(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, f(a, a)) → f(f(f(a, a), a), 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(x, f(a, a)) → f(f(f(a, a), a), f(x, a))
The set Q consists of the following terms:
f(x0, f(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, a)) → F(f(f(a, a), a), f(x, a))
F(x, f(a, a)) → F(f(a, a), a)
F(x, f(a, a)) → F(x, a)
The TRS R consists of the following rules:
f(x, f(a, a)) → f(f(f(a, a), a), f(x, a))
The set Q consists of the following terms:
f(x0, f(a, 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 2 less nodes.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(x, f(a, a)) → F(f(f(a, a), a), f(x, a))
The TRS R consists of the following rules:
f(x, f(a, a)) → f(f(f(a, a), a), f(x, a))
The set Q consists of the following terms:
f(x0, f(a, a))
We have to consider all minimal (P,Q,R)-chains.
(7) 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.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(x, f(a, a)) → F(f(f(a, a), a), f(x, a))
R is empty.
The set Q consists of the following terms:
f(x0, f(a, a))
We have to consider all minimal (P,Q,R)-chains.
(9) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
F(x, f(a, a)) → F(f(f(a, a), a), 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:
none
(10) Obligation:
Q DP problem:
P is empty.
R is empty.
The set Q consists of the following terms:
f(x0, f(a, a))
We have to consider all minimal (P,Q,R)-chains.
(11) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(12) TRUE