(0) Obligation:

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

f(f(x)) → f(g(f(x), x))
f(f(x)) → f(h(f(x), f(x)))
g(x, y) → y
h(x, x) → g(x, 0)

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)) → F(g(f(x), x))
F(f(x)) → G(f(x), x)
F(f(x)) → F(h(f(x), f(x)))
F(f(x)) → H(f(x), f(x))
H(x, x) → G(x, 0)

The TRS R consists of the following rules:

f(f(x)) → f(g(f(x), x))
f(f(x)) → f(h(f(x), f(x)))
g(x, y) → y
h(x, x) → g(x, 0)

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 1 SCC with 3 less nodes.

(4) Obligation:

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

F(f(x)) → F(h(f(x), f(x)))
F(f(x)) → F(g(f(x), x))

The TRS R consists of the following rules:

f(f(x)) → f(g(f(x), x))
f(f(x)) → f(h(f(x), f(x)))
g(x, y) → y
h(x, x) → g(x, 0)

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)) → F(g(f(x), x))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(x1)  =  F(x1)
f(x1)  =  f(x1)
h(x1, x2)  =  x2
g(x1, x2)  =  x2
0  =  0

Lexicographic path order with status [LPO].
Quasi-Precedence:
F1 > [f1, 0]

Status:
F1: [1]
f1: [1]
0: []


The following usable rules [FROCOS05] were oriented:

f(f(x)) → f(h(f(x), f(x)))
f(f(x)) → f(g(f(x), x))
h(x, x) → g(x, 0)
g(x, y) → y

(6) Obligation:

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

F(f(x)) → F(h(f(x), f(x)))

The TRS R consists of the following rules:

f(f(x)) → f(g(f(x), x))
f(f(x)) → f(h(f(x), f(x)))
g(x, y) → y
h(x, x) → g(x, 0)

Q is empty.
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(f(x)) → F(h(f(x), 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)
f(x1)  =  f
h(x1, x2)  =  h
g(x1, x2)  =  x2
0  =  0

Lexicographic path order with status [LPO].
Quasi-Precedence:
f > [h, 0] > F1

Status:
F1: [1]
f: []
h: []
0: []


The following usable rules [FROCOS05] were oriented:

h(x, x) → g(x, 0)
g(x, y) → y

(8) Obligation:

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

f(f(x)) → f(g(f(x), x))
f(f(x)) → f(h(f(x), f(x)))
g(x, y) → y
h(x, x) → g(x, 0)

Q is empty.
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