(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, f(a, f(f(a, a), a))) → f(f(a, x), x)
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(x, f(a, f(f(a, a), a))) → f(f(a, x), x)
The set Q consists of the following terms:
f(x0, f(a, f(f(a, a), 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(x, f(a, f(f(a, a), a))) → F(f(a, x), x)
F(x, f(a, f(f(a, a), a))) → F(a, x)
The TRS R consists of the following rules:
f(x, f(a, f(f(a, a), a))) → f(f(a, x), x)
The set Q consists of the following terms:
f(x0, f(a, f(f(a, a), a)))
We have to consider all minimal (P,Q,R)-chains.
(5) TransformationProof (EQUIVALENT transformation)
By narrowing [LPAR04] the rule
F(
x,
f(
a,
f(
f(
a,
a),
a))) →
F(
f(
a,
x),
x) at position [] we obtained the following new rules [LPAR04]:
F(f(a, f(f(a, a), a)), f(a, f(f(a, a), a))) → F(f(f(a, a), a), f(a, f(f(a, a), a))) → F(f(a, f(f(a, a), a)), f(a, f(f(a, a), a))) → F(f(f(a, a), a), f(a, f(f(a, a), a)))
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(x, f(a, f(f(a, a), a))) → F(a, x)
F(f(a, f(f(a, a), a)), f(a, f(f(a, a), a))) → F(f(f(a, a), a), f(a, f(f(a, a), a)))
The TRS R consists of the following rules:
f(x, f(a, f(f(a, a), a))) → f(f(a, x), x)
The set Q consists of the following terms:
f(x0, f(a, f(f(a, a), a)))
We have to consider all minimal (P,Q,R)-chains.
(7) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(x, f(a, f(f(a, a), a))) → F(a, x)
The TRS R consists of the following rules:
f(x, f(a, f(f(a, a), a))) → f(f(a, x), x)
The set Q consists of the following terms:
f(x0, f(a, f(f(a, a), a)))
We have to consider all minimal (P,Q,R)-chains.
(9) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(x, f(a, f(f(a, a), a))) → F(a, x)
R is empty.
The set Q consists of the following terms:
f(x0, f(a, f(f(a, a), a)))
We have to consider all minimal (P,Q,R)-chains.
(11) TransformationProof (EQUIVALENT transformation)
By instantiating [LPAR04] the rule
F(
x,
f(
a,
f(
f(
a,
a),
a))) →
F(
a,
x) we obtained the following new rules [LPAR04]:
F(a, f(a, f(f(a, a), a))) → F(a, a) → F(a, f(a, f(f(a, a), a))) → F(a, a)
(12) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(a, f(a, f(f(a, a), a))) → F(a, a)
R is empty.
The set Q consists of the following terms:
f(x0, f(a, f(f(a, a), a)))
We have to consider all minimal (P,Q,R)-chains.
(13) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
(14) TRUE