(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