(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
w(r(x)) → r(w(x))
b(r(x)) → r(b(x))
b(w(x)) → w(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:
W(r(x)) → W(x)
B(r(x)) → B(x)
B(w(x)) → W(b(x))
B(w(x)) → B(x)
The TRS R consists of the following rules:
w(r(x)) → r(w(x))
b(r(x)) → r(b(x))
b(w(x)) → w(b(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
W(r(x)) → W(x)
The TRS R consists of the following rules:
w(r(x)) → r(w(x))
b(r(x)) → r(b(x))
b(w(x)) → w(b(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(6) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
W(r(x)) → W(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:
r1 > W1
The following usable rules [FROCOS05] were oriented:
none
(7) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
w(r(x)) → r(w(x))
b(r(x)) → r(b(x))
b(w(x)) → w(b(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(8) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(9) TRUE
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(w(x)) → B(x)
B(r(x)) → B(x)
The TRS R consists of the following rules:
w(r(x)) → r(w(x))
b(r(x)) → r(b(x))
b(w(x)) → w(b(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(11) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
B(w(x)) → B(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
B(
x1) =
x1
w(
x1) =
w(
x1)
r(
x1) =
x1
Recursive Path Order [RPO].
Precedence:
trivial
The following usable rules [FROCOS05] were oriented:
none
(12) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(r(x)) → B(x)
The TRS R consists of the following rules:
w(r(x)) → r(w(x))
b(r(x)) → r(b(x))
b(w(x)) → w(b(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(13) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
B(r(x)) → B(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:
r1 > B1
The following usable rules [FROCOS05] were oriented:
none
(14) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
w(r(x)) → r(w(x))
b(r(x)) → r(b(x))
b(w(x)) → w(b(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(15) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(16) TRUE