(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(f(a, f(x, a)), a) → f(a, f(f(x, a), a))
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(f(a, f(x, a)), a) → F(a, f(f(x, a), a))
F(f(a, f(x, a)), a) → F(f(x, a), a)
The TRS R consists of the following rules:
f(f(a, f(x, a)), a) → f(a, f(f(x, a), a))
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 1 SCC with 1 less node.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(f(a, f(x, a)), a) → F(f(x, a), a)
The TRS R consists of the following rules:
f(f(a, f(x, a)), a) → f(a, f(f(x, a), a))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) 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:Recursive path order with status [RPO].
Quasi-Precedence:
[f2, a]
Status:
f2: multiset
a: multiset
AFS:
f(x1, x2) = f(x1, x2)
a = a
From the DPs we obtained the following set of size-change graphs:
- F(f(a, f(x, a)), a) → F(f(x, a), a) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2
We oriented the following set of usable rules [AAECC05,FROCOS05].
f(f(a, f(x, a)), a) → f(a, f(f(x, a), a))
(6) TRUE