(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(a, f(x, a)) → f(a, f(f(a, x), f(a, a)))
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, x), f(a, a)))
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, x), f(a, a)))
F(a, f(x, a)) → F(f(a, x), f(a, a))
F(a, f(x, a)) → F(a, x)
F(a, f(x, a)) → F(a, a)
The TRS R consists of the following rules:
f(a, f(x, a)) → f(a, f(f(a, x), f(a, a)))
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, x), f(a, a)))
The set Q consists of the following terms:
f(a, f(x0, a))
We have to consider all minimal (P,Q,R)-chains.
(7) 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: Combined order from the following AFS and order.
F(
x1,
x2) =
x2
a =
a
f(
x1,
x2) =
f(
x1)
Lexicographic path order with status [LPO].
Quasi-Precedence:
f1 > a
Status:
a: []
f1: [1]
The following usable rules [FROCOS05] were oriented:
f(a, f(x, a)) → f(a, f(f(a, x), f(a, a)))
(8) 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, x), f(a, a)))
The set Q consists of the following terms:
f(a, f(x0, a))
We have to consider all minimal (P,Q,R)-chains.
(9) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(10) TRUE