(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(f(f(a, f(a, a)), a), x) → f(x, f(x, 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(f(a, f(a, a)), a), x) → f(x, f(x, a))
The set Q consists of the following terms:
f(f(f(a, f(a, a)), a), x0)
(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(f(a, f(a, a)), a), x) → F(x, f(x, a))
F(f(f(a, f(a, a)), a), x) → F(x, a)
The TRS R consists of the following rules:
f(f(f(a, f(a, a)), a), x) → f(x, f(x, a))
The set Q consists of the following terms:
f(f(f(a, f(a, a)), a), x0)
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(f(f(a, f(a, a)), a), x) → F(x, a)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:
| POL(F(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(f(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
The following usable rules [FROCOS05] were oriented:
f(f(f(a, f(a, a)), a), x) → f(x, f(x, a))
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(f(f(a, f(a, a)), a), x) → F(x, f(x, a))
The TRS R consists of the following rules:
f(f(f(a, f(a, a)), a), x) → f(x, f(x, a))
The set Q consists of the following terms:
f(f(f(a, f(a, a)), a), x0)
We have to consider all minimal (P,Q,R)-chains.
(7) Narrowing (EQUIVALENT transformation)
By narrowing [LPAR04] the rule
F(
f(
f(
a,
f(
a,
a)),
a),
x) →
F(
x,
f(
x,
a)) at position [] we obtained the following new rules [LPAR04]:
F(f(f(a, f(a, a)), a), f(f(a, f(a, a)), a)) → F(f(f(a, f(a, a)), a), f(a, f(a, a)))
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(f(f(a, f(a, a)), a), f(f(a, f(a, a)), a)) → F(f(f(a, f(a, a)), a), f(a, f(a, a)))
The TRS R consists of the following rules:
f(f(f(a, f(a, a)), a), x) → f(x, f(x, a))
The set Q consists of the following terms:
f(f(f(a, f(a, a)), a), x0)
We have to consider all minimal (P,Q,R)-chains.
(9) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
(10) TRUE