(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(f(a, f(a, a)), x) → f(x, f(f(a, 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, f(a, a)), x) → f(x, f(f(a, a), a))
The set Q consists of the following terms:
f(f(a, f(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(a, f(a, a)), x) → F(x, f(f(a, a), a))
F(f(a, f(a, a)), x) → F(f(a, a), a)
The TRS R consists of the following rules:
f(f(a, f(a, a)), x) → f(x, f(f(a, a), a))
The set Q consists of the following terms:
f(f(a, f(a, a)), x0)
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 1 less node.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(f(a, f(a, a)), x) → F(x, f(f(a, a), a))
The TRS R consists of the following rules:
f(f(a, f(a, a)), x) → f(x, f(f(a, a), a))
The set Q consists of the following terms:
f(f(a, f(a, a)), x0)
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(f(a, f(a, a)), x) → F(x, f(f(a, a), a))
R is empty.
The set Q consists of the following terms:
f(f(a, f(a, a)), x0)
We have to consider all minimal (P,Q,R)-chains.
(9) MRRProof (EQUIVALENT transformation)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:
F(f(a, f(a, a)), x) → F(x, f(f(a, a), a))
Used ordering: Polynomial interpretation [POLO]:
POL(F(x1, x2)) = 2·x1 + 2·x2
POL(a) = 2
POL(f(x1, x2)) = 2 + x1 + 2·x2
(10) Obligation:
Q DP problem:
P is empty.
R is empty.
The set Q consists of the following terms:
f(f(a, f(a, a)), x0)
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