(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(x)) → b(b(b(x)))
b(x) → c(c(d(x)))
c(x) → d(d(d(x)))
b(c(x)) → c(b(x))
b(c(d(x))) → a(x)
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a(x1)) = 47 + x1
POL(b(x1)) = 31 + x1
POL(c(x1)) = 13 + x1
POL(d(x1)) = 4 + 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(b(x)))
b(x) → c(c(d(x)))
c(x) → d(d(d(x)))
b(c(d(x))) → a(x)
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
b(c(x)) → c(b(x))
Q is empty.
(3) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
b(c(x)) → c(b(x))
The set Q consists of the following terms:
b(c(x0))
(5) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(c(x)) → B(x)
The TRS R consists of the following rules:
b(c(x)) → c(b(x))
The set Q consists of the following terms:
b(c(x0))
We have to consider all minimal (P,Q,R)-chains.
(7) 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.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(c(x)) → B(x)
R is empty.
The set Q consists of the following terms:
b(c(x0))
We have to consider all minimal (P,Q,R)-chains.
(9) 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].
b(c(x0))
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(c(x)) → B(x)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(11) 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:
- B(c(x)) → B(x)
The graph contains the following edges 1 > 1
(12) YES