(0) Obligation:

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

f(g(X), Y) → f(X, f(g(X), Y))

Q is empty.

(1) Overlay + Local Confluence (EQUIVALENT transformation)

The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.

(2) Obligation:

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

f(g(X), Y) → f(X, f(g(X), Y))

The set Q consists of the following terms:

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, f(g(X), Y))
F(g(X), Y) → F(g(X), Y)

The TRS R consists of the following rules:

f(g(X), Y) → f(X, f(g(X), Y))

The set Q consists of the following terms:

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

Lexicographic Path Order [LPO].
Precedence:
F1 > f
g1 > f

The following usable rules [FROCOS05] were oriented:

f(g(X), Y) → f(X, f(g(X), Y))

(6) Obligation:

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

F(g(X), Y) → F(g(X), Y)

The TRS R consists of the following rules:

f(g(X), Y) → f(X, f(g(X), Y))

The set Q consists of the following terms:

f(g(x0), x1)

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