(0) Obligation:

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

f(f(x, y, z), u, f(x, y, v)) → f(x, y, f(z, u, v))
f(x, y, y) → y
f(x, y, g(y)) → x
f(x, x, y) → x
f(g(x), x, y) → y

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:

F(f(x, y, z), u, f(x, y, v)) → F(x, y, f(z, u, v))
F(f(x, y, z), u, f(x, y, v)) → F(z, u, v)

The TRS R consists of the following rules:

f(f(x, y, z), u, f(x, y, v)) → f(x, y, f(z, u, v))
f(x, y, y) → y
f(x, y, g(y)) → x
f(x, x, y) → x
f(g(x), x, y) → y

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

(3) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


F(f(x, y, z), u, f(x, y, v)) → F(x, y, f(z, u, v))
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
F(x0, x1, x2, x3)  =  F(x1, x2)

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

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
F(x1, x2, x3)  =  F
f(x1, x2, x3)  =  f(x1, x2, x3)
g(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:
[F, f3]

Status:
F: []
f3: [3,1,2]


The following usable rules [FROCOS05] were oriented: none

(4) Obligation:

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

F(f(x, y, z), u, f(x, y, v)) → F(z, u, v)

The TRS R consists of the following rules:

f(f(x, y, z), u, f(x, y, v)) → f(x, y, f(z, u, v))
f(x, y, y) → y
f(x, y, g(y)) → x
f(x, x, y) → x
f(g(x), x, y) → y

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

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


F(f(x, y, z), u, f(x, y, v)) → F(z, u, v)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
F(x0, x1, x2, x3)  =  F(x1, x2, x3)

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

Comparison: MS
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
F(x1, x2, x3)  =  F
f(x1, x2, x3)  =  f(x1, x2, x3)

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

Status:
F: []
f3: [1,2,3]


The following usable rules [FROCOS05] were oriented: none

(6) Obligation:

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

f(f(x, y, z), u, f(x, y, v)) → f(x, y, f(z, u, v))
f(x, y, y) → y
f(x, y, g(y)) → x
f(x, x, y) → x
f(g(x), x, y) → y

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

(7) PisEmptyProof (EQUIVALENT transformation)

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

(8) TRUE