21, U22, isList, U41, U42, U51, U52, U61, U71, A_{_ISNEPAL}, a_{_isQid}, and, isPal, isNePal, A_{_ISPAL}, a_{_isNePal}, a_{_U11}, a_{_U21}, a_{_U22}, a_{_U41}, a_{_U42}, a_{_U51}, a_{_U52}, a_{_U61}, a_{_U71}, a_{_isPal}, u] > i
A_{_U52₂} > [A, , MARK, mark, a, U11, tt, A_{_ISNELIST}, a_{_{isPalListKind}}, isNeList, a_{_and}, a_{_isList}, a_{_isNeList}, A_{_ISLIST}, A_{_U21}, U21, U22, isList, U41, U42, U51, U52, U61, U71, A_{_ISNEPAL}, a_{_isQid}, and, isPal, isNePal, A_{_ISPAL}, a_{_isNePal}, a_{_U11}, a_{_U21}, a_{_U22}, a_{_U41}, a_{_U42}, a_{_U51}, a_{_U52}, a_{_U61}, a_{_U71}, a_{_isPal}, u] > o

Status:

A: []
_: []
MARK: []
mark: []
a: []
U11: []
A_{_U11}: []
tt: []
A_{_ISNELIST}: []
A_{_{ISPALLISTKIND}}: []
a_{_{isPalListKind}}: []
isPalListKind: []
isNeList: []
a_{_and}: []
A_{_U42₁}: [1]
a_{_isList}: []
A_{_U51₃}: [1,2,3]
A_{_U52₂}: [1,2]
a_{_isNeList}: []
A_{_ISLIST}: []
A_{_U21}: []
A_{_U22₂}: [1,2]
U21: []
U22: []
isList: []
U41: []
U42: []
U51: []
U52: []
U61: []
U71: []
A_{_ISNEPAL}: []
a_{_isQid}: []
and: []
isPal: []
isNePal: []
A_{_ISPAL}: []
nil: []
a_{_isNePal}: []
a_{_U11}: []
a_{_U21}: []
a_{_U22}: []
isQid: []
a_{_U41}: []
a_{_U42}: []
a_{_U51}: []
a_{_U52}: []
a_{_U61}: []
a_{_U71}: []
a_{_isPal}: []
a: []
e: []
i: []
o: []
u: []

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____(__(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)
A__U11(tt, V) → A__ISNELIST(V)
A__ISPALLISTKIND(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__AND(tt, X) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(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__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(__(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__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)) → MARK(X1)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
MARK(U51(X1, X2, X3)) → MARK(X1)
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)))
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.

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Complex Obligation (AND)

(9) Obligation:

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

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__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__U11(tt, V) → A__ISNELIST(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__U21(tt, V1, V2) → A__ISLIST(V1)
A__U51(tt, V1, V2) → A__ISNELIST(V1)
A__U41(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__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.

(10) 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__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__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__U11(tt, V) → A__ISNELIST(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__U21(tt, V1, V2) → A__ISLIST(V1)
A__U51(tt, V1, V2) → A__ISNELIST(V1)
A__U41(tt, V1, V2) → A__ISLIST(V1)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A__ISNELIST(x0, x1) = A__ISNELIST(x0, x1)
A__U41(x0, x1, x2, x3) = A__U41(x0, x2, x3)
A__U42(x0, x1, x2) = A__U42(x1, x2)
A__U51(x0, x1, x2, x3) = A__U51(x0, x1)
A__U52(x0, x1, x2) = A__U52(x0, x1, x2)
A__ISLIST(x0, x1) = A__ISLIST(x1)
A__U11(x0, x1, x2) = A__U11(x0, x1)
A__U21(x0, x1, x2, x3) = A__U21(x0, x1)
A__U22(x0, x1, x2) = A__U22(x0, x1)

Tags:
A__ISNELIST has argument tags [0,1] and root tag 4
A__U41 has argument tags [30,30,20,5] and root tag 4
A__U42 has argument tags [26,0,2] and root tag 8
A__U51 has argument tags [1,1,2,4] and root tag 4
A__U52 has argument tags [16,7,21] and root tag 8
A__ISLIST has argument tags [8,21] and root tag 5
A__U11 has argument tags [21,21,10] and root tag 1
A__U21 has argument tags [21,21,16,1] and root tag 0
A__U22 has argument tags [21,22,2] 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__ISNELIST(x1) = x1
__(x1, x2) = __(x1, x2)
A__U41(x1, x2, x3) = A__U41(x2)
a__and(x1, x2) = a__and(x1, x2)
a__isPalListKind(x1) = x1
isPalListKind(x1) = x1
tt = tt
A__U42(x1, x2) = A__U42(x1, x2)
a__isList(x1) = a__isList
A__U51(x1, x2, x3) = A__U51(x2, x3)
A__U52(x1, x2) = x1
a__isNeList(x1) = a__isNeList
A__ISLIST(x1) = x1
A__U11(x1, x2) = x2
A__U21(x1, x2, x3) = A__U21(x2, x3)
A__U22(x1, x2) = x2
a = a
e = e
i = i
nil = nil
o = o
u = u
a____(x1, x2) = a____(x1, x2)
mark(x1) = x1
isNePal(x1) = isNePal(x1)
a__isNePal(x1) = a__isNePal(x1)
a__isQid(x1) = a__isQid
and(x1, x2) = and(x1, x2)
isPal(x1) = x1
a__U11(x1, x2) = a__U11
a__U21(x1, x2, x3) = a__U21
isList(x1) = isList
a__U31(x1, x2) = a__U31
a__U41(x1, x2, x3) = a__U41
a__U51(x1, x2, x3) = a__U51
isNeList(x1) = isNeList
U11(x1, x2) = U11
U12(x1) = x1
a__U12(x1) = x1
U21(x1, x2, x3) = U21
U22(x1, x2) = U22
a__U22(x1, x2) = a__U22
U23(x1) = x1
a__U23(x1) = x1
U31(x1, x2) = U31
U32(x1) = x1
a__U32(x1) = x1
U41(x1, x2, x3) = U41
U42(x1, x2) = U42
a__U42(x1, x2) = a__U42
U43(x1) = x1
a__U43(x1) = x1
U51(x1, x2, x3) = U51
U52(x1, x2) = x1
a__U52(x1, x2) = x1
U53(x1) = x1
a__U53(x1) = x1
U61(x1, x2) = U61(x1)
a__U61(x1, x2) = a__U61(x1)
U62(x1) = U62
a__U62(x1) = a__U62
U71(x1, x2) = x1
a__U71(x1, x2) = x1
U72(x1) = U72
a__U72(x1) = a__U72
a__isPal(x1) = x1
isQid(x1) = isQid

Lexicographic path order with status [LPO].
Quasi-Precedence:

e > [A_{_U41₁}, tt, a_{_isList}, a_{_isNeList}, a, a_{_isQid}, a_{_U11}, a_{_U21}, isList, a_{_U31}, a_{_U41}, a_{_U51}, isNeList, U11, U21, U22, a_{_U22}, U31, U41, U42, a_{_U42}, U51, U72, a_{_U72}, isQid] > A_{_U42₂}
i > [A_{_U41₁}, tt, a_{_isList}, a_{_isNeList}, a, a_{_isQid}, a_{_U11}, a_{_U21}, isList, a_{_U31}, a_{_U41}, a_{_U51}, isNeList, U11, U21, U22, a_{_U22}, U31, U41, U42, a_{_U42}, U51, U72, a_{_U72}, isQid] > A_{_U42₂}
nil > [A_{_U41₁}, tt, a_{_isList}, a_{_isNeList}, a, a_{_isQid}, a_{_U11}, a_{_U21}, isList, a_{_U31}, a_{_U41}, a_{_U51}, isNeList, U11, U21, U22, a_{_U22}, U31, U41, U42, a_{_U42}, U51, U72, a_{_U72}, isQid] > A_{_U42₂}
o > [A_{_U41₁}, tt, a_{_isList}, a_{_isNeList}, a, a_{_isQid}, a_{_U11}, a_{_U21}, isList, a_{_U31}, a_{_U41}, a_{_U51}, isNeList, U11, U21, U22, a_{_U22}, U31, U41, U42, a_{_U42}, U51, U72, a_{_U72}, isQid] > A_{_U42₂}
u > [A_{_U41₁}, tt, a_{_isList}, a_{_isNeList}, a, a_{_isQid}, a_{_U11}, a_{_U21}, isList, a_{_U31}, a_{_U41}, a_{_U51}, isNeList, U11, U21, U22, a_{_U22}, U31, U41, U42, a_{_U42}, U51, U72, a_{_U72}, isQid] > A_{_U42₂}
[isNePal₁, a_{_isNePal₁}, U61₁, a_{_U61₁}] > [_{_₂}, a_{_and₂}, A_{_U51₂}, A_{_U21₂}, a_{_{_{_₂}}}, and₂] > [A_{_U41₁}, tt, a_{_isList}, a_{_isNeList}, a, a_{_isQid}, a_{_U11}, a_{_U21}, isList, a_{_U31}, a_{_U41}, a_{_U51}, isNeList, U11, U21, U22, a_{_U22}, U31, U41, U42, a_{_U42}, U51, U72, a_{_U72}, isQid] > A_{_U42₂}
[isNePal₁, a_{_isNePal₁}, U61₁, a_{_U61₁}] > [U62, a_{_U62}] > [A_{_U41₁}, tt, a_{_isList}, a_{_isNeList}, a, a_{_isQid}, a_{_U11}, a_{_U21}, isList, a_{_U31}, a_{_U41}, a_{_U51}, isNeList, U11, U21, U22, a_{_U22}, U31, U41, U42, a_{_U42}, U51, U72, a_{_U72}, isQid] > A_{_U42₂}

Status:

__₂: [1,2]
A_{_U41₁}: [1]
a_{_and₂}: [1,2]
tt: []
A_{_U42₂}: [1,2]
a_{_isList}: []
A_{_U51₂}: [1,2]
a_{_isNeList}: []
A_{_U21₂}: [1,2]
a: []
e: []
i: []
nil: []
o: []
u: []
a_{_{_{_₂}}}: [1,2]
isNePal₁: [1]
a_{_isNePal₁}: [1]
a_{_isQid}: []
and₂: [1,2]
a_{_U11}: []
a_{_U21}: []
isList: []
a_{_U31}: []
a_{_U41}: []
a_{_U51}: []
isNeList: []
U11: []
U21: []
U22: []
a_{_U22}: []
U31: []
U41: []
U42: []
a_{_U42}: []
U51: []
U61₁: [1]
a_{_U61₁}: [1]
U62: []
a_{_U62}: []
U72: []
a_{_U72}: []
isQid: []

The following usable rules [FROCOS05] were oriented:

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__U11(tt, V) → a__U12(a__isNeList(V))
a__U41(tt, V1, V2) → a__U42(a__isList(V1), V2)
a__U21(tt, V1, V2) → a__U22(a__isList(V1), V2)
a__U51(tt, V1, V2) → a__U52(a__isNeList(V1), V2)
a__U22(tt, V2) → a__U23(a__isList(V2))
a__U42(tt, V2) → a__U43(a__isNeList(V2))
a__U52(tt, V2) → a__U53(a__isList(V2))
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__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__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)

(11) Obligation:

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

A__ISNELIST(__(V1, V2)) → A__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, 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__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.

(12) DependencyGraphProof (EQUIVALENT transformation)

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

(13) TRUE

(14) Obligation:

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

MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
A____(__(X, Y), Z) → MARK(X)
MARK(__(X1, X2)) → MARK(X1)
MARK(__(X1, X2)) → MARK(X2)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U21(X1, X2, X3)) → MARK(X1)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
MARK(U51(X1, X2, X3)) → MARK(X1)
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__ISPALLISTKIND(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__AND(tt, X) → MARK(X)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(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)))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isPalListKind(X)) → A__ISPALLISTKIND(X)
A__ISPALLISTKIND(__(V1, V2)) → A__ISPALLISTKIND(V1)
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) → 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)

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.

A__ISPAL(V) → A__ISPALLISTKIND(V)

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)
A____(x0, x1, x2) = A____(x0, x1)
A__U71(x0, x1, x2) = A__U71(x0, x2)
A__ISNEPAL(x0, x1) = A__ISNEPAL(x0, x1)
A__ISPALLISTKIND(x0, x1) = A__ISPALLISTKIND(x0, x1)
A__AND(x0, x1, x2) = A__AND(x0, x2)
A__ISPAL(x0, x1) = A__ISPAL(x0, x1)

Tags:
MARK has argument tags [0,0] and root tag 0
A____ has argument tags [0,0,27] and root tag 0
A__U71 has argument tags [0,31,16] and root tag 0
A__ISNEPAL has argument tags [0,8] and root tag 0
A__ISPALLISTKIND has argument tags [0,8] and root tag 0
A__AND has argument tags [0,27,0] and root tag 0
A__ISPAL has argument tags [0,16] and root tag 0

Comparison: MIN
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
MARK(x1) = MARK
__(x1, x2) = __
A____(x1, x2) = A____
mark(x1) = mark
a____(x1, x2) = a____
U11(x1, x2) = U11
U12(x1) = U12
U21(x1, x2, x3) = x1
U22(x1, x2) = U22
U23(x1) = U23
U31(x1, x2) = U31
U32(x1) = x1
U41(x1, x2, x3) = U41
U42(x1, x2) = U42
U43(x1) = x1
U51(x1, x2, x3) = U51
U52(x1, x2) = U52
U53(x1) = U53
U61(x1, x2) = x1
U62(x1) = U62
U71(x1, x2) = U71
A__U71(x1, x2) = A__U71
tt = tt
A__ISNEPAL(x1) = A__ISNEPAL
A__ISPALLISTKIND(x1) = A__ISPALLISTKIND
A__AND(x1, x2) = A__AND
a__isPalListKind(x1) = a__isPalListKind
isPalListKind(x1) = isPalListKind
U72(x1) = U72
isNePal(x1) = isNePal
a__and(x1, x2) = a__and
a__isQid(x1) = a__isQid
and(x1, x2) = and
isPal(x1) = isPal
A__ISPAL(x1) = A__ISPAL
nil = nil
a__isNePal(x1) = a__isNePal
a__U11(x1, x2) = a__U11
a__U12(x1) = a__U12
isNeList(x1) = isNeList
a__isNeList(x1) = a__isNeList
a__U21(x1, x2, x3) = x1
a__U22(x1, x2) = a__U22
isList(x1) = isList
a__isList(x1) = a__isList
a__U23(x1) = a__U23
a__U31(x1, x2) = a__U31
a__U32(x1) = x1
isQid(x1) = isQid
a__U41(x1, x2, x3) = a__U41
a__U42(x1, x2) = a__U42
a__U43(x1) = x1
a__U51(x1, x2, x3) = a__U51
a__U52(x1, x2) = a__U52
a__U53(x1) = a__U53
a__U61(x1, x2) = x1
a__U62(x1) = a__U62
a__U71(x1, x2) = a__U71
a__U72(x1) = a__U72
a__isPal(x1) = a__isPal
a = a
e = e
i = i
o = o
u = u

Lexicographic path order with status [LPO].
Quasi-Precedence:

[mark, a, a_{_{isPalListKind}}, a_{_and}, a_{_isNePal}, isList, a_{_isList}] > and > [MARK, , A, U11, U22, U23, U31, U41, U42, U51, U52, U53, U62, U71, A_{_U71}, tt, A_{_ISNEPAL}, A_{_AND}, U72, isNePal, A_{_ISPAL}, nil, a_{_U22}, a_{_U23}, a_{_U62}] > [A_{_{ISPALLISTKIND}}, isPalListKind]
[mark, a, a_{_{isPalListKind}}, a_{_and}, a_{_isNePal}, isList, a_{_isList}] > a_{_U11} > a_{_U12} > U12 > [MARK, , A, U11, U22, U23, U31, U41, U42, U51, U52, U53, U62, U71, A_{_U71}, tt, A_{_ISNEPAL}, A_{_AND}, U72, isNePal, A_{_ISPAL}, nil, a_{_U22}, a_{_U23}, a_{_U62}] > [A_{_{ISPALLISTKIND}}, isPalListKind]
[mark, a, a_{_{isPalListKind}}, a_{_and}, a_{_isNePal}, isList, a_{_isList}] > a_{_U11} > [a_{_isNeList}, a_{_U41}, a_{_U42}] > isNeList
[mark, a, a_{_{isPalListKind}}, a_{_and}, a_{_isNePal}, isList, a_{_isList}] > a_{_U11} > [a_{_isNeList}, a_{_U41}, a_{_U42}] > a_{_U31} > a_{_isQid} > [MARK, , A, U11, U22, U23, U31, U41, U42, U51, U52, U53, U62, U71, A_{_U71}, tt, A_{_ISNEPAL}, A_{_AND}, U72, isNePal, A_{_ISPAL}, nil, a_{_U22}, a_{_U23}, a_{_U62}] > [A_{_{ISPALLISTKIND}}, isPalListKind]
[mark, a, a_{_{isPalListKind}}, a_{_and}, a_{_isNePal}, isList, a_{_isList}] > a_{_U11} > [a_{_isNeList}, a_{_U41}, a_{_U42}] > a_{_U31} > a_{_isQid} > isQid
[mark, a, a_{_{isPalListKind}}, a_{_and}, a_{_isNePal}, isList, a_{_isList}] > a_{_U11} > [a_{_isNeList}, a_{_U41}, a_{_U42}] > [a_{_U51}, a_{_U52}, a_{_U53}] > [MARK, , A, U11, U22, U23, U31, U41, U42, U51, U52, U53, U62, U71, A_{_U71}, tt, A_{_ISNEPAL}, A_{_AND}, U72, isNePal, A_{_ISPAL}, nil, a_{_U22}, a_{_U23}, a_{_U62}] > [A_{_{ISPALLISTKIND}}, isPalListKind]
[mark, a, a_{_{isPalListKind}}, a_{_and}, a_{_isNePal}, isList, a_{_isList}] > a_{_isPal} > isPal > [MARK, , A, U11, U22, U23, U31, U41, U42, U51, U52, U53, U62, U71, A_{_U71}, tt, A_{_ISNEPAL}, A_{_AND}, U72, isNePal, A_{_ISPAL}, nil, a_{_U22}, a_{_U23}, a_{_U62}] > [A_{_{ISPALLISTKIND}}, isPalListKind]
[mark, a, a_{_{isPalListKind}}, a_{_and}, a_{_isNePal}, isList, a_{_isList}] > a_{_isPal} > [a_{_U71}, a_{_U72}] > [MARK, , A, U11, U22, U23, U31, U41, U42, U51, U52, U53, U62, U71, A_{_U71}, tt, A_{_ISNEPAL}, A_{_AND}, U72, isNePal, A_{_ISPAL}, nil, a_{_U22}, a_{_U23}, a_{_U62}] > [A_{_{ISPALLISTKIND}}, isPalListKind]
[mark, a, a_{_{isPalListKind}}, a_{_and}, a_{_isNePal}, isList, a_{_isList}] > a
[mark, a, a_{_{isPalListKind}}, a_{_and}, a_{_isNePal}, isList, a_{_isList}] > e
[mark, a, a_{_{isPalListKind}}, a_{_and}, a_{_isNePal}, isList, a_{_isList}] > i
[mark, a, a_{_{isPalListKind}}, a_{_and}, a_{_isNePal}, isList, a_{_isList}] > o > [MARK, , A, U11, U22, U23, U31, U41, U42, U51, U52, U53, U62, U71, A_{_U71}, tt, A_{_ISNEPAL}, A_{_AND}, U72, isNePal, A_{_ISPAL}, nil, a_{_U22}, a_{_U23}, a_{_U62}] > [A_{_{ISPALLISTKIND}}, isPalListKind]
[mark, a, a_{_{isPalListKind}}, a_{_and}, a_{_isNePal}, isList, a_{_isList}] > u

Status:

MARK: []
_: []
A: []
mark: []
a: []
U11: []
U12: []
U22: []
U23: []
U31: []
U41: []
U42: []
U51: []
U52: []
U53: []
U62: []
U71: []
A_{_U71}: []
tt: []
A_{_ISNEPAL}: []
A_{_{ISPALLISTKIND}}: []
A_{_AND}: []
a_{_{isPalListKind}}: []
isPalListKind: []
U72: []
isNePal: []
a_{_and}: []
a_{_isQid}: []
and: []
isPal: []
A_{_ISPAL}: []
nil: []
a_{_isNePal}: []
a_{_U11}: []
a_{_U12}: []
isNeList: []
a_{_isNeList}: []
a_{_U22}: []
isList: []
a_{_isList}: []
a_{_U23}: []
a_{_U31}: []
isQid: []
a_{_U41}: []
a_{_U42}: []
a_{_U51}: []
a_{_U52}: []
a_{_U53}: []
a_{_U62}: []
a_{_U71}: []
a_{_U72}: []
a_{_isPal}: []
a: []
e: []
i: []
o: []
u: []

The following usable rules [FROCOS05] were oriented: none

(16) Obligation:

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

MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
A____(__(X, Y), Z) → MARK(X)
MARK(__(X1, X2)) → MARK(X1)
MARK(__(X1, X2)) → MARK(X2)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U21(X1, X2, X3)) → MARK(X1)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
MARK(U51(X1, X2, X3)) → MARK(X1)
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__ISPALLISTKIND(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__AND(tt, X) → MARK(X)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(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)))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isPalListKind(X)) → A__ISPALLISTKIND(X)
A__ISPALLISTKIND(__(V1, V2)) → A__ISPALLISTKIND(V1)
MARK(isPal(X)) → A__ISPAL(X)
A__ISPAL(V) → A__U71(a__isPalListKind(V), V)
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)

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(__(X1, X2)) → A____(mark(X1), mark(X2))
A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
A____(__(X, Y), Z) → MARK(X)
MARK(__(X1, X2)) → MARK(X1)
MARK(__(X1, X2)) → MARK(X2)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U21(X1, X2, X3)) → MARK(X1)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → A__U71(mark(X1), X2)
A__U71(tt, V) → A__ISNEPAL(V)
A__ISPALLISTKIND(__(V1, V2)) → A__AND(a__isPalListKind(V1), isPalListKind(V2))
A__AND(tt, X) → MARK(X)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(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)))
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(isPalListKind(X)) → A__ISPALLISTKIND(X)
A__ISPALLISTKIND(__(V1, V2)) → A__ISPALLISTKIND(V1)
MARK(isPal(X)) → A__ISPAL(X)
A__ISPAL(V) → A__U71(a__isPalListKind(V), V)
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)

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(x1)
A____(x0, x1, x2) = A____(x0)
A__U71(x0, x1, x2) = A__U71(x0, x2)
A__ISNEPAL(x0, x1) = A__ISNEPAL(x0)
A__ISPALLISTKIND(x0, x1) = A__ISPALLISTKIND(x0, x1)
A__AND(x0, x1, x2) = A__AND(x2)
A__ISPAL(x0, x1) = A__ISPAL(x1)

Tags:
MARK has argument tags [8,8] and root tag 0
A____ has argument tags [0,23,16] and root tag 4
A__U71 has argument tags [7,24,16] and root tag 4
A__ISNEPAL has argument tags [6,18] and root tag 1
A__ISPALLISTKIND has argument tags [8,6] and root tag 1
A__AND has argument tags [2,23,19] and root tag 6
A__ISPAL has argument tags [16,12] and root tag 3

Comparison: MIN
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
MARK(x1) = MARK
__(x1, x2) = __(x1, x2)
A____(x1, x2) = A____(x1, x2)
mark(x1) = x1
a____(x1, x2) = a____(x1, x2)
U11(x1, x2) = U11(x1, x2)
U12(x1) = x1
U21(x1, x2, x3) = U21(x1, x2, x3)
U22(x1, x2) = U22(x1, x2)
U23(x1) = x1
U31(x1, x2) = x1
U32(x1) = x1
U41(x1, x2, x3) = U41(x1, x2, x3)
U42(x1, x2) = U42(x1, x2)
U43(x1) = U43(x1)
U51(x1, x2, x3) = U51(x1, x2, x3)
U52(x1, x2) = U52(x1, x2)
U53(x1) = x1
U61(x1, x2) = x1
U62(x1) = x1
U71(x1, x2) = U71(x1, x2)
A__U71(x1, x2) = x2
tt = tt
A__ISNEPAL(x1) = x1
A__ISPALLISTKIND(x1) = x1
A__AND(x1, x2) = x2
a__isPalListKind(x1) = x1
isPalListKind(x1) = x1
U72(x1) = U72(x1)
isNePal(x1) = x1
a__and(x1, x2) = a__and(x1, x2)
a__isQid(x1) = a__isQid
and(x1, x2) = and(x1, x2)
isPal(x1) = isPal(x1)
A__ISPAL(x1) = A__ISPAL
nil = nil
a__isNePal(x1) = x1
a__U11(x1, x2) = a__U11(x1, x2)
a__U12(x1) = x1
isNeList(x1) = x1
a__isNeList(x1) = x1
a__U21(x1, x2, x3) = a__U21(x1, x2, x3)
a__U22(x1, x2) = a__U22(x1, x2)
isList(x1) = isList(x1)
a__isList(x1) = a__isList(x1)
a__U23(x1) = x1
a__U31(x1, x2) = x1
a__U32(x1) = x1
isQid(x1) = isQid
a__U41(x1, x2, x3) = a__U41(x1, x2, x3)
a__U42(x1, x2) = a__U42(x1, x2)
a__U43(x1) = a__U43(x1)
a__U51(x1, x2, x3) = a__U51(x1, x2, x3)
a__U52(x1, x2) = a__U52(x1, x2)
a__U53(x1) = x1
a__U61(x1, x2) = x1
a__U62(x1) = x1
a__U71(x1, x2) = a__U71(x1, x2)
a__U72(x1) = a__U72(x1)
a__isPal(x1) = a__isPal(x1)
a = a
e = e
i = i
o = o
u = u

Lexicographic path order with status [LPO].
Quasi-Precedence:

[MARK, A_{_ISPAL}] > [_{_₂}, A_{_{_{_₂}}}, a_{_{_{_₂}}}] > [U21₃, a_{_U21₃}] > [U22₂, U41₃, a_{_U22₂}, isList₁, a_{_isList₁}, a_{_U41₃}] > [U11₂, a_{_U11₂}]
[MARK, A_{_ISPAL}] > [_{_₂}, A_{_{_{_₂}}}, a_{_{_{_₂}}}] > [U21₃, a_{_U21₃}] > [U22₂, U41₃, a_{_U22₂}, isList₁, a_{_isList₁}, a_{_U41₃}] > [tt, a_{_isQid}, isQid, u] > [U42₂, a_{_U42₂}] > [U43₁, a_{_U43₁}]
[MARK, A_{_ISPAL}] > [_{_₂}, A_{_{_{_₂}}}, a_{_{_{_₂}}}] > [U51₃, a_{_U51₃}] > [U52₂, a_{_U52₂}] > [U22₂, U41₃, a_{_U22₂}, isList₁, a_{_isList₁}, a_{_U41₃}] > [U11₂, a_{_U11₂}]
[MARK, A_{_ISPAL}] > [_{_₂}, A_{_{_{_₂}}}, a_{_{_{_₂}}}] > [U51₃, a_{_U51₃}] > [U52₂, a_{_U52₂}] > [U22₂, U41₃, a_{_U22₂}, isList₁, a_{_isList₁}, a_{_U41₃}] > [tt, a_{_isQid}, isQid, u] > [U42₂, a_{_U42₂}] > [U43₁, a_{_U43₁}]
[MARK, A_{_ISPAL}] > [_{_₂}, A_{_{_{_₂}}}, a_{_{_{_₂}}}] > [a_{_and₂}, and₂]
[MARK, A_{_ISPAL}] > [_{_₂}, A_{_{_{_₂}}}, a_{_{_{_₂}}}] > [isPal₁, a_{_isPal₁}] > [U71₂, a_{_U71₂}] > [U72₁, a_{_U72₁}] > [tt, a_{_isQid}, isQid, u] > [U42₂, a_{_U42₂}] > [U43₁, a_{_U43₁}]
nil > [tt, a_{_isQid}, isQid, u] > [U42₂, a_{_U42₂}] > [U43₁, a_{_U43₁}]
a > [tt, a_{_isQid}, isQid, u] > [U42₂, a_{_U42₂}] > [U43₁, a_{_U43₁}]
e > [tt, a_{_isQid}, isQid, u] > [U42₂, a_{_U42₂}] > [U43₁, a_{_U43₁}]
i > [tt, a_{_isQid}, isQid, u] > [U42₂, a_{_U42₂}] > [U43₁, a_{_U43₁}]
o > [tt, a_{_isQid}, isQid, u] > [U42₂, a_{_U42₂}] > [U43₁, a_{_U43₁}]

Status:

MARK: []
__₂: [1,2]
A_{_{_{_₂}}}: [1,2]
a_{_{_{_₂}}}: [1,2]
U11₂: [2,1]
U21₃: [1,2,3]
U22₂: [2,1]
U41₃: [2,1,3]
U42₂: [1,2]
U43₁: [1]
U51₃: [1,3,2]
U52₂: [1,2]
U71₂: [2,1]
tt: []
U72₁: [1]
a_{_and₂}: [1,2]
a_{_isQid}: []
and₂: [1,2]
isPal₁: [1]
A_{_ISPAL}: []
nil: []
a_{_U11₂}: [2,1]
a_{_U21₃}: [1,2,3]
a_{_U22₂}: [2,1]
isList₁: [1]
a_{_isList₁}: [1]
isQid: []
a_{_U41₃}: [2,1,3]
a_{_U42₂}: [1,2]
a_{_U43₁}: [1]
a_{_U51₃}: [1,3,2]
a_{_U52₂}: [1,2]
a_{_U71₂}: [2,1]
a_{_U72₁}: [1]
a_{_isPal₁}: [1]
a: []
e: []
i: []
o: []
u: []

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__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__and(X1, X2) → and(X1, X2)
a__U11(tt, V) → a__U12(a__isNeList(V))
a__isNeList(V) → a__U31(a__isPalListKind(V), V)
a__isNeList(__(V1, V2)) → a__U41(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
a__isNeList(__(V1, V2)) → a__U51(a__and(a__isPalListKind(V1), isPalListKind(V2)), V1, V2)
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__isList(V) → a__U11(a__isPalListKind(V), V)
a__isList(__(V1, V2)) → a__U21(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__U43(a__isNeList(V2))
a__U51(tt, V1, V2) → a__U52(a__isNeList(V1), 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__isNeList(X) → isNeList(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__isList(nil) → tt
a__isList(X) → isList(X)
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)

(18) Obligation:

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

MARK(U12(X)) → MARK(X)
MARK(U23(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U53(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U62(X)) → MARK(X)
A__ISNEPAL(V) → A__ISPALLISTKIND(V)

The TRS R consists of the following rules:

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

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

(19) DependencyGraphProof (EQUIVALENT transformation)

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

(20) Obligation:

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

MARK(U23(X)) → MARK(X)
MARK(U12(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U53(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
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(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 [1,0] 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
U31(x1, x2) = x1
U32(x1) = x1
U53(x1) = x1
U61(x1, x2) = x1
U62(x1) = x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

U23₁ > MARK

Status:

MARK: []
U23₁: [1]

The following usable rules [FROCOS05] were oriented: none

(22) Obligation:

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

MARK(U12(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U53(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
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.

(23) 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)

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
U31(x1, x2) = x1
U32(x1) = x1
U53(x1) = x1
U61(x1, x2) = x1
U62(x1) = U62(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

MARK₁: [1]
U62₁: [1]

The following usable rules [FROCOS05] were oriented: none

(24) Obligation:

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

MARK(U12(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U53(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)

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.

(25) 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 [0,0] 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) = U12(x1)
U31(x1, x2) = x1
U32(x1) = x1
U53(x1) = x1
U61(x1, x2) = x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

U12₁: [1]

The following usable rules [FROCOS05] were oriented: none

(26) Obligation:

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

MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U53(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)

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.

(27) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U31(X1, X2)) → MARK(X1)

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: 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)
U31(x1, x2) = U31(x1, x2)
U32(x1) = x1
U53(x1) = x1
U61(x1, x2) = x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

MARK₁: [1]
U31₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(28) Obligation:

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

MARK(U32(X)) → MARK(X)
MARK(U53(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)

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.

(29) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(U61(X1, X2)) → MARK(X1)

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: MS
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
MARK(x1) = MARK
U32(x1) = x1
U53(x1) = x1
U61(x1, x2) = U61(x1, x2)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

MARK: []
U61₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(30) Obligation:

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

MARK(U32(X)) → MARK(X)
MARK(U53(X)) → MARK(X)

The TRS R consists of the following rules:

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

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

(31) 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, x1)

Tags:
MARK has argument tags [1,0] 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
U32(x1) = U32(x1)
U53(x1) = x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

MARK: []
U32₁: [1]

The following usable rules [FROCOS05] were oriented: none

(32) Obligation:

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

MARK(U53(X)) → MARK(X)

The TRS R consists of the following rules:

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

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

(33) 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)

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
U53(x1) = U53(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

U53₁: [1]

The following usable rules [FROCOS05] were oriented: none

(34) 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.

(35) PisEmptyProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Obligation:

(5) QDPOrderProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) QDPOrderProof (EQUIVALENT transformation)

(11) Obligation:

(12) DependencyGraphProof (EQUIVALENT transformation)

(13) TRUE

(14) Obligation:

(15) QDPOrderProof (EQUIVALENT transformation)

(16) Obligation:

(17) QDPOrderProof (EQUIVALENT transformation)

(18) Obligation:

(19) DependencyGraphProof (EQUIVALENT transformation)

(20) Obligation:

(21) QDPOrderProof (EQUIVALENT transformation)

(22) Obligation:

(23) QDPOrderProof (EQUIVALENT transformation)

(24) Obligation:

(25) QDPOrderProof (EQUIVALENT transformation)

(26) Obligation:

(27) QDPOrderProof (EQUIVALENT transformation)

(28) Obligation:

(29) QDPOrderProof (EQUIVALENT transformation)

(30) Obligation:

(31) QDPOrderProof (EQUIVALENT transformation)

(32) Obligation:

(33) QDPOrderProof (EQUIVALENT transformation)

(34) Obligation:

(35) PisEmptyProof (EQUIVALENT transformation)

(36) TRUE