Termination w.r.t. Q proof of /tmp/tmp-rQPuK/PALINDROME_nokinds

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

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

    ↳5 QDP

        ↳6 QDPOrderProof (⇔)

        ↳7 QDP

        ↳8 PisEmptyProof (⇔)

        ↳9 TRUE

    ↳10 QDP

        ↳11 QDPOrderProof (⇔)

        ↳12 QDP

        ↳13 QDPOrderProof (⇔)

        ↳14 QDP

        ↳15 DependencyGraphProof (⇔)

        ↳16 AND

            ↳17 QDP

                ↳18 QDPOrderProof (⇔)

                ↳19 QDP

                ↳20 PisEmptyProof (⇔)

                ↳21 TRUE

            ↳22 QDP

                ↳23 QDPOrderProof (⇔)

                ↳24 QDP

                ↳25 PisEmptyProof (⇔)

                ↳26 TRUE

(0) Obligation:

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

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, n____(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → isNePal(activate(V))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → 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:

__¹(__(X, Y), Z) → __¹(X, __(Y, Z))
__¹(__(X, Y), Z) → __¹(Y, Z)
AND(tt, X) → ACTIVATE(X)
ISLIST(V) → ISNELIST(activate(V))
ISLIST(V) → ACTIVATE(V)
ISLIST(n____(V1, V2)) → AND(isList(activate(V1)), n__isList(activate(V2)))
ISLIST(n____(V1, V2)) → ISLIST(activate(V1))
ISLIST(n____(V1, V2)) → ACTIVATE(V1)
ISLIST(n____(V1, V2)) → ACTIVATE(V2)
ISNELIST(V) → ISQID(activate(V))
ISNELIST(V) → ACTIVATE(V)
ISNELIST(n____(V1, V2)) → AND(isList(activate(V1)), n__isNeList(activate(V2)))
ISNELIST(n____(V1, V2)) → ISLIST(activate(V1))
ISNELIST(n____(V1, V2)) → ACTIVATE(V1)
ISNELIST(n____(V1, V2)) → ACTIVATE(V2)
ISNELIST(n____(V1, V2)) → AND(isNeList(activate(V1)), n__isList(activate(V2)))
ISNELIST(n____(V1, V2)) → ISNELIST(activate(V1))
ISNEPAL(V) → ISQID(activate(V))
ISNEPAL(V) → ACTIVATE(V)
ISNEPAL(n____(I, n____(P, I))) → AND(isQid(activate(I)), n__isPal(activate(P)))
ISNEPAL(n____(I, n____(P, I))) → ISQID(activate(I))
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(I)
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(P)
ISPAL(V) → ISNEPAL(activate(V))
ISPAL(V) → ACTIVATE(V)
ACTIVATE(n__nil) → NIL
ACTIVATE(n____(X1, X2)) → __¹(activate(X1), activate(X2))
ACTIVATE(n____(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n____(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__isList(X)) → ISLIST(X)
ACTIVATE(n__isNeList(X)) → ISNELIST(X)
ACTIVATE(n__isPal(X)) → ISPAL(X)
ACTIVATE(n__a) → A
ACTIVATE(n__e) → E
ACTIVATE(n__i) → I
ACTIVATE(n__o) → O
ACTIVATE(n__u) → U

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, n____(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → isNePal(activate(V))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → 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 2 SCCs with 10 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

__¹(__(X, Y), Z) → __¹(Y, Z)
__¹(__(X, Y), Z) → __¹(X, __(Y, Z))

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, n____(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → isNePal(activate(V))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → 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.

__¹(__(X, Y), Z) → __¹(Y, Z)
__¹(__(X, Y), Z) → __¹(X, __(Y, Z))

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
__¹(x0, x1, x2) = __¹(x1)

Tags:
__¹ has argument tags [1,3,2] and root tag 0

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

POL(__(x₁, x₂)) = 1 + x₁ + x₂
POL(__¹(x₁, x₂)) = x₂
POL(n____(x₁, x₂)) = x₂
POL(nil) = 0

The following usable rules [FROCOS05] were oriented:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
__(X1, X2) → n____(X1, X2)

(7) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, n____(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → isNePal(activate(V))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → 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:

ACTIVATE(n____(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n____(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__isList(X)) → ISLIST(X)
ISLIST(V) → ISNELIST(activate(V))
ISNELIST(V) → ACTIVATE(V)
ACTIVATE(n__isNeList(X)) → ISNELIST(X)
ISNELIST(n____(V1, V2)) → AND(isList(activate(V1)), n__isNeList(activate(V2)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__isPal(X)) → ISPAL(X)
ISPAL(V) → ISNEPAL(activate(V))
ISNEPAL(V) → ACTIVATE(V)
ISNEPAL(n____(I, n____(P, I))) → AND(isQid(activate(I)), n__isPal(activate(P)))
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(I)
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(P)
ISPAL(V) → ACTIVATE(V)
ISNELIST(n____(V1, V2)) → ISLIST(activate(V1))
ISLIST(V) → ACTIVATE(V)
ISLIST(n____(V1, V2)) → AND(isList(activate(V1)), n__isList(activate(V2)))
ISLIST(n____(V1, V2)) → ISLIST(activate(V1))
ISLIST(n____(V1, V2)) → ACTIVATE(V1)
ISLIST(n____(V1, V2)) → ACTIVATE(V2)
ISNELIST(n____(V1, V2)) → ACTIVATE(V1)
ISNELIST(n____(V1, V2)) → ACTIVATE(V2)
ISNELIST(n____(V1, V2)) → AND(isNeList(activate(V1)), n__isList(activate(V2)))
ISNELIST(n____(V1, V2)) → ISNELIST(activate(V1))

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, n____(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → isNePal(activate(V))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → 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.

ISNELIST(n____(V1, V2)) → AND(isList(activate(V1)), n__isNeList(activate(V2)))
AND(tt, X) → ACTIVATE(X)
ISNEPAL(n____(I, n____(P, I))) → AND(isQid(activate(I)), n__isPal(activate(P)))
ISLIST(n____(V1, V2)) → AND(isList(activate(V1)), n__isList(activate(V2)))
ISNELIST(n____(V1, V2)) → AND(isNeList(activate(V1)), n__isList(activate(V2)))

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
ACTIVATE(x0, x1) = ACTIVATE(x1)
ISLIST(x0, x1) = ISLIST(x1)
ISNELIST(x0, x1) = ISNELIST(x0, x1)
AND(x0, x1, x2) = AND(x0, x1)
ISPAL(x0, x1) = ISPAL(x0, x1)
ISNEPAL(x0, x1) = ISNEPAL(x0, x1)

Tags:
ACTIVATE has argument tags [6,6] and root tag 4
ISLIST has argument tags [2,6] and root tag 4
ISNELIST has argument tags [6,0] and root tag 4
AND has argument tags [1,6,1] and root tag 0
ISPAL has argument tags [6,6] and root tag 4
ISNEPAL has argument tags [6,6] and root tag 4

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

POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₁ + x₂
POL(ISLIST(x₁)) = 1
POL(ISNELIST(x₁)) = x₁
POL(ISNEPAL(x₁)) = 0
POL(ISPAL(x₁)) = 0
POL(__(x₁, x₂)) = x₁ + x₂
POL(a) = 1
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₂
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(n____(x₁, x₂)) = x₁ + x₂
POL(n__a) = 1
POL(n__e) = 1
POL(n__i) = 1
POL(n__isList(x₁)) = x₁
POL(n__isNeList(x₁)) = x₁
POL(n__isPal(x₁)) = x₁
POL(n__nil) = 1
POL(n__o) = 1
POL(n__u) = 1
POL(nil) = 1
POL(o) = 1
POL(tt) = 1
POL(u) = 1

The following usable rules [FROCOS05] were oriented:

activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isList(X)) → isList(X)
isList(V) → isNeList(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
and(tt, X) → activate(X)
activate(n__isNeList(X)) → isNeList(X)
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
activate(n__isPal(X)) → isPal(X)
isPal(V) → isNePal(activate(V))
isNePal(n____(I, n____(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X
isList(n__nil) → tt
isList(X) → n__isList(X)
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
isNeList(V) → isQid(activate(V))
isNeList(X) → n__isNeList(X)
isNePal(V) → isQid(activate(V))
isPal(n__nil) → tt
isPal(X) → n__isPal(X)
__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
__(X1, X2) → n____(X1, X2)
nil → n__nil
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u

(12) Obligation:

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

ACTIVATE(n____(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n____(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__isList(X)) → ISLIST(X)
ISLIST(V) → ISNELIST(activate(V))
ISNELIST(V) → ACTIVATE(V)
ACTIVATE(n__isNeList(X)) → ISNELIST(X)
ACTIVATE(n__isPal(X)) → ISPAL(X)
ISPAL(V) → ISNEPAL(activate(V))
ISNEPAL(V) → ACTIVATE(V)
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(I)
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(P)
ISPAL(V) → ACTIVATE(V)
ISNELIST(n____(V1, V2)) → ISLIST(activate(V1))
ISLIST(V) → ACTIVATE(V)
ISLIST(n____(V1, V2)) → ISLIST(activate(V1))
ISLIST(n____(V1, V2)) → ACTIVATE(V1)
ISLIST(n____(V1, V2)) → ACTIVATE(V2)
ISNELIST(n____(V1, V2)) → ACTIVATE(V1)
ISNELIST(n____(V1, V2)) → ACTIVATE(V2)
ISNELIST(n____(V1, V2)) → ISNELIST(activate(V1))

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, n____(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → isNePal(activate(V))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

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

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__isList(X)) → ISLIST(X)
ISNELIST(V) → ACTIVATE(V)
ACTIVATE(n__isNeList(X)) → ISNELIST(X)
ACTIVATE(n__isPal(X)) → ISPAL(X)
ISNEPAL(V) → ACTIVATE(V)
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(I)
ISNEPAL(n____(I, n____(P, I))) → ACTIVATE(P)
ISPAL(V) → ACTIVATE(V)
ISLIST(V) → ACTIVATE(V)
ISLIST(n____(V1, V2)) → ACTIVATE(V1)
ISLIST(n____(V1, V2)) → ACTIVATE(V2)
ISNELIST(n____(V1, V2)) → ACTIVATE(V1)
ISNELIST(n____(V1, V2)) → ACTIVATE(V2)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
ACTIVATE(x0, x1) = ACTIVATE(x1)
ISLIST(x0, x1) = ISLIST(x0, x1)
ISNELIST(x0, x1) = ISNELIST(x0, x1)
ISPAL(x0, x1) = ISPAL(x0, x1)
ISNEPAL(x0, x1) = ISNEPAL(x0, x1)

Tags:
ACTIVATE has argument tags [1,0] and root tag 2
ISLIST has argument tags [15,0] and root tag 5
ISNELIST has argument tags [15,0] and root tag 5
ISPAL has argument tags [0,0] and root tag 0
ISNEPAL 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(ACTIVATE(x₁)) = 0
POL(ISLIST(x₁)) = 0
POL(ISNELIST(x₁)) = 0
POL(ISNEPAL(x₁)) = 0
POL(ISPAL(x₁)) = 0
POL(__(x₁, x₂)) = x₁ + x₂
POL(a) = 0
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₂
POL(e) = 0
POL(i) = 1
POL(isList(x₁)) = 1 + x₁
POL(isNeList(x₁)) = 1 + x₁
POL(isNePal(x₁)) = 1 + x₁
POL(isPal(x₁)) = 1 + x₁
POL(isQid(x₁)) = x₁
POL(n____(x₁, x₂)) = x₁ + x₂
POL(n__a) = 0
POL(n__e) = 0
POL(n__i) = 1
POL(n__isList(x₁)) = 1 + x₁
POL(n__isNeList(x₁)) = 1 + x₁
POL(n__isPal(x₁)) = 1 + x₁
POL(n__nil) = 1
POL(n__o) = 1
POL(n__u) = 0
POL(nil) = 1
POL(o) = 1
POL(tt) = 0
POL(u) = 0

The following usable rules [FROCOS05] were oriented: none

(14) Obligation:

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

ACTIVATE(n____(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n____(X1, X2)) → ACTIVATE(X2)
ISLIST(V) → ISNELIST(activate(V))
ISPAL(V) → ISNEPAL(activate(V))
ISNELIST(n____(V1, V2)) → ISLIST(activate(V1))
ISLIST(n____(V1, V2)) → ISLIST(activate(V1))
ISNELIST(n____(V1, V2)) → ISNELIST(activate(V1))

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, n____(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → isNePal(activate(V))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

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

(15) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.

(16) Complex Obligation (AND)

(17) Obligation:

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

ISNELIST(n____(V1, V2)) → ISLIST(activate(V1))
ISLIST(V) → ISNELIST(activate(V))
ISNELIST(n____(V1, V2)) → ISNELIST(activate(V1))
ISLIST(n____(V1, V2)) → ISLIST(activate(V1))

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, n____(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → isNePal(activate(V))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → 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.

ISNELIST(n____(V1, V2)) → ISLIST(activate(V1))
ISLIST(V) → ISNELIST(activate(V))
ISNELIST(n____(V1, V2)) → ISNELIST(activate(V1))
ISLIST(n____(V1, V2)) → ISLIST(activate(V1))

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

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

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

POL(ISLIST(x₁)) = 0
POL(ISNELIST(x₁)) = 0
POL(__(x₁, x₂)) = 1 + x₁ + x₂
POL(a) = 0
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₂
POL(e) = 0
POL(i) = 0
POL(isList(x₁)) = 0
POL(isNeList(x₁)) = 0
POL(isNePal(x₁)) = 0
POL(isPal(x₁)) = 0
POL(isQid(x₁)) = 0
POL(n____(x₁, x₂)) = 1 + x₁ + x₂
POL(n__a) = 0
POL(n__e) = 0
POL(n__i) = 0
POL(n__isList(x₁)) = 0
POL(n__isNeList(x₁)) = 0
POL(n__isPal(x₁)) = 0
POL(n__nil) = 0
POL(n__o) = 0
POL(n__u) = 0
POL(nil) = 0
POL(o) = 0
POL(tt) = 0
POL(u) = 0

The following usable rules [FROCOS05] were oriented: none

(19) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, n____(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → isNePal(activate(V))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

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

(20) PisEmptyProof (EQUIVALENT transformation)

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

(21) TRUE

(22) Obligation:

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

ACTIVATE(n____(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n____(X1, X2)) → ACTIVATE(X1)

The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, n____(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → isNePal(activate(V))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

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

(23) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n____(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n____(X1, X2)) → ACTIVATE(X1)

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

Tags:
ACTIVATE 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(ACTIVATE(x₁)) = 1
POL(n____(x₁, x₂)) = 1 + x₁ + x₂

The following usable rules [FROCOS05] were oriented: none

(24) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
and(tt, X) → activate(X)
isList(V) → isNeList(activate(V))
isList(n__nil) → tt
isList(n____(V1, V2)) → and(isList(activate(V1)), n__isList(activate(V2)))
isNeList(V) → isQid(activate(V))
isNeList(n____(V1, V2)) → and(isList(activate(V1)), n__isNeList(activate(V2)))
isNeList(n____(V1, V2)) → and(isNeList(activate(V1)), n__isList(activate(V2)))
isNePal(V) → isQid(activate(V))
isNePal(n____(I, n____(P, I))) → and(isQid(activate(I)), n__isPal(activate(P)))
isPal(V) → isNePal(activate(V))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
isList(X) → n__isList(X)
isNeList(X) → n__isNeList(X)
isPal(X) → n__isPal(X)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(activate(X1), activate(X2))
activate(n__isList(X)) → isList(X)
activate(n__isNeList(X)) → isNeList(X)
activate(n__isPal(X)) → isPal(X)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

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

(25) PisEmptyProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) QDPOrderProof (EQUIVALENT transformation)

(7) Obligation:

(8) PisEmptyProof (EQUIVALENT transformation)

(9) TRUE

(10) Obligation:

(11) QDPOrderProof (EQUIVALENT transformation)

(12) Obligation:

(13) QDPOrderProof (EQUIVALENT transformation)

(14) Obligation:

(15) DependencyGraphProof (EQUIVALENT transformation)

(16) Complex Obligation (AND)

(17) Obligation:

(18) QDPOrderProof (EQUIVALENT transformation)

(19) Obligation:

(20) PisEmptyProof (EQUIVALENT transformation)

(21) TRUE

(22) Obligation:

(23) QDPOrderProof (EQUIVALENT transformation)

(24) Obligation:

(25) PisEmptyProof (EQUIVALENT transformation)

(26) TRUE