(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) 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(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)))
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 3 less nodes.
(4) 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)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
F(a, f(x, a)) → F(a, x)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
F(
x1,
x2) =
F(
x2)
Tags:
F has tags [1,1]
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:
POL(a) = 1
POL(f(x1, x2)) = 1 + x1
The following usable rules [FROCOS05] were oriented:
none
(6) Obligation:
Q DP problem:
P is empty.
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.
We have to consider all minimal (P,Q,R)-chains.
(7) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(8) TRUE