(0) Obligation:

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

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

Q is empty.

(1) AAECC Innermost (EQUIVALENT transformation)

We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is

f(h(x)) → h(g(x))
f(g(x)) → g(f(f(x)))

The TRS R 2 is

f'(s(x), y, y) → f'(y, x, s(x))

The signature Sigma is {f'}

(2) Obligation:

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

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

The set Q consists of the following terms:

f(g(x0))
f(h(x0))
f'(s(x0), x1, x1)

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(4) Obligation:

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

f(g(x0))
f(h(x0))
f'(s(x0), x1, x1)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Obligation:

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

F(g(x)) → F(x)
F(g(x)) → F(f(x))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

f(g(x0))
f(h(x0))
f'(s(x0), x1, x1)

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

(7) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Recursive Path Order [RPO].
Precedence:
g1 > [F1, h]
f'2 > [F1, h]


The following usable rules [FROCOS05] were oriented:

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

(8) Obligation:

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

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

The set Q consists of the following terms:

f(g(x0))
f(h(x0))
f'(s(x0), x1, x1)

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

(9) PisEmptyProof (EQUIVALENT transformation)

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

(10) TRUE