(0) Obligation:

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

g(x, y) → x
g(x, y) → y
f(s(x), y, y) → f(y, x, s(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:

F(s(x), y, y) → F(y, x, s(x))

The TRS R consists of the following rules:

g(x, y) → x
g(x, y) → y
f(s(x), y, y) → f(y, x, s(x))

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(s(x), y, y) → F(y, x, s(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(x1, x2, x3)  =  F(x1, x2)
s(x1)  =  s(x1)
g(x1, x2)  =  g(x1, x2)
f(x1, x2, x3)  =  f(x1, x3)

Recursive path order with status [RPO].
Quasi-Precedence:
g2 > [F2, s1, f2]

Status:
F2: multiset
s1: multiset
g2: multiset
f2: multiset


The following usable rules [FROCOS05] were oriented:

g(x, y) → x
g(x, y) → y
f(s(x), y, y) → f(y, x, s(x))

(4) Obligation:

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

g(x, y) → x
g(x, y) → y
f(s(x), y, y) → f(y, x, s(x))

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

(5) PisEmptyProof (EQUIVALENT transformation)

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

(6) TRUE