(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(x)) → a(b(b(c(x))))
b(a(x)) → x
c(b(x)) → a(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(b(c(x))))
A(a(x)) → B(b(c(x)))
A(a(x)) → B(c(x))
A(a(x)) → C(x)
C(b(x)) → A(c(x))
C(b(x)) → C(x)
The TRS R consists of the following rules:
a(a(x)) → a(b(b(c(x))))
b(a(x)) → x
c(b(x)) → a(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 1 SCC with 2 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(a(x)) → C(x)
C(b(x)) → A(c(x))
A(a(x)) → A(b(b(c(x))))
C(b(x)) → C(x)
The TRS R consists of the following rules:
a(a(x)) → a(b(b(c(x))))
b(a(x)) → x
c(b(x)) → a(c(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(a(x)) → C(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
| POL(a(x1)) = | | + | | / | -I | -I | -I | \ |
| | | 0A | 0A | 1A | | |
| \ | 0A | 0A | 1A | / |
| · | x1 |
| POL(b(x1)) = | | + | | / | 1A | 0A | 0A | \ |
| | | 0A | 0A | 0A | | |
| \ | 0A | 0A | 0A | / |
| · | x1 |
| POL(c(x1)) = | | + | | / | -I | -I | -I | \ |
| | | 0A | 0A | 0A | | |
| \ | 0A | -I | -I | / |
| · | x1 |
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
c(b(x)) → a(c(x))
b(a(x)) → x
a(a(x)) → a(b(b(c(x))))
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
C(b(x)) → A(c(x))
A(a(x)) → A(b(b(c(x))))
C(b(x)) → C(x)
The TRS R consists of the following rules:
a(a(x)) → a(b(b(c(x))))
b(a(x)) → x
c(b(x)) → a(c(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 2 SCCs with 1 less node.
(8) Complex Obligation (AND)
(9) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(a(x)) → A(b(b(c(x))))
The TRS R consists of the following rules:
a(a(x)) → a(b(b(c(x))))
b(a(x)) → x
c(b(x)) → a(c(x))
Q is empty.
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(b(c(x))))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
| POL(a(x1)) = | | + | | / | 0A | 0A | 0A | \ |
| | | 1A | 1A | 1A | | |
| \ | -I | -I | -I | / |
| · | x1 |
| POL(b(x1)) = | | + | | / | 0A | -I | -I | \ |
| | | 0A | -I | 0A | | |
| \ | 0A | -I | 1A | / |
| · | x1 |
| POL(c(x1)) = | | + | | / | 0A | -I | 0A | \ |
| | | 0A | -I | 1A | | |
| \ | -I | -I | -I | / |
| · | x1 |
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
c(b(x)) → a(c(x))
b(a(x)) → x
a(a(x)) → a(b(b(c(x))))
(11) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
a(a(x)) → a(b(b(c(x))))
b(a(x)) → x
c(b(x)) → a(c(x))
Q is empty.
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)
The TRS R consists of the following rules:
a(a(x)) → a(b(b(c(x))))
b(a(x)) → x
c(b(x)) → a(c(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(15) 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.
(16) 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.
(17) 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
(18) YES