(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(g(x), y, y) → g(f(x, 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, y) → g(f(x, x, y))
The set Q consists of the following terms:
f(g(x0), x1, 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, y) → F(x, x, y)
The TRS R consists of the following rules:
f(g(x), y, y) → g(f(x, x, y))
The set Q consists of the following terms:
f(g(x0), x1, x1)
We have to consider all minimal (P,Q,R)-chains.
(5) 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.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(g(x), y, y) → F(x, x, y)
R is empty.
The set Q consists of the following terms:
f(g(x0), x1, x1)
We have to consider all minimal (P,Q,R)-chains.
(7) QReductionProof (EQUIVALENT transformation)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
f(g(x0), x1, x1)
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(g(x), y, y) → F(x, x, y)
R is empty.
Q is empty.
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(g(x), y, y) → F(x, x, y)
The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3, 3 >= 3
(10) TRUE