(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) → tt
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt, P) → a__U72(a__isPal(P))
a__U72(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
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(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isPal(X) → isPal(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(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__U21(tt, V2) → A__U22(a__isList(V2))
A__U21(tt, V2) → A__ISLIST(V2)
A__U41(tt, V2) → A__U42(a__isNeList(V2))
A__U41(tt, V2) → A__ISNELIST(V2)
A__U51(tt, V2) → A__U52(a__isList(V2))
A__U51(tt, V2) → A__ISLIST(V2)
A__U71(tt, P) → A__U72(a__isPal(P))
A__U71(tt, P) → A__ISPAL(P)
A__ISLIST(V) → A__U11(a__isNeList(V))
A__ISLIST(V) → A__ISNELIST(V)
A__ISLIST(__(V1, V2)) → A__U21(a__isList(V1), V2)
A__ISLIST(__(V1, V2)) → A__ISLIST(V1)
A__ISNELIST(V) → A__U31(a__isQid(V))
A__ISNELIST(V) → A__ISQID(V)
A__ISNELIST(__(V1, V2)) → A__U41(a__isList(V1), V2)
A__ISNELIST(__(V1, V2)) → A__ISLIST(V1)
A__ISNELIST(__(V1, V2)) → A__U51(a__isNeList(V1), V2)
A__ISNELIST(__(V1, V2)) → A__ISNELIST(V1)
A__ISNEPAL(V) → A__U61(a__isQid(V))
A__ISNEPAL(V) → A__ISQID(V)
A__ISNEPAL(__(I, __(P, I))) → A__U71(a__isQid(I), P)
A__ISNEPAL(__(I, __(P, I))) → A__ISQID(I)
A__ISPAL(V) → A__U81(a__isNePal(V))
A__ISPAL(V) → A__ISNEPAL(V)
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
MARK(__(X1, X2)) → MARK(X1)
MARK(__(X1, X2)) → MARK(X2)
MARK(U11(X)) → A__U11(mark(X))
MARK(U11(X)) → MARK(X)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X)) → A__U22(mark(X))
MARK(U22(X)) → MARK(X)
MARK(isList(X)) → A__ISLIST(X)
MARK(U31(X)) → A__U31(mark(X))
MARK(U31(X)) → MARK(X)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → A__U42(mark(X))
MARK(U42(X)) → MARK(X)
MARK(isNeList(X)) → A__ISNELIST(X)
MARK(U51(X1, X2)) → A__U51(mark(X1), X2)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X)) → A__U52(mark(X))
MARK(U52(X)) → MARK(X)
MARK(U61(X)) → A__U61(mark(X))
MARK(U61(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(isPal(X)) → A__ISPAL(X)
MARK(U81(X)) → A__U81(mark(X))
MARK(U81(X)) → MARK(X)
MARK(isQid(X)) → A__ISQID(X)
MARK(isNePal(X)) → A__ISNEPAL(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) → tt
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt, P) → a__U72(a__isPal(P))
a__U72(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
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(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isPal(X) → isPal(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(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 3 SCCs with 28 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

A__U71(tt, P) → A__ISPAL(P)
A__ISPAL(V) → A__ISNEPAL(V)
A__ISNEPAL(__(I, __(P, I))) → A__U71(a__isQid(I), P)

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) → tt
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt, P) → a__U72(a__isPal(P))
a__U72(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
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(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isPal(X) → isPal(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


A__U71(tt, P) → A__ISPAL(P)
A__ISPAL(V) → A__ISNEPAL(V)
A__ISNEPAL(__(I, __(P, I))) → A__U71(a__isQid(I), P)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A__U71(x0, x1, x2)  =  A__U71(x2)
A__ISPAL(x0, x1)  =  A__ISPAL(x1)
A__ISNEPAL(x0, x1)  =  A__ISNEPAL(x0, x1)

Tags:
A__U71 has argument tags [0,3,1] and root tag 1
A__ISPAL has argument tags [0,0] and root tag 2
A__ISNEPAL has argument tags [0,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(A__ISNEPAL(x1)) = 0   
POL(A__ISPAL(x1)) = 0   
POL(A__U71(x1, x2)) = 1 + x1   
POL(__(x1, x2)) = 1 + x1 + x2   
POL(a) = 0   
POL(a__isQid(x1)) = 1 + x1   
POL(e) = 0   
POL(i) = 1   
POL(isQid(x1)) = 0   
POL(o) = 1   
POL(tt) = 1   
POL(u) = 1   

The following usable rules [FROCOS05] were oriented:

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)

(7) 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) → tt
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt, P) → a__U72(a__isPal(P))
a__U72(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
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(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isPal(X) → isPal(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)

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

(8) PisEmptyProof (EQUIVALENT transformation)

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

(9) TRUE

(10) Obligation:

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

A__U21(tt, V2) → A__ISLIST(V2)
A__ISLIST(V) → A__ISNELIST(V)
A__ISNELIST(__(V1, V2)) → A__U41(a__isList(V1), V2)
A__U41(tt, V2) → A__ISNELIST(V2)
A__ISNELIST(__(V1, V2)) → A__ISLIST(V1)
A__ISLIST(__(V1, V2)) → A__U21(a__isList(V1), V2)
A__ISLIST(__(V1, V2)) → A__ISLIST(V1)
A__ISNELIST(__(V1, V2)) → A__U51(a__isNeList(V1), V2)
A__U51(tt, V2) → A__ISLIST(V2)
A__ISNELIST(__(V1, V2)) → A__ISNELIST(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) → tt
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt, P) → a__U72(a__isPal(P))
a__U72(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
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(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isPal(X) → isPal(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)

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

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


A__U21(tt, V2) → A__ISLIST(V2)
A__ISLIST(V) → A__ISNELIST(V)
A__ISNELIST(__(V1, V2)) → A__U41(a__isList(V1), V2)
A__U41(tt, V2) → A__ISNELIST(V2)
A__ISNELIST(__(V1, V2)) → A__ISLIST(V1)
A__ISLIST(__(V1, V2)) → A__U21(a__isList(V1), V2)
A__ISLIST(__(V1, V2)) → A__ISLIST(V1)
A__ISNELIST(__(V1, V2)) → A__U51(a__isNeList(V1), V2)
A__U51(tt, V2) → A__ISLIST(V2)
A__ISNELIST(__(V1, V2)) → A__ISNELIST(V1)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A__U21(x0, x1, x2)  =  A__U21(x0, x1)
A__ISLIST(x0, x1)  =  A__ISLIST(x0, x1)
A__ISNELIST(x0, x1)  =  A__ISNELIST(x0, x1)
A__U41(x0, x1, x2)  =  A__U41(x0, x1)
A__U51(x0, x1, x2)  =  A__U51(x0)

Tags:
A__U21 has argument tags [8,8,0] and root tag 1
A__ISLIST has argument tags [0,8] and root tag 2
A__ISNELIST has argument tags [0,8] and root tag 0
A__U41 has argument tags [0,7,0] and root tag 2
A__U51 has argument tags [0,4,0] and root tag 4

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(A__ISLIST(x1)) = 1   
POL(A__ISNELIST(x1)) = 1   
POL(A__U21(x1, x2)) = 1 + x1 + x2   
POL(A__U41(x1, x2)) = 1 + x2   
POL(A__U51(x1, x2)) = 1 + x2   
POL(U11(x1)) = 0   
POL(U21(x1, x2)) = x2   
POL(U22(x1)) = 0   
POL(U31(x1)) = x1   
POL(U41(x1, x2)) = 1 + x1   
POL(U42(x1)) = 1 + x1   
POL(U51(x1, x2)) = 1 + x1   
POL(U52(x1)) = x1   
POL(__(x1, x2)) = 1 + x1 + x2   
POL(a) = 0   
POL(a__U11(x1)) = 0   
POL(a__U21(x1, x2)) = x2   
POL(a__U22(x1)) = x1   
POL(a__U31(x1)) = 0   
POL(a__U41(x1, x2)) = x2   
POL(a__U42(x1)) = 0   
POL(a__U51(x1, x2)) = x2   
POL(a__U52(x1)) = 1   
POL(a__isList(x1)) = x1   
POL(a__isNeList(x1)) = 0   
POL(a__isQid(x1)) = 0   
POL(e) = 0   
POL(i) = 0   
POL(isList(x1)) = 0   
POL(isNeList(x1)) = 1 + x1   
POL(isQid(x1)) = 1 + x1   
POL(nil) = 1   
POL(o) = 0   
POL(tt) = 0   
POL(u) = 0   

The following usable rules [FROCOS05] were oriented:

a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isList(X) → isList(X)
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U11(tt) → tt
a__U11(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(tt) → tt
a__U22(X) → U22(X)

(12) 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) → tt
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt, P) → a__U72(a__isPal(P))
a__U72(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
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(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isPal(X) → isPal(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)

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

(13) PisEmptyProof (EQUIVALENT transformation)

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

(14) TRUE

(15) 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(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X)) → MARK(X)
MARK(U31(X)) → MARK(X)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U61(X)) → MARK(X)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X)) → MARK(X)
MARK(U81(X)) → MARK(X)
A____(__(X, Y), Z) → MARK(Z)
A____(X, nil) → MARK(X)
A____(nil, X) → MARK(X)

The TRS R consists of the following rules:

a____(__(X, Y), Z) → a____(mark(X), a____(mark(Y), mark(Z)))
a____(X, nil) → mark(X)
a____(nil, X) → mark(X)
a__U11(tt) → tt
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt, P) → a__U72(a__isPal(P))
a__U72(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
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(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isPal(X) → isPal(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)

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

(16) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


A____(X, nil) → MARK(X)
A____(nil, X) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A____(x0, x1, x2)  =  A____(x0, x1)
MARK(x0, x1)  =  MARK(x0, x1)

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

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(A____(x1, x2)) = x1 + x2   
POL(MARK(x1)) = 0   
POL(U11(x1)) = x1   
POL(U21(x1, x2)) = x1 + x2   
POL(U22(x1)) = x1   
POL(U31(x1)) = x1   
POL(U41(x1, x2)) = x1 + x2   
POL(U42(x1)) = x1   
POL(U51(x1, x2)) = x1 + x2   
POL(U52(x1)) = x1   
POL(U61(x1)) = x1   
POL(U71(x1, x2)) = x1 + x2   
POL(U72(x1)) = x1   
POL(U81(x1)) = x1   
POL(__(x1, x2)) = x1 + x2   
POL(a) = 1   
POL(a__U11(x1)) = x1   
POL(a__U21(x1, x2)) = x1 + x2   
POL(a__U22(x1)) = x1   
POL(a__U31(x1)) = x1   
POL(a__U41(x1, x2)) = x1 + x2   
POL(a__U42(x1)) = x1   
POL(a__U51(x1, x2)) = x1 + x2   
POL(a__U52(x1)) = x1   
POL(a__U61(x1)) = x1   
POL(a__U71(x1, x2)) = x1 + x2   
POL(a__U72(x1)) = x1   
POL(a__U81(x1)) = x1   
POL(a____(x1, x2)) = x1 + x2   
POL(a__isList(x1)) = x1   
POL(a__isNeList(x1)) = x1   
POL(a__isNePal(x1)) = x1   
POL(a__isPal(x1)) = x1   
POL(a__isQid(x1)) = x1   
POL(e) = 1   
POL(i) = 1   
POL(isList(x1)) = x1   
POL(isNeList(x1)) = x1   
POL(isNePal(x1)) = x1   
POL(isPal(x1)) = x1   
POL(isQid(x1)) = x1   
POL(mark(x1)) = x1   
POL(nil) = 1   
POL(o) = 1   
POL(tt) = 1   
POL(u) = 1   

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(U11(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(tt) → tt
a__U11(X) → U11(X)
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U21(X1, X2) → U21(X1, X2)
a__U22(tt) → tt
a__U22(X) → U22(X)
a__U31(tt) → tt
a__U31(X) → U31(X)
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U41(X1, X2) → U41(X1, X2)
a__U42(tt) → tt
a__U42(X) → U42(X)
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U51(X1, X2) → U51(X1, X2)
a__U52(tt) → tt
a__U52(X) → U52(X)
a__U61(tt) → tt
a__U61(X) → U61(X)
a__U71(tt, P) → a__U72(a__isPal(P))
a__U71(X1, X2) → U71(X1, X2)
a__U72(tt) → tt
a__U72(X) → U72(X)
a__U81(tt) → tt
a__U81(X) → U81(X)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNeList(X) → isNeList(X)
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isList(X) → isList(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__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isNePal(X) → isNePal(X)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
a__isPal(X) → isPal(X)

(17) 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(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X)) → MARK(X)
MARK(U31(X)) → MARK(X)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U61(X)) → MARK(X)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X)) → MARK(X)
MARK(U81(X)) → MARK(X)
A____(__(X, Y), Z) → MARK(Z)

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) → tt
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt, P) → a__U72(a__isPal(P))
a__U72(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
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(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isPal(X) → isPal(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)

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

(18) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(U72(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A____(x0, x1, x2)  =  A____(x0)
MARK(x0, x1)  =  MARK(x0, x1)

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

Comparison: MIN
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(A____(x1, x2)) = x1 + x2   
POL(MARK(x1)) = x1   
POL(U11(x1)) = x1   
POL(U21(x1, x2)) = x1 + x2   
POL(U22(x1)) = x1   
POL(U31(x1)) = x1   
POL(U41(x1, x2)) = x1 + x2   
POL(U42(x1)) = x1   
POL(U51(x1, x2)) = x1 + x2   
POL(U52(x1)) = x1   
POL(U61(x1)) = x1   
POL(U71(x1, x2)) = x1 + x2   
POL(U72(x1)) = 1 + x1   
POL(U81(x1)) = x1   
POL(__(x1, x2)) = x1 + x2   
POL(a) = 1   
POL(a__U11(x1)) = x1   
POL(a__U21(x1, x2)) = x1 + x2   
POL(a__U22(x1)) = x1   
POL(a__U31(x1)) = x1   
POL(a__U41(x1, x2)) = x1 + x2   
POL(a__U42(x1)) = x1   
POL(a__U51(x1, x2)) = x1 + x2   
POL(a__U52(x1)) = x1   
POL(a__U61(x1)) = x1   
POL(a__U71(x1, x2)) = x1 + x2   
POL(a__U72(x1)) = 1 + x1   
POL(a__U81(x1)) = x1   
POL(a____(x1, x2)) = x1 + x2   
POL(a__isList(x1)) = x1   
POL(a__isNeList(x1)) = x1   
POL(a__isNePal(x1)) = x1   
POL(a__isPal(x1)) = x1   
POL(a__isQid(x1)) = x1   
POL(e) = 1   
POL(i) = 1   
POL(isList(x1)) = x1   
POL(isNeList(x1)) = x1   
POL(isNePal(x1)) = x1   
POL(isPal(x1)) = x1   
POL(isQid(x1)) = x1   
POL(mark(x1)) = x1   
POL(nil) = 1   
POL(o) = 1   
POL(tt) = 1   
POL(u) = 1   

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(U11(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(tt) → tt
a__U11(X) → U11(X)
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U21(X1, X2) → U21(X1, X2)
a__U22(tt) → tt
a__U22(X) → U22(X)
a__U31(tt) → tt
a__U31(X) → U31(X)
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U41(X1, X2) → U41(X1, X2)
a__U42(tt) → tt
a__U42(X) → U42(X)
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U51(X1, X2) → U51(X1, X2)
a__U52(tt) → tt
a__U52(X) → U52(X)
a__U61(tt) → tt
a__U61(X) → U61(X)
a__U71(tt, P) → a__U72(a__isPal(P))
a__U71(X1, X2) → U71(X1, X2)
a__U72(tt) → tt
a__U72(X) → U72(X)
a__U81(tt) → tt
a__U81(X) → U81(X)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNeList(X) → isNeList(X)
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isList(X) → isList(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__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isNePal(X) → isNePal(X)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
a__isPal(X) → isPal(X)

(19) 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(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X)) → MARK(X)
MARK(U31(X)) → MARK(X)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U61(X)) → MARK(X)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U81(X)) → MARK(X)
A____(__(X, Y), Z) → MARK(Z)

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) → tt
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt, P) → a__U72(a__isPal(P))
a__U72(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
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(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isPal(X) → isPal(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)

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

(20) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(U22(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A____(x0, x1, x2)  =  A____(x0, x1, x2)
MARK(x0, x1)  =  MARK(x0, x1)

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

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(A____(x1, x2)) = x1 + x2   
POL(MARK(x1)) = 0   
POL(U11(x1)) = x1   
POL(U21(x1, x2)) = x1 + x2   
POL(U22(x1)) = 1 + x1   
POL(U31(x1)) = x1   
POL(U41(x1, x2)) = x1 + x2   
POL(U42(x1)) = x1   
POL(U51(x1, x2)) = x1 + x2   
POL(U52(x1)) = x1   
POL(U61(x1)) = x1   
POL(U71(x1, x2)) = x1   
POL(U72(x1)) = 1   
POL(U81(x1)) = x1   
POL(__(x1, x2)) = x1 + x2   
POL(a) = 1   
POL(a__U11(x1)) = x1   
POL(a__U21(x1, x2)) = x1 + x2   
POL(a__U22(x1)) = 1 + x1   
POL(a__U31(x1)) = x1   
POL(a__U41(x1, x2)) = x1 + x2   
POL(a__U42(x1)) = x1   
POL(a__U51(x1, x2)) = x1 + x2   
POL(a__U52(x1)) = x1   
POL(a__U61(x1)) = x1   
POL(a__U71(x1, x2)) = x1   
POL(a__U72(x1)) = 1   
POL(a__U81(x1)) = x1   
POL(a____(x1, x2)) = x1 + x2   
POL(a__isList(x1)) = x1   
POL(a__isNeList(x1)) = x1   
POL(a__isNePal(x1)) = x1   
POL(a__isPal(x1)) = 1 + x1   
POL(a__isQid(x1)) = x1   
POL(e) = 1   
POL(i) = 1   
POL(isList(x1)) = x1   
POL(isNeList(x1)) = x1   
POL(isNePal(x1)) = x1   
POL(isPal(x1)) = 1 + x1   
POL(isQid(x1)) = x1   
POL(mark(x1)) = x1   
POL(nil) = 1   
POL(o) = 1   
POL(tt) = 1   
POL(u) = 1   

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(U11(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(tt) → tt
a__U11(X) → U11(X)
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U21(X1, X2) → U21(X1, X2)
a__U22(tt) → tt
a__U22(X) → U22(X)
a__U31(tt) → tt
a__U31(X) → U31(X)
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U41(X1, X2) → U41(X1, X2)
a__U42(tt) → tt
a__U42(X) → U42(X)
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U51(X1, X2) → U51(X1, X2)
a__U52(tt) → tt
a__U52(X) → U52(X)
a__U61(tt) → tt
a__U61(X) → U61(X)
a__U71(tt, P) → a__U72(a__isPal(P))
a__U71(X1, X2) → U71(X1, X2)
a__U72(tt) → tt
a__U72(X) → U72(X)
a__U81(tt) → tt
a__U81(X) → U81(X)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNeList(X) → isNeList(X)
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isList(X) → isList(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__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isNePal(X) → isNePal(X)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
a__isPal(X) → isPal(X)

(21) 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(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U31(X)) → MARK(X)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U61(X)) → MARK(X)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U81(X)) → MARK(X)
A____(__(X, Y), Z) → MARK(Z)

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) → tt
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt, P) → a__U72(a__isPal(P))
a__U72(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
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(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isPal(X) → isPal(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)

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

(22) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(U52(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A____(x0, x1, x2)  =  A____(x0, x1, x2)
MARK(x0, x1)  =  MARK(x0, x1)

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

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(A____(x1, x2)) = x1 + x2   
POL(MARK(x1)) = 0   
POL(U11(x1)) = x1   
POL(U21(x1, x2)) = x1   
POL(U22(x1)) = 1   
POL(U31(x1)) = x1   
POL(U41(x1, x2)) = x1 + x2   
POL(U42(x1)) = x1   
POL(U51(x1, x2)) = x1 + x2   
POL(U52(x1)) = 1 + x1   
POL(U61(x1)) = x1   
POL(U71(x1, x2)) = x1   
POL(U72(x1)) = 1   
POL(U81(x1)) = x1   
POL(__(x1, x2)) = x1 + x2   
POL(a) = 1   
POL(a__U11(x1)) = x1   
POL(a__U21(x1, x2)) = x1   
POL(a__U22(x1)) = 1   
POL(a__U31(x1)) = x1   
POL(a__U41(x1, x2)) = x1 + x2   
POL(a__U42(x1)) = x1   
POL(a__U51(x1, x2)) = x1 + x2   
POL(a__U52(x1)) = 1 + x1   
POL(a__U61(x1)) = x1   
POL(a__U71(x1, x2)) = x1   
POL(a__U72(x1)) = 1   
POL(a__U81(x1)) = x1   
POL(a____(x1, x2)) = x1 + x2   
POL(a__isList(x1)) = x1   
POL(a__isNeList(x1)) = x1   
POL(a__isNePal(x1)) = x1   
POL(a__isPal(x1)) = x1   
POL(a__isQid(x1)) = x1   
POL(e) = 1   
POL(i) = 1   
POL(isList(x1)) = x1   
POL(isNeList(x1)) = x1   
POL(isNePal(x1)) = x1   
POL(isPal(x1)) = x1   
POL(isQid(x1)) = x1   
POL(mark(x1)) = x1   
POL(nil) = 1   
POL(o) = 1   
POL(tt) = 1   
POL(u) = 1   

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(U11(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(tt) → tt
a__U11(X) → U11(X)
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U21(X1, X2) → U21(X1, X2)
a__U22(tt) → tt
a__U22(X) → U22(X)
a__U31(tt) → tt
a__U31(X) → U31(X)
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U41(X1, X2) → U41(X1, X2)
a__U42(tt) → tt
a__U42(X) → U42(X)
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U51(X1, X2) → U51(X1, X2)
a__U52(tt) → tt
a__U52(X) → U52(X)
a__U61(tt) → tt
a__U61(X) → U61(X)
a__U71(tt, P) → a__U72(a__isPal(P))
a__U71(X1, X2) → U71(X1, X2)
a__U72(tt) → tt
a__U72(X) → U72(X)
a__U81(tt) → tt
a__U81(X) → U81(X)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNeList(X) → isNeList(X)
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isList(X) → isList(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__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isNePal(X) → isNePal(X)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
a__isPal(X) → isPal(X)

(23) 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(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U31(X)) → MARK(X)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U61(X)) → MARK(X)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U81(X)) → MARK(X)
A____(__(X, Y), Z) → MARK(Z)

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) → tt
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt, P) → a__U72(a__isPal(P))
a__U72(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
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(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isPal(X) → isPal(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)

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

(24) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(U42(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A____(x0, x1, x2)  =  A____(x0, x1)
MARK(x0, x1)  =  MARK(x0, x1)

Tags:
A____ has argument tags [0,5,7] and root tag 0
MARK has argument tags [0,4] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(A____(x1, x2)) = 1 + x1 + x2   
POL(MARK(x1)) = 1 + x1   
POL(U11(x1)) = x1   
POL(U21(x1, x2)) = x1   
POL(U22(x1)) = 1   
POL(U31(x1)) = x1   
POL(U41(x1, x2)) = x1 + x2   
POL(U42(x1)) = 1 + x1   
POL(U51(x1, x2)) = x1 + x2   
POL(U52(x1)) = 1   
POL(U61(x1)) = x1   
POL(U71(x1, x2)) = x1 + x2   
POL(U72(x1)) = x1   
POL(U81(x1)) = x1   
POL(__(x1, x2)) = x1 + x2   
POL(a) = 1   
POL(a__U11(x1)) = x1   
POL(a__U21(x1, x2)) = x1   
POL(a__U22(x1)) = 1   
POL(a__U31(x1)) = x1   
POL(a__U41(x1, x2)) = x1 + x2   
POL(a__U42(x1)) = 1 + x1   
POL(a__U51(x1, x2)) = x1 + x2   
POL(a__U52(x1)) = 1   
POL(a__U61(x1)) = x1   
POL(a__U71(x1, x2)) = x1 + x2   
POL(a__U72(x1)) = x1   
POL(a__U81(x1)) = x1   
POL(a____(x1, x2)) = x1 + x2   
POL(a__isList(x1)) = x1   
POL(a__isNeList(x1)) = x1   
POL(a__isNePal(x1)) = x1   
POL(a__isPal(x1)) = x1   
POL(a__isQid(x1)) = x1   
POL(e) = 1   
POL(i) = 1   
POL(isList(x1)) = x1   
POL(isNeList(x1)) = x1   
POL(isNePal(x1)) = x1   
POL(isPal(x1)) = x1   
POL(isQid(x1)) = x1   
POL(mark(x1)) = x1   
POL(nil) = 1   
POL(o) = 1   
POL(tt) = 1   
POL(u) = 1   

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(U11(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(tt) → tt
a__U11(X) → U11(X)
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U21(X1, X2) → U21(X1, X2)
a__U22(tt) → tt
a__U22(X) → U22(X)
a__U31(tt) → tt
a__U31(X) → U31(X)
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U41(X1, X2) → U41(X1, X2)
a__U42(tt) → tt
a__U42(X) → U42(X)
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U51(X1, X2) → U51(X1, X2)
a__U52(tt) → tt
a__U52(X) → U52(X)
a__U61(tt) → tt
a__U61(X) → U61(X)
a__U71(tt, P) → a__U72(a__isPal(P))
a__U71(X1, X2) → U71(X1, X2)
a__U72(tt) → tt
a__U72(X) → U72(X)
a__U81(tt) → tt
a__U81(X) → U81(X)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNeList(X) → isNeList(X)
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isList(X) → isList(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__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isNePal(X) → isNePal(X)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
a__isPal(X) → isPal(X)

(25) 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(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U31(X)) → MARK(X)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U61(X)) → MARK(X)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U81(X)) → MARK(X)
A____(__(X, Y), Z) → MARK(Z)

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) → tt
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt, P) → a__U72(a__isPal(P))
a__U72(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
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(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isPal(X) → isPal(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)

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

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(U31(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A____(x0, x1, x2)  =  A____(x0, x2)
MARK(x0, x1)  =  MARK(x1)

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

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(A____(x1, x2)) = x1 + x2   
POL(MARK(x1)) = 1   
POL(U11(x1)) = x1   
POL(U21(x1, x2)) = x1   
POL(U22(x1)) = 0   
POL(U31(x1)) = 1 + x1   
POL(U41(x1, x2)) = x1   
POL(U42(x1)) = 0   
POL(U51(x1, x2)) = x1   
POL(U52(x1)) = 0   
POL(U61(x1)) = x1   
POL(U71(x1, x2)) = x1   
POL(U72(x1)) = 0   
POL(U81(x1)) = x1   
POL(__(x1, x2)) = x1 + x2   
POL(a) = 0   
POL(a__U11(x1)) = x1   
POL(a__U21(x1, x2)) = x1   
POL(a__U22(x1)) = 0   
POL(a__U31(x1)) = 1 + x1   
POL(a__U41(x1, x2)) = x1   
POL(a__U42(x1)) = 0   
POL(a__U51(x1, x2)) = x1   
POL(a__U52(x1)) = 0   
POL(a__U61(x1)) = x1   
POL(a__U71(x1, x2)) = x1   
POL(a__U72(x1)) = 0   
POL(a__U81(x1)) = x1   
POL(a____(x1, x2)) = x1 + x2   
POL(a__isList(x1)) = 1   
POL(a__isNeList(x1)) = 1   
POL(a__isNePal(x1)) = 0   
POL(a__isPal(x1)) = 1   
POL(a__isQid(x1)) = 0   
POL(e) = 0   
POL(i) = 0   
POL(isList(x1)) = 1   
POL(isNeList(x1)) = 1   
POL(isNePal(x1)) = 0   
POL(isPal(x1)) = 1   
POL(isQid(x1)) = 0   
POL(mark(x1)) = x1   
POL(nil) = 0   
POL(o) = 0   
POL(tt) = 0   
POL(u) = 0   

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(U11(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(tt) → tt
a__U11(X) → U11(X)
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U21(X1, X2) → U21(X1, X2)
a__U22(tt) → tt
a__U22(X) → U22(X)
a__U31(tt) → tt
a__U31(X) → U31(X)
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U41(X1, X2) → U41(X1, X2)
a__U42(tt) → tt
a__U42(X) → U42(X)
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U51(X1, X2) → U51(X1, X2)
a__U52(tt) → tt
a__U52(X) → U52(X)
a__U61(tt) → tt
a__U61(X) → U61(X)
a__U71(tt, P) → a__U72(a__isPal(P))
a__U71(X1, X2) → U71(X1, X2)
a__U72(tt) → tt
a__U72(X) → U72(X)
a__U81(tt) → tt
a__U81(X) → U81(X)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNeList(X) → isNeList(X)
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isList(X) → isList(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__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isNePal(X) → isNePal(X)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
a__isPal(X) → isPal(X)

(27) 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(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U61(X)) → MARK(X)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U81(X)) → MARK(X)
A____(__(X, Y), Z) → MARK(Z)

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) → tt
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt, P) → a__U72(a__isPal(P))
a__U72(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
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(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isPal(X) → isPal(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)

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

(28) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(U71(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:
A____(x0, x1, x2)  =  A____(x0, x1, x2)
MARK(x0, x1)  =  MARK(x0, x1)

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

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(A____(x1, x2)) = x1 + x2   
POL(MARK(x1)) = 0   
POL(U11(x1)) = x1   
POL(U21(x1, x2)) = x1   
POL(U22(x1)) = 0   
POL(U31(x1)) = 0   
POL(U41(x1, x2)) = x1   
POL(U42(x1)) = 0   
POL(U51(x1, x2)) = x1   
POL(U52(x1)) = 0   
POL(U61(x1)) = x1   
POL(U71(x1, x2)) = 1 + x1   
POL(U72(x1)) = 0   
POL(U81(x1)) = x1   
POL(__(x1, x2)) = x1 + x2   
POL(a) = 0   
POL(a__U11(x1)) = x1   
POL(a__U21(x1, x2)) = x1   
POL(a__U22(x1)) = 0   
POL(a__U31(x1)) = 0   
POL(a__U41(x1, x2)) = x1   
POL(a__U42(x1)) = 0   
POL(a__U51(x1, x2)) = x1   
POL(a__U52(x1)) = 0   
POL(a__U61(x1)) = x1   
POL(a__U71(x1, x2)) = 1 + x1   
POL(a__U72(x1)) = 0   
POL(a__U81(x1)) = x1   
POL(a____(x1, x2)) = x1 + x2   
POL(a__isList(x1)) = 0   
POL(a__isNeList(x1)) = 0   
POL(a__isNePal(x1)) = 1   
POL(a__isPal(x1)) = 1   
POL(a__isQid(x1)) = 0   
POL(e) = 0   
POL(i) = 0   
POL(isList(x1)) = 0   
POL(isNeList(x1)) = 0   
POL(isNePal(x1)) = 1   
POL(isPal(x1)) = 1   
POL(isQid(x1)) = 0   
POL(mark(x1)) = x1   
POL(nil) = 1   
POL(o) = 0   
POL(tt) = 0   
POL(u) = 0   

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(U11(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(tt) → tt
a__U11(X) → U11(X)
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U21(X1, X2) → U21(X1, X2)
a__U22(tt) → tt
a__U22(X) → U22(X)
a__U31(tt) → tt
a__U31(X) → U31(X)
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U41(X1, X2) → U41(X1, X2)
a__U42(tt) → tt
a__U42(X) → U42(X)
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U51(X1, X2) → U51(X1, X2)
a__U52(tt) → tt
a__U52(X) → U52(X)
a__U61(tt) → tt
a__U61(X) → U61(X)
a__U71(tt, P) → a__U72(a__isPal(P))
a__U71(X1, X2) → U71(X1, X2)
a__U72(tt) → tt
a__U72(X) → U72(X)
a__U81(tt) → tt
a__U81(X) → U81(X)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNeList(X) → isNeList(X)
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isList(X) → isList(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__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isNePal(X) → isNePal(X)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
a__isPal(X) → isPal(X)

(29) 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(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U61(X)) → MARK(X)
MARK(U81(X)) → MARK(X)
A____(__(X, Y), Z) → MARK(Z)

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) → tt
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt, P) → a__U72(a__isPal(P))
a__U72(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
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(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isPal(X) → isPal(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)

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

(30) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(U61(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A____(x0, x1, x2)  =  A____(x0, x1)
MARK(x0, x1)  =  MARK(x0, x1)

Tags:
A____ has argument tags [5,4,0] and root tag 0
MARK has argument tags [0,5] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(A____(x1, x2)) = x1 + x2   
POL(MARK(x1)) = 0   
POL(U11(x1)) = x1   
POL(U21(x1, x2)) = x1   
POL(U22(x1)) = 0   
POL(U31(x1)) = 0   
POL(U41(x1, x2)) = x1   
POL(U42(x1)) = 0   
POL(U51(x1, x2)) = x1   
POL(U52(x1)) = 0   
POL(U61(x1)) = 1 + x1   
POL(U71(x1, x2)) = 1   
POL(U72(x1)) = 0   
POL(U81(x1)) = x1   
POL(__(x1, x2)) = x1 + x2   
POL(a) = 1   
POL(a__U11(x1)) = x1   
POL(a__U21(x1, x2)) = x1   
POL(a__U22(x1)) = 0   
POL(a__U31(x1)) = 0   
POL(a__U41(x1, x2)) = x1   
POL(a__U42(x1)) = 0   
POL(a__U51(x1, x2)) = x1   
POL(a__U52(x1)) = 0   
POL(a__U61(x1)) = 1 + x1   
POL(a__U71(x1, x2)) = 1   
POL(a__U72(x1)) = 0   
POL(a__U81(x1)) = x1   
POL(a____(x1, x2)) = x1 + x2   
POL(a__isList(x1)) = 0   
POL(a__isNeList(x1)) = 0   
POL(a__isNePal(x1)) = 1   
POL(a__isPal(x1)) = 1   
POL(a__isQid(x1)) = 0   
POL(e) = 0   
POL(i) = 0   
POL(isList(x1)) = 0   
POL(isNeList(x1)) = 0   
POL(isNePal(x1)) = 1   
POL(isPal(x1)) = 1   
POL(isQid(x1)) = 0   
POL(mark(x1)) = x1   
POL(nil) = 1   
POL(o) = 1   
POL(tt) = 0   
POL(u) = 1   

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(U11(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(tt) → tt
a__U11(X) → U11(X)
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U21(X1, X2) → U21(X1, X2)
a__U22(tt) → tt
a__U22(X) → U22(X)
a__U31(tt) → tt
a__U31(X) → U31(X)
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U41(X1, X2) → U41(X1, X2)
a__U42(tt) → tt
a__U42(X) → U42(X)
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U51(X1, X2) → U51(X1, X2)
a__U52(tt) → tt
a__U52(X) → U52(X)
a__U61(tt) → tt
a__U61(X) → U61(X)
a__U71(tt, P) → a__U72(a__isPal(P))
a__U71(X1, X2) → U71(X1, X2)
a__U72(tt) → tt
a__U72(X) → U72(X)
a__U81(tt) → tt
a__U81(X) → U81(X)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNeList(X) → isNeList(X)
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isList(X) → isList(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__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isNePal(X) → isNePal(X)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
a__isPal(X) → isPal(X)

(31) 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(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U81(X)) → MARK(X)
A____(__(X, Y), Z) → MARK(Z)

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) → tt
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt, P) → a__U72(a__isPal(P))
a__U72(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
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(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isPal(X) → isPal(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)

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

(32) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(U81(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A____(x0, x1, x2)  =  A____(x0, x1, x2)
MARK(x0, x1)  =  MARK(x1)

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

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(A____(x1, x2)) = x1 + x2   
POL(MARK(x1)) = 0   
POL(U11(x1)) = x1   
POL(U21(x1, x2)) = x1   
POL(U22(x1)) = 0   
POL(U31(x1)) = 0   
POL(U41(x1, x2)) = x1   
POL(U42(x1)) = 0   
POL(U51(x1, x2)) = x1   
POL(U52(x1)) = x1   
POL(U61(x1)) = 0   
POL(U71(x1, x2)) = 0   
POL(U72(x1)) = 0   
POL(U81(x1)) = 1 + x1   
POL(__(x1, x2)) = x1 + x2   
POL(a) = 0   
POL(a__U11(x1)) = x1   
POL(a__U21(x1, x2)) = x1   
POL(a__U22(x1)) = 0   
POL(a__U31(x1)) = 0   
POL(a__U41(x1, x2)) = x1   
POL(a__U42(x1)) = 0   
POL(a__U51(x1, x2)) = x1   
POL(a__U52(x1)) = x1   
POL(a__U61(x1)) = 0   
POL(a__U71(x1, x2)) = 0   
POL(a__U72(x1)) = 0   
POL(a__U81(x1)) = 1 + x1   
POL(a____(x1, x2)) = x1 + x2   
POL(a__isList(x1)) = 0   
POL(a__isNeList(x1)) = 0   
POL(a__isNePal(x1)) = 0   
POL(a__isPal(x1)) = 1   
POL(a__isQid(x1)) = x1   
POL(e) = 0   
POL(i) = 0   
POL(isList(x1)) = 0   
POL(isNeList(x1)) = 0   
POL(isNePal(x1)) = 0   
POL(isPal(x1)) = 1   
POL(isQid(x1)) = x1   
POL(mark(x1)) = x1   
POL(nil) = 0   
POL(o) = 0   
POL(tt) = 0   
POL(u) = 0   

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(U11(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(tt) → tt
a__U11(X) → U11(X)
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U21(X1, X2) → U21(X1, X2)
a__U22(tt) → tt
a__U22(X) → U22(X)
a__U31(tt) → tt
a__U31(X) → U31(X)
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U41(X1, X2) → U41(X1, X2)
a__U42(tt) → tt
a__U42(X) → U42(X)
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U51(X1, X2) → U51(X1, X2)
a__U52(tt) → tt
a__U52(X) → U52(X)
a__U61(tt) → tt
a__U61(X) → U61(X)
a__U71(tt, P) → a__U72(a__isPal(P))
a__U71(X1, X2) → U71(X1, X2)
a__U72(tt) → tt
a__U72(X) → U72(X)
a__U81(tt) → tt
a__U81(X) → U81(X)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNeList(X) → isNeList(X)
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isList(X) → isList(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__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isNePal(X) → isNePal(X)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
a__isPal(X) → isPal(X)

(33) 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(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U51(X1, X2)) → MARK(X1)
A____(__(X, Y), Z) → MARK(Z)

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) → tt
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt, P) → a__U72(a__isPal(P))
a__U72(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
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(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isPal(X) → isPal(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)

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

(34) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


A____(__(X, Y), Z) → MARK(X)
MARK(__(X1, X2)) → A____(mark(X1), mark(X2))
A____(__(X, Y), Z) → A____(mark(Y), mark(Z))
A____(__(X, Y), Z) → MARK(Y)
MARK(__(X1, X2)) → MARK(X1)
MARK(__(X1, X2)) → MARK(X2)
A____(__(X, Y), Z) → MARK(Z)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A____(x0, x1, x2)  =  A____(x0, x2)
MARK(x0, x1)  =  MARK(x0, x1)

Tags:
A____ has argument tags [4,0,0] and root tag 1
MARK has argument tags [0,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(A____(x1, x2)) = x1 + x2   
POL(MARK(x1)) = 0   
POL(U11(x1)) = x1   
POL(U21(x1, x2)) = x1   
POL(U22(x1)) = x1   
POL(U31(x1)) = 0   
POL(U41(x1, x2)) = x1   
POL(U42(x1)) = 0   
POL(U51(x1, x2)) = x1   
POL(U52(x1)) = 0   
POL(U61(x1)) = 0   
POL(U71(x1, x2)) = 0   
POL(U72(x1)) = 0   
POL(U81(x1)) = x1   
POL(__(x1, x2)) = 1 + x1 + x2   
POL(a) = 0   
POL(a__U11(x1)) = x1   
POL(a__U21(x1, x2)) = x1   
POL(a__U22(x1)) = x1   
POL(a__U31(x1)) = 0   
POL(a__U41(x1, x2)) = x1   
POL(a__U42(x1)) = 0   
POL(a__U51(x1, x2)) = x1   
POL(a__U52(x1)) = 0   
POL(a__U61(x1)) = 0   
POL(a__U71(x1, x2)) = 0   
POL(a__U72(x1)) = 0   
POL(a__U81(x1)) = x1   
POL(a____(x1, x2)) = 1 + x1 + x2   
POL(a__isList(x1)) = 0   
POL(a__isNeList(x1)) = 0   
POL(a__isNePal(x1)) = 0   
POL(a__isPal(x1)) = 0   
POL(a__isQid(x1)) = x1   
POL(e) = 1   
POL(i) = 0   
POL(isList(x1)) = 0   
POL(isNeList(x1)) = 0   
POL(isNePal(x1)) = 0   
POL(isPal(x1)) = 0   
POL(isQid(x1)) = x1   
POL(mark(x1)) = x1   
POL(nil) = 0   
POL(o) = 0   
POL(tt) = 0   
POL(u) = 0   

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(U11(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(tt) → tt
a__U11(X) → U11(X)
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U21(X1, X2) → U21(X1, X2)
a__U22(tt) → tt
a__U22(X) → U22(X)
a__U31(tt) → tt
a__U31(X) → U31(X)
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U41(X1, X2) → U41(X1, X2)
a__U42(tt) → tt
a__U42(X) → U42(X)
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U51(X1, X2) → U51(X1, X2)
a__U52(tt) → tt
a__U52(X) → U52(X)
a__U61(tt) → tt
a__U61(X) → U61(X)
a__U71(tt, P) → a__U72(a__isPal(P))
a__U71(X1, X2) → U71(X1, X2)
a__U72(tt) → tt
a__U72(X) → U72(X)
a__U81(tt) → tt
a__U81(X) → U81(X)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNeList(X) → isNeList(X)
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isList(X) → isList(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__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isNePal(X) → isNePal(X)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
a__isPal(X) → isPal(X)

(35) Obligation:

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

A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
MARK(U11(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U51(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) → tt
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt, P) → a__U72(a__isPal(P))
a__U72(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
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(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isPal(X) → isPal(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)

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

(36) DependencyGraphProof (EQUIVALENT transformation)

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

(37) Complex Obligation (AND)

(38) Obligation:

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

MARK(U21(X1, X2)) → MARK(X1)
MARK(U11(X)) → MARK(X)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U51(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) → tt
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt, P) → a__U72(a__isPal(P))
a__U72(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
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(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isPal(X) → isPal(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)

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

(39) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(U11(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:
Polynomial interpretation [POLO]:

POL(MARK(x1)) = 0   
POL(U11(x1)) = 1 + x1   
POL(U21(x1, x2)) = x1   
POL(U41(x1, x2)) = x1   
POL(U51(x1, x2)) = x1   

The following usable rules [FROCOS05] were oriented: none

(40) Obligation:

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

MARK(U21(X1, X2)) → MARK(X1)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U51(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) → tt
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt, P) → a__U72(a__isPal(P))
a__U72(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
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(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isPal(X) → isPal(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)

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

(41) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(U51(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:
Polynomial interpretation [POLO]:

POL(MARK(x1)) = 1   
POL(U21(x1, x2)) = x1   
POL(U41(x1, x2)) = x1   
POL(U51(x1, x2)) = 1 + x1   

The following usable rules [FROCOS05] were oriented: none

(42) Obligation:

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

MARK(U21(X1, X2)) → MARK(X1)
MARK(U41(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) → tt
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt, P) → a__U72(a__isPal(P))
a__U72(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
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(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isPal(X) → isPal(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)

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

(43) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(U41(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(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:
Polynomial interpretation [POLO]:

POL(MARK(x1)) = 1   
POL(U21(x1, x2)) = x1   
POL(U41(x1, x2)) = 1 + x1   

The following usable rules [FROCOS05] were oriented: none

(44) Obligation:

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

MARK(U21(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) → tt
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt, P) → a__U72(a__isPal(P))
a__U72(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
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(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isPal(X) → isPal(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)

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

(45) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MARK(U21(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(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:
Polynomial interpretation [POLO]:

POL(MARK(x1)) = 1   
POL(U21(x1, x2)) = 1 + x1   

The following usable rules [FROCOS05] were oriented: none

(46) 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) → tt
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt, P) → a__U72(a__isPal(P))
a__U72(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
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(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isPal(X) → isPal(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)

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

(47) PisEmptyProof (EQUIVALENT transformation)

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

(48) TRUE

(49) Obligation:

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

A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))

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) → tt
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt, P) → a__U72(a__isPal(P))
a__U72(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
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(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isPal(X) → isPal(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)

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

(50) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


A____(__(X, Y), Z) → A____(mark(X), a____(mark(Y), mark(Z)))
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____(x1)

Tags:
A____ has argument tags [1,0,3] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(A____(x1, x2)) = 1   
POL(U11(x1)) = 0   
POL(U21(x1, x2)) = 0   
POL(U22(x1)) = x1   
POL(U31(x1)) = 0   
POL(U41(x1, x2)) = 1 + x2   
POL(U42(x1)) = 1   
POL(U51(x1, x2)) = x1 + x2   
POL(U52(x1)) = 0   
POL(U61(x1)) = 0   
POL(U71(x1, x2)) = 0   
POL(U72(x1)) = 0   
POL(U81(x1)) = x1   
POL(__(x1, x2)) = 1 + x1 + x2   
POL(a) = 0   
POL(a__U11(x1)) = 0   
POL(a__U21(x1, x2)) = 0   
POL(a__U22(x1)) = x1   
POL(a__U31(x1)) = 0   
POL(a__U41(x1, x2)) = 1 + x2   
POL(a__U42(x1)) = 1   
POL(a__U51(x1, x2)) = x1 + x2   
POL(a__U52(x1)) = 0   
POL(a__U61(x1)) = 0   
POL(a__U71(x1, x2)) = 0   
POL(a__U72(x1)) = 0   
POL(a__U81(x1)) = x1   
POL(a____(x1, x2)) = 1 + x1 + x2   
POL(a__isList(x1)) = 0   
POL(a__isNeList(x1)) = x1   
POL(a__isNePal(x1)) = 0   
POL(a__isPal(x1)) = 1   
POL(a__isQid(x1)) = 1   
POL(e) = 0   
POL(i) = 0   
POL(isList(x1)) = 0   
POL(isNeList(x1)) = x1   
POL(isNePal(x1)) = 0   
POL(isPal(x1)) = 1   
POL(isQid(x1)) = 1   
POL(mark(x1)) = x1   
POL(nil) = 0   
POL(o) = 0   
POL(tt) = 0   
POL(u) = 0   

The following usable rules [FROCOS05] were oriented: none

(51) 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) → tt
a__U21(tt, V2) → a__U22(a__isList(V2))
a__U22(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNeList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isList(V2))
a__U52(tt) → tt
a__U61(tt) → tt
a__U71(tt, P) → a__U72(a__isPal(P))
a__U72(tt) → tt
a__U81(tt) → tt
a__isList(V) → a__U11(a__isNeList(V))
a__isList(nil) → tt
a__isList(__(V1, V2)) → a__U21(a__isList(V1), V2)
a__isNeList(V) → a__U31(a__isQid(V))
a__isNeList(__(V1, V2)) → a__U41(a__isList(V1), V2)
a__isNeList(__(V1, V2)) → a__U51(a__isNeList(V1), V2)
a__isNePal(V) → a__U61(a__isQid(V))
a__isNePal(__(I, __(P, I))) → a__U71(a__isQid(I), P)
a__isPal(V) → a__U81(a__isNePal(V))
a__isPal(nil) → tt
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(X)) → a__U11(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isList(X)) → a__isList(X)
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNeList(X)) → a__isNeList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X)) → a__U61(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(U81(X)) → a__U81(mark(X))
mark(isQid(X)) → a__isQid(X)
mark(isNePal(X)) → a__isNePal(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(X) → U11(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isList(X) → isList(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNeList(X) → isNeList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__U61(X) → U61(X)
a__U71(X1, X2) → U71(X1, X2)
a__U72(X) → U72(X)
a__isPal(X) → isPal(X)
a__U81(X) → U81(X)
a__isQid(X) → isQid(X)
a__isNePal(X) → isNePal(X)

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

(52) PisEmptyProof (EQUIVALENT transformation)

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

(53) TRUE