(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(b(b(x))) → b(a(a(x)))
a(a(b(x))) → b(b(a(x)))
Q is empty.
(1) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(2) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(b(b(x))) → A(a(x))
A(b(b(x))) → A(x)
A(a(b(x))) → A(x)
The TRS R consists of the following rules:
a(b(b(x))) → b(a(a(x)))
a(a(b(x))) → b(b(a(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) 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:
A(b(b(x))) → A(a(x))
A(b(b(x))) → A(x)
A(a(b(x))) → A(x)
Used ordering: Polynomial interpretation [POLO]:
POL(A(x1)) = x1
POL(a(x1)) = 1 + x1
POL(b(x1)) = 1 + x1
(4) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
a(b(b(x))) → b(a(a(x)))
a(a(b(x))) → b(b(a(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(6) YES