(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) QTRS Reverse (EQUIVALENT transformation)
We applied the QTRS Reverse Processor [REVERSE].
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
r(w(x)) → w(r(x))
r(b(x)) → b(r(x))
w(b(x)) → b(w(x))
Q is empty.
(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:
R(w(x)) → W(r(x))
R(w(x)) → R(x)
R(b(x)) → R(x)
W(b(x)) → W(x)
The TRS R consists of the following rules:
r(w(x)) → w(r(x))
r(b(x)) → b(r(x))
w(b(x)) → b(w(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
(6) Complex Obligation (AND)
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
W(b(x)) → W(x)
The TRS R consists of the following rules:
r(w(x)) → w(r(x))
r(b(x)) → b(r(x))
w(b(x)) → b(w(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(8) MNOCProof (EQUIVALENT transformation)
We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
(9) Obligation:
Q DP problem:
The TRS P consists of the following rules:
W(b(x)) → W(x)
The TRS R consists of the following rules:
r(w(x)) → w(r(x))
r(b(x)) → b(r(x))
w(b(x)) → b(w(x))
The set Q consists of the following terms:
r(w(x0))
r(b(x0))
w(b(x0))
We have to consider all minimal (P,Q,R)-chains.
(10) 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.
(11) Obligation:
Q DP problem:
The TRS P consists of the following rules:
W(b(x)) → W(x)
R is empty.
The set Q consists of the following terms:
r(w(x0))
r(b(x0))
w(b(x0))
We have to consider all minimal (P,Q,R)-chains.
(12) QReductionProof (EQUIVALENT transformation)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
r(w(x0))
r(b(x0))
w(b(x0))
(13) Obligation:
Q DP problem:
The TRS P consists of the following rules:
W(b(x)) → W(x)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(14) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- W(b(x)) → W(x)
The graph contains the following edges 1 > 1
(15) YES
(16) Obligation:
Q DP problem:
The TRS P consists of the following rules:
R(b(x)) → R(x)
R(w(x)) → R(x)
The TRS R consists of the following rules:
r(w(x)) → w(r(x))
r(b(x)) → b(r(x))
w(b(x)) → b(w(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(17) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(18) Obligation:
Q DP problem:
The TRS P consists of the following rules:
R(b(x)) → R(x)
R(w(x)) → R(x)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(19) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- R(b(x)) → R(x)
The graph contains the following edges 1 > 1
- R(w(x)) → R(x)
The graph contains the following edges 1 > 1
(20) YES