(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(x) → x
a(a(x)) → a(b(x))
b(x) → x
c(b(x)) → b(a(c(c(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:
A(a(x)) → A(b(x))
A(a(x)) → B(x)
C(b(x)) → B(a(c(c(x))))
C(b(x)) → A(c(c(x)))
C(b(x)) → C(c(x))
C(b(x)) → C(x)
The TRS R consists of the following rules:
a(x) → x
a(a(x)) → a(b(x))
b(x) → x
c(b(x)) → b(a(c(c(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 3 less nodes.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(a(x)) → A(b(x))
The TRS R consists of the following rules:
a(x) → x
a(a(x)) → a(b(x))
b(x) → x
c(b(x)) → b(a(c(c(x))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(6) 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.
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(a(x)) → A(b(x))
The TRS R consists of the following rules:
b(x) → 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:
A(a(x)) → A(b(x))
The TRS R consists of the following rules:
b(x) → x
The set Q consists of the following terms:
b(x0)
We have to consider all minimal (P,Q,R)-chains.
(10) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
A(a(x)) → A(b(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(A(x1)) = x1
POL(a(x1)) = 1 + x1
POL(b(x1)) = x1
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
b(x) → x
(11) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
b(x) → x
The set Q consists of the following terms:
b(x0)
We have to consider all minimal (P,Q,R)-chains.
(12) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(13) YES
(14) Obligation:
Q DP problem:
The TRS P consists of the following rules:
C(b(x)) → C(x)
C(b(x)) → C(c(x))
The TRS R consists of the following rules:
a(x) → x
a(a(x)) → a(b(x))
b(x) → x
c(b(x)) → b(a(c(c(x))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(15) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
C(b(x)) → C(c(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
| POL(b(x1)) = | | + | | / | 0A | 0A | 1A | \ |
| | | 0A | 0A | 0A | | |
| \ | 1A | -I | 0A | / |
| · | x1 |
| POL(c(x1)) = | | + | | / | 0A | -I | 0A | \ |
| | | 0A | -I | 0A | | |
| \ | 0A | -I | 0A | / |
| · | x1 |
| POL(a(x1)) = | | + | | / | 0A | 0A | 0A | \ |
| | | 1A | 1A | 1A | | |
| \ | 0A | 0A | 0A | / |
| · | x1 |
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
c(b(x)) → b(a(c(c(x))))
a(x) → x
a(a(x)) → a(b(x))
b(x) → x
(16) Obligation:
Q DP problem:
The TRS P consists of the following rules:
C(b(x)) → C(x)
The TRS R consists of the following rules:
a(x) → x
a(a(x)) → a(b(x))
b(x) → x
c(b(x)) → b(a(c(c(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:
C(b(x)) → C(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:
- C(b(x)) → C(x)
The graph contains the following edges 1 > 1
(20) YES