(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(a, f(x, a)) → f(a, f(f(a, a), f(a, x)))
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(a, f(x, a)) → f(a, f(f(a, a), f(a, x)))
The set Q consists of the following terms:
f(a, f(x0, a))
(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(a, f(x, a)) → F(a, f(f(a, a), f(a, x)))
F(a, f(x, a)) → F(f(a, a), f(a, x))
F(a, f(x, a)) → F(a, a)
F(a, f(x, a)) → F(a, x)
The TRS R consists of the following rules:
f(a, f(x, a)) → f(a, f(f(a, a), f(a, x)))
The set Q consists of the following terms:
f(a, f(x0, a))
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(a, f(x, a)) → F(a, x)
The TRS R consists of the following rules:
f(a, f(x, a)) → f(a, f(f(a, a), f(a, x)))
The set Q consists of the following terms:
f(a, f(x0, a))
We have to consider all minimal (P,Q,R)-chains.
(7) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(a, f(x, a)) → F(a, x)
R is empty.
The set Q consists of the following terms:
f(a, f(x0, a))
We have to consider all minimal (P,Q,R)-chains.
(9) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- F(a, f(x, a)) → F(a, x)
The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2
(10) YES