(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(x) → b(x)
a(b(x)) → b(c(a(x)))
b(x) → c(x)
c(b(x)) → a(x)
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a(x1)) = 1 + x1
POL(b(x1)) = 1 + x1
POL(c(x1)) = x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
b(x) → c(x)
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(x) → b(x)
a(b(x)) → b(c(a(x)))
c(b(x)) → a(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:
A(b(x)) → C(a(x))
A(b(x)) → A(x)
C(b(x)) → A(x)
The TRS R consists of the following rules:
a(x) → b(x)
a(b(x)) → b(c(a(x)))
c(b(x)) → a(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,JAR06].
The following pairs can be oriented strictly and are deleted.
A(b(x)) → A(x)
C(b(x)) → A(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(A(x1)) = 1 + x1
POL(C(x1)) = 1 + x1
POL(a(x1)) = 1 + x1
POL(b(x1)) = 1 + x1
POL(c(x1)) = x1
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
a(x) → b(x)
a(b(x)) → b(c(a(x)))
c(b(x)) → a(x)
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(b(x)) → C(a(x))
The TRS R consists of the following rules:
a(x) → b(x)
a(b(x)) → b(c(a(x)))
c(b(x)) → a(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
(8) TRUE