(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(f(a, x), a) → f(f(f(a, a), f(x, 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(f(a, x), a) → f(f(f(a, a), f(x, a)), a)
The set Q consists of the following terms:
f(f(a, 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(f(a, x), a) → F(f(f(a, a), f(x, a)), a)
F(f(a, x), a) → F(f(a, a), f(x, a))
F(f(a, x), a) → F(a, a)
F(f(a, x), a) → F(x, a)
The TRS R consists of the following rules:
f(f(a, x), a) → f(f(f(a, a), f(x, a)), a)
The set Q consists of the following terms:
f(f(a, 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(f(a, x), a) → F(x, a)
The TRS R consists of the following rules:
f(f(a, x), a) → f(f(f(a, a), f(x, a)), a)
The set Q consists of the following terms:
f(f(a, 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(f(a, x), a) → F(x, a)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Precedence:
F2 > a
f2 > a
Status:
F2: [1,2]
f2: [1,2]
a: []
The following usable rules [FROCOS05] were oriented:
none
(8) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f(f(a, x), a) → f(f(f(a, a), f(x, a)), a)
The set Q consists of the following terms:
f(f(a, 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