(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))

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(g(X)) → F(f(X))
F(g(X)) → F(X)

The TRS R consists of the following rules:

f(g(X)) → g(f(f(X)))
f(h(X)) → h(g(X))

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

(3) QDPSizeChangeProof (EQUIVALENT transformation)

We used the following order and afs together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.

Order:Combined order from the following AFS and order.
f(x1)  =  x1
g(x1)  =  g(x1)
h(x1)  =  h

Lexicographic path order with status [LPO].
Quasi-Precedence:

[g1, h]

Status:
g1: [1]
h: []

AFS:
f(x1)  =  x1
g(x1)  =  g(x1)
h(x1)  =  h

From the DPs we obtained the following set of size-change graphs:

  • F(g(X)) → F(f(X)) (allowed arguments on rhs = {1})
    The graph contains the following edges 1 > 1

  • F(g(X)) → F(X) (allowed arguments on rhs = {1})
    The graph contains the following edges 1 > 1

We oriented the following set of usable rules [AAECC05,FROCOS05].


f(g(X)) → g(f(f(X)))
f(h(X)) → h(g(X))

(4) TRUE