(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: SCNP Order with the following components:
Level mapping:
Top level AFS:
A(
x0,
x1) =
A(
x0,
x1)
B(
x0,
x1) =
B(
x0,
x1)
Tags:
A has argument tags [2,0] and root tag 1
B has argument tags [2,0] and root tag 1
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
A(
x1) =
A
a(
x1) =
a(
x1)
B(
x1) =
B
b(
x1) =
b(
x1)
Recursive path order with status [RPO].
Quasi-Precedence:
[A, a1, B, b1]
Status:
A: multiset
a1: multiset
B: multiset
b1: multiset
The following usable rules [FROCOS05] were oriented:
none
(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: SCNP Order with the following components:
Level mapping:
Top level AFS:
A(
x0,
x1) =
A(
x0)
B(
x0,
x1) =
B(
x1)
Tags:
A has argument tags [0,3] and root tag 0
B has argument tags [0,0] and root tag 1
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
A(
x1) =
A(
x1)
a(
x1) =
a(
x1)
B(
x1) =
B
b(
x1) =
x1
Recursive path order with status [RPO].
Quasi-Precedence:
[A1, a1, B]
Status:
A1: [1]
a1: [1]
B: multiset
The following usable rules [FROCOS05] were oriented:
b(b(a(x))) → a(b(b(x)))
a(a(x)) → 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