(0) Obligation:

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

f(x, x) → a
f(g(x), y) → f(x, y)

Q is empty.

(1) AAECC Innermost (EQUIVALENT transformation)

We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is none

The TRS R 2 is

f(x, x) → a
f(g(x), y) → f(x, y)

The signature Sigma is {a, f}

(2) Obligation:

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

f(x, x) → a
f(g(x), y) → f(x, y)

The set Q consists of the following terms:

f(x0, x0)
f(g(x0), 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), y) → F(x, y)

The TRS R consists of the following rules:

f(x, x) → a
f(g(x), y) → f(x, y)

The set Q consists of the following terms:

f(x0, x0)
f(g(x0), x1)

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(g(x), y) → F(x, y)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:
f2 > a

Status:
a: []
f2: [1,2]
g1: [1]
F2: [2,1]


The following usable rules [FROCOS05] were oriented:

f(x, x) → a
f(g(x), y) → f(x, y)

(6) Obligation:

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

f(x, x) → a
f(g(x), y) → f(x, y)

The set Q consists of the following terms:

f(x0, x0)
f(g(x0), x1)

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