Termination w.r.t. Q proof of /tmp/tmpYfCbSB/PALINDROME_nosorts_noand

Termination w.r.t. Q of the given QTRS could not be shown:

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

  ↳5 QDP

    ↳6 QDPOrderProof (⇔)

    ↳7 QDP

    ↳8 QDPOrderProof (⇔)

    ↳9 QDP

    ↳10 PisEmptyProof (⇔)

    ↳11 TRUE

  ↳12 QDP

    ↳13 QDPOrderProof (⇔)

    ↳14 QDP

    ↳15 QDPOrderProof (⇔)

    ↳16 QDP

    ↳17 PisEmptyProof (⇔)

    ↳18 TRUE

  ↳19 QDP

    ↳20 QDPOrderProof (⇔)

    ↳21 QDP

    ↳22 QDPOrderProof (⇔)

    ↳23 QDP

    ↳24 PisEmptyProof (⇔)

    ↳25 TRUE

  ↳26 QDP

    ↳27 QDPOrderProof (⇔)

    ↳28 QDP

    ↳29 QDPOrderProof (⇔)

    ↳30 QDP

    ↳31 QDPOrderProof (⇔)

    ↳32 QDP

    ↳33 PisEmptyProof (⇔)

    ↳34 TRUE

  ↳35 QDP

    ↳36 QDPOrderProof (⇔)

    ↳37 QDP

    ↳38 QDPOrderProof (⇔)

    ↳39 QDP

    ↳40 PisEmptyProof (⇔)

    ↳41 TRUE

  ↳42 QDP

    ↳43 QDPOrderProof (⇔)

    ↳44 QDP

    ↳45 QDPOrderProof (⇔)

    ↳46 QDP

    ↳47 PisEmptyProof (⇔)

    ↳48 TRUE

  ↳49 QDP

(0) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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:

ACTIVE(__(__(X, Y), Z)) → __¹(X, __(Y, Z))
ACTIVE(__(__(X, Y), Z)) → __¹(Y, Z)
ACTIVE(U11(tt)) → U12¹(tt)
ACTIVE(isNePal(__(I, __(P, I)))) → U11¹(tt)
ACTIVE(__(X1, X2)) → __¹(active(X1), X2)
ACTIVE(__(X1, X2)) → ACTIVE(X1)
ACTIVE(__(X1, X2)) → __¹(X1, active(X2))
ACTIVE(__(X1, X2)) → ACTIVE(X2)
ACTIVE(U11(X)) → U11¹(active(X))
ACTIVE(U11(X)) → ACTIVE(X)
ACTIVE(U12(X)) → U12¹(active(X))
ACTIVE(U12(X)) → ACTIVE(X)
ACTIVE(isNePal(X)) → ISNEPAL(active(X))
ACTIVE(isNePal(X)) → ACTIVE(X)
__¹(mark(X1), X2) → __¹(X1, X2)
__¹(X1, mark(X2)) → __¹(X1, X2)
U11¹(mark(X)) → U11¹(X)
U12¹(mark(X)) → U12¹(X)
ISNEPAL(mark(X)) → ISNEPAL(X)
PROPER(__(X1, X2)) → __¹(proper(X1), proper(X2))
PROPER(__(X1, X2)) → PROPER(X1)
PROPER(__(X1, X2)) → PROPER(X2)
PROPER(U11(X)) → U11¹(proper(X))
PROPER(U11(X)) → PROPER(X)
PROPER(U12(X)) → U12¹(proper(X))
PROPER(U12(X)) → PROPER(X)
PROPER(isNePal(X)) → ISNEPAL(proper(X))
PROPER(isNePal(X)) → PROPER(X)
__¹(ok(X1), ok(X2)) → __¹(X1, X2)
U11¹(ok(X)) → U11¹(X)
U12¹(ok(X)) → U12¹(X)
ISNEPAL(ok(X)) → ISNEPAL(X)
TOP(mark(X)) → TOP(proper(X))
TOP(mark(X)) → PROPER(X)
TOP(ok(X)) → TOP(active(X))
TOP(ok(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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 7 SCCs with 15 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

ISNEPAL(ok(X)) → ISNEPAL(X)
ISNEPAL(mark(X)) → ISNEPAL(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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.

ISNEPAL(mark(X)) → ISNEPAL(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNEPAL(x1) = x1
ok(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = x1
tt = tt
U12(x1) = x1
isNePal(x1) = x1
proper(x1) = proper(x1)
top(x1) = top

Recursive Path Order [RPO].
Precedence:

proper₁ > [active₁, _{_₂}] > tt > mark₁ > top
proper₁ > nil

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(7) Obligation:

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

ISNEPAL(ok(X)) → ISNEPAL(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNEPAL(ok(X)) → ISNEPAL(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNEPAL(x1) = ISNEPAL(x1)
ok(x1) = ok(x1)
active(x1) = x1
__(x1, x2) = x2
mark(x1) = mark
nil = nil
U11(x1) = x1
tt = tt
U12(x1) = x1
isNePal(x1) = x1
proper(x1) = proper(x1)
top(x1) = top

Recursive Path Order [RPO].
Precedence:

tt > [ISNEPAL₁, ok₁, proper₁] > nil > mark
top > [ISNEPAL₁, ok₁, proper₁] > nil > mark

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(9) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(10) PisEmptyProof (EQUIVALENT transformation)

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

(11) TRUE

(12) Obligation:

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

U12¹(ok(X)) → U12¹(X)
U12¹(mark(X)) → U12¹(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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.

U12¹(mark(X)) → U12¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1) = x1
ok(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = x1
tt = tt
U12(x1) = x1
isNePal(x1) = x1
proper(x1) = proper(x1)
top(x1) = top

Recursive Path Order [RPO].
Precedence:

proper₁ > [active₁, _{_₂}] > tt > mark₁ > top
proper₁ > nil

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(14) Obligation:

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

U12¹(ok(X)) → U12¹(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(ok(X)) → U12¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1) = U12¹(x1)
ok(x1) = ok(x1)
active(x1) = x1
__(x1, x2) = x2
mark(x1) = mark
nil = nil
U11(x1) = x1
tt = tt
U12(x1) = x1
isNePal(x1) = x1
proper(x1) = proper(x1)
top(x1) = top

Recursive Path Order [RPO].
Precedence:

tt > [U12^1₁, ok₁, proper₁] > nil > mark
top > [U12^1₁, ok₁, proper₁] > nil > mark

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(16) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(17) PisEmptyProof (EQUIVALENT transformation)

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

(18) TRUE

(19) Obligation:

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

U11¹(ok(X)) → U11¹(X)
U11¹(mark(X)) → U11¹(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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.

U11¹(mark(X)) → U11¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1) = x1
ok(x1) = x1
mark(x1) = mark(x1)
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = x1
tt = tt
U12(x1) = x1
isNePal(x1) = x1
proper(x1) = proper(x1)
top(x1) = top

Recursive Path Order [RPO].
Precedence:

proper₁ > [active₁, _{_₂}] > tt > mark₁ > top
proper₁ > nil

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(21) Obligation:

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

U11¹(ok(X)) → U11¹(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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.

U11¹(ok(X)) → U11¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1) = U11¹(x1)
ok(x1) = ok(x1)
active(x1) = x1
__(x1, x2) = x2
mark(x1) = mark
nil = nil
U11(x1) = x1
tt = tt
U12(x1) = x1
isNePal(x1) = x1
proper(x1) = proper(x1)
top(x1) = top

Recursive Path Order [RPO].
Precedence:

tt > [U11^1₁, ok₁, proper₁] > nil > mark
top > [U11^1₁, ok₁, proper₁] > nil > mark

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(23) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(24) PisEmptyProof (EQUIVALENT transformation)

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

(25) TRUE

(26) Obligation:

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

__¹(X1, mark(X2)) → __¹(X1, X2)
__¹(mark(X1), X2) → __¹(X1, X2)
__¹(ok(X1), ok(X2)) → __¹(X1, X2)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(27) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

__¹(ok(X1), ok(X2)) → __¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
__¹(x1, x2) = __¹(x1, x2)
mark(x1) = x1
ok(x1) = ok(x1)
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = x1
tt = tt
U12(x1) = x1
isNePal(x1) = x1
proper(x1) = proper(x1)
top(x1) = top

Recursive Path Order [RPO].
Precedence:

active₁ > [_{_₂}, proper₁] > ok₁
active₁ > tt > ok₁
nil > ok₁
top > [_{_₂}, proper₁] > ok₁

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(28) Obligation:

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

__¹(X1, mark(X2)) → __¹(X1, X2)
__¹(mark(X1), X2) → __¹(X1, X2)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(29) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

__¹(mark(X1), X2) → __¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
__¹(x1, x2) = __¹(x1)
mark(x1) = mark(x1)
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = x1
tt = tt
U12(x1) = x1
isNePal(x1) = x1
proper(x1) = x1
ok(x1) = ok
top(x1) = top

Recursive Path Order [RPO].
Precedence:

active₁ > tt > [_{_₂}, ok] > [mark₁, top]
nil > [_{_₂}, ok] > [mark₁, top]

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(30) Obligation:

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

__¹(X1, mark(X2)) → __¹(X1, X2)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(31) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

__¹(X1, mark(X2)) → __¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
__¹(x1, x2) = __¹(x1, x2)
mark(x1) = mark(x1)
active(x1) = active(x1)
__(x1, x2) = __(x1, x2)
nil = nil
U11(x1) = U11(x1)
tt = tt
U12(x1) = x1
isNePal(x1) = isNePal(x1)
proper(x1) = proper(x1)
ok(x1) = x1
top(x1) = top

Recursive Path Order [RPO].
Precedence:

[active₁, tt, proper₁] > __₂ > U11₁ > [mark₁, isNePal₁]
[active₁, tt, proper₁] > nil > [mark₁, isNePal₁]

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(32) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(33) PisEmptyProof (EQUIVALENT transformation)

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

(34) TRUE

(35) Obligation:

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

PROPER(__(X1, X2)) → PROPER(X2)
PROPER(__(X1, X2)) → PROPER(X1)
PROPER(U11(X)) → PROPER(X)
PROPER(U12(X)) → PROPER(X)
PROPER(isNePal(X)) → PROPER(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(36) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(__(X1, X2)) → PROPER(X2)
PROPER(__(X1, X2)) → PROPER(X1)
PROPER(U12(X)) → PROPER(X)
PROPER(isNePal(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1) = PROPER(x1)
__(x1, x2) = __(x1, x2)
U11(x1) = x1
U12(x1) = U12(x1)
isNePal(x1) = isNePal(x1)
active(x1) = active(x1)
mark(x1) = x1
nil = nil
tt = tt
proper(x1) = proper(x1)
ok(x1) = x1
top(x1) = top

Recursive Path Order [RPO].
Precedence:

[active₁, top] > isNePal₁ > __₂
[active₁, top] > tt > [PROPER₁, U12₁] > __₂
proper₁ > isNePal₁ > __₂
proper₁ > nil > __₂
proper₁ > tt > [PROPER₁, U12₁] > __₂

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(37) Obligation:

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

PROPER(U11(X)) → PROPER(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(38) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(U11(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1) = PROPER(x1)
U11(x1) = U11(x1)
active(x1) = x1
__(x1, x2) = x1
mark(x1) = mark
nil = nil
tt = tt
U12(x1) = U12
isNePal(x1) = isNePal
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Recursive Path Order [RPO].
Precedence:

PROPER₁ > [mark, nil]
U11₁ > [tt, isNePal] > U12 > [mark, nil]
top > [mark, nil]

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(39) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(40) PisEmptyProof (EQUIVALENT transformation)

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

(41) TRUE

(42) Obligation:

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

ACTIVE(__(X1, X2)) → ACTIVE(X2)
ACTIVE(__(X1, X2)) → ACTIVE(X1)
ACTIVE(U11(X)) → ACTIVE(X)
ACTIVE(U12(X)) → ACTIVE(X)
ACTIVE(isNePal(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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.

ACTIVE(__(X1, X2)) → ACTIVE(X2)
ACTIVE(__(X1, X2)) → ACTIVE(X1)
ACTIVE(U12(X)) → ACTIVE(X)
ACTIVE(isNePal(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1) = ACTIVE(x1)
__(x1, x2) = __(x1, x2)
U11(x1) = x1
U12(x1) = U12(x1)
isNePal(x1) = isNePal(x1)
active(x1) = active(x1)
mark(x1) = x1
nil = nil
tt = tt
proper(x1) = proper(x1)
ok(x1) = x1
top(x1) = top

Recursive Path Order [RPO].
Precedence:

[active₁, top] > isNePal₁ > __₂
[active₁, top] > tt > [ACTIVE₁, U12₁] > __₂
proper₁ > isNePal₁ > __₂
proper₁ > nil > __₂
proper₁ > tt > [ACTIVE₁, U12₁] > __₂

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(44) Obligation:

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

ACTIVE(U11(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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.

ACTIVE(U11(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1) = ACTIVE(x1)
U11(x1) = U11(x1)
active(x1) = x1
__(x1, x2) = x1
mark(x1) = mark
nil = nil
tt = tt
U12(x1) = U12
isNePal(x1) = isNePal
proper(x1) = x1
ok(x1) = x1
top(x1) = top

Recursive Path Order [RPO].
Precedence:

ACTIVE₁ > [mark, nil]
U11₁ > [tt, isNePal] > U12 > [mark, nil]
top > [mark, nil]

The following usable rules [FROCOS05] were oriented:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

(46) Obligation:

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

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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:

TOP(ok(X)) → TOP(active(X))
TOP(mark(X)) → TOP(proper(X))

The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(U11(tt)) → mark(U12(tt))
active(U12(tt)) → mark(tt)
active(isNePal(__(I, __(P, I)))) → mark(U11(tt))
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(U11(X)) → U11(active(X))
active(U12(X)) → U12(active(X))
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
U11(mark(X)) → mark(U11(X))
U12(mark(X)) → mark(U12(X))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
U11(ok(X)) → ok(U11(X))
U12(ok(X)) → ok(U12(X))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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