(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) 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:
a(a(x)) → b(b(x))
a(b(b(x))) → b(b(a(x)))
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a(x1)) = 1 + x1
POL(b(x1)) = x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a(a(x)) → b(b(x))
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(b(b(x))) → b(b(a(x)))
Q is empty.
(5) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(6) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(b(b(x))) → b(b(a(x)))
The set Q consists of the following terms:
a(b(b(x0)))
(7) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(b(b(x))) → A(x)
The TRS R consists of the following rules:
a(b(b(x))) → b(b(a(x)))
The set Q consists of the following terms:
a(b(b(x0)))
We have to consider all minimal (P,Q,R)-chains.
(9) 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.
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(b(b(x))) → A(x)
R is empty.
The set Q consists of the following terms:
a(b(b(x0)))
We have to consider all minimal (P,Q,R)-chains.
(11) 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].
a(b(b(x0)))
(12) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(b(b(x))) → A(x)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(13) 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:
- A(b(b(x))) → A(x)
The graph contains the following edges 1 > 1
(14) YES