(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) 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)
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))

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 2 SCCs.

(4) Complex Obligation (AND)

(5) Obligation:

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

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))

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

(6) 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:Homeomorphic Embedding Order

AFS:
s(x1)  =  s(x1)

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

  • F'(s(x), y, y) → F'(y, x, s(x)) (allowed arguments on rhs = {1, 2, 3})
    The graph contains the following edges 2 >= 1, 3 >= 1, 1 > 2, 1 >= 3

We oriented the following set of usable rules [AAECC05,FROCOS05]. none

(7) TRUE

(8) 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))

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

(9) 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

Homeomorphic Embedding Order

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(x) (allowed arguments on rhs = {1})
    The graph contains the following edges 1 > 1

  • F(g(x)) → F(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))

(10) TRUE