(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(x)) → b(b(x))
b(b(a(x))) → a(b(b(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(a(x)) → B(b(x))
A(a(x)) → B(x)
B(b(a(x))) → A(b(b(x)))
B(b(a(x))) → B(b(x))
B(b(a(x))) → B(x)
The TRS R consists of the following rules:
a(a(x)) → b(b(x))
b(b(a(x))) → a(b(b(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
A(a(x)) → B(x)
B(b(a(x))) → B(b(x))
B(b(a(x))) → B(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic Path Order [LPO].
Precedence:
[a1, b1] > [A1, B1]
The following usable rules [FROCOS05] were oriented:
a(a(x)) → b(b(x))
b(b(a(x))) → a(b(b(x)))
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(a(x)) → B(b(x))
B(b(a(x))) → A(b(b(x)))
The TRS R consists of the following rules:
a(a(x)) → b(b(x))
b(b(a(x))) → a(b(b(x)))
Q is empty.
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.
A(a(x)) → B(b(x))
B(b(a(x))) → A(b(b(x)))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A(
x1) =
x1
a(
x1) =
a(
x1)
B(
x1) =
x1
b(
x1) =
x1
Lexicographic Path Order [LPO].
Precedence:
trivial
The following usable rules [FROCOS05] were oriented:
a(a(x)) → b(b(x))
b(b(a(x))) → a(b(b(x)))
(6) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
a(a(x)) → b(b(x))
b(b(a(x))) → a(b(b(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(8) TRUE