(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(b(b(x1))) → P(a(b(x1)))
a(P(x1)) → P(a(x(x1)))
a(x(x1)) → x(a(x1))
b(P(x1)) → b(Q(x1))
Q(x(x1)) → a(Q(x1))
Q(a(x1)) → b(b(a(x1)))
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(b(b(x1))) → A(b(x1))
A(P(x1)) → A(x(x1))
A(x(x1)) → A(x1)
B(P(x1)) → B(Q(x1))
B(P(x1)) → Q1(x1)
Q1(x(x1)) → A(Q(x1))
Q1(x(x1)) → Q1(x1)
Q1(a(x1)) → B(b(a(x1)))
Q1(a(x1)) → B(a(x1))
The TRS R consists of the following rules:
a(b(b(x1))) → P(a(b(x1)))
a(P(x1)) → P(a(x(x1)))
a(x(x1)) → x(a(x1))
b(P(x1)) → b(Q(x1))
Q(x(x1)) → a(Q(x1))
Q(a(x1)) → b(b(a(x1)))
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 1 less node.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(P(x1)) → A(x(x1))
A(x(x1)) → A(x1)
A(b(b(x1))) → A(b(x1))
The TRS R consists of the following rules:
a(b(b(x1))) → P(a(b(x1)))
a(P(x1)) → P(a(x(x1)))
a(x(x1)) → x(a(x1))
b(P(x1)) → b(Q(x1))
Q(x(x1)) → a(Q(x1))
Q(a(x1)) → b(b(a(x1)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(6) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
A(P(x1)) → A(x(x1))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(A(x1)) = x1
POL(P(x1)) = 1 + x1
POL(Q(x1)) = 0
POL(a(x1)) = 0
POL(b(x1)) = 0
POL(x(x1)) = x1
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
b(P(x1)) → b(Q(x1))
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(x(x1)) → A(x1)
A(b(b(x1))) → A(b(x1))
The TRS R consists of the following rules:
a(b(b(x1))) → P(a(b(x1)))
a(P(x1)) → P(a(x(x1)))
a(x(x1)) → x(a(x1))
b(P(x1)) → b(Q(x1))
Q(x(x1)) → a(Q(x1))
Q(a(x1)) → b(b(a(x1)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(8) 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(x(x1)) → A(x1)
The graph contains the following edges 1 > 1
- A(b(b(x1))) → A(b(x1))
The graph contains the following edges 1 > 1
(9) YES
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
Q1(x(x1)) → Q1(x1)
Q1(a(x1)) → B(b(a(x1)))
B(P(x1)) → B(Q(x1))
B(P(x1)) → Q1(x1)
Q1(a(x1)) → B(a(x1))
The TRS R consists of the following rules:
a(b(b(x1))) → P(a(b(x1)))
a(P(x1)) → P(a(x(x1)))
a(x(x1)) → x(a(x1))
b(P(x1)) → b(Q(x1))
Q(x(x1)) → a(Q(x1))
Q(a(x1)) → b(b(a(x1)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(11) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
Q1(a(x1)) → B(b(a(x1)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(B(x1)) = x1
POL(P(x1)) = 1
POL(Q(x1)) = 1
POL(Q1(x1)) = 1
POL(a(x1)) = 1
POL(b(x1)) = 0
POL(x(x1)) = x1
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
a(b(b(x1))) → P(a(b(x1)))
a(P(x1)) → P(a(x(x1)))
a(x(x1)) → x(a(x1))
b(P(x1)) → b(Q(x1))
Q(x(x1)) → a(Q(x1))
Q(a(x1)) → b(b(a(x1)))
(12) Obligation:
Q DP problem:
The TRS P consists of the following rules:
Q1(x(x1)) → Q1(x1)
B(P(x1)) → B(Q(x1))
B(P(x1)) → Q1(x1)
Q1(a(x1)) → B(a(x1))
The TRS R consists of the following rules:
a(b(b(x1))) → P(a(b(x1)))
a(P(x1)) → P(a(x(x1)))
a(x(x1)) → x(a(x1))
b(P(x1)) → b(Q(x1))
Q(x(x1)) → a(Q(x1))
Q(a(x1)) → b(b(a(x1)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(13) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
B(P(x1)) → B(Q(x1))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
| POL(x(x1)) = | | + | | / | 0A | 0A | -I | \ |
| | | -I | 1A | -I | | |
| \ | -I | -I | 0A | / |
| · | x1 |
| POL(P(x1)) = | | + | | / | 0A | 0A | -I | \ |
| | | -I | -I | -I | | |
| \ | 0A | 1A | 0A | / |
| · | x1 |
| POL(Q(x1)) = | | + | | / | -I | 0A | -I | \ |
| | | -I | 0A | -I | | |
| \ | -I | 0A | -I | / |
| · | x1 |
| POL(a(x1)) = | | + | | / | 0A | 0A | 0A | \ |
| | | -I | 0A | -I | | |
| \ | 0A | 0A | 1A | / |
| · | x1 |
| POL(b(x1)) = | | + | | / | -I | -I | -I | \ |
| | | -I | -I | -I | | |
| \ | -I | -I | -I | / |
| · | x1 |
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
Q(x(x1)) → a(Q(x1))
Q(a(x1)) → b(b(a(x1)))
a(b(b(x1))) → P(a(b(x1)))
a(P(x1)) → P(a(x(x1)))
a(x(x1)) → x(a(x1))
b(P(x1)) → b(Q(x1))
(14) Obligation:
Q DP problem:
The TRS P consists of the following rules:
Q1(x(x1)) → Q1(x1)
B(P(x1)) → Q1(x1)
Q1(a(x1)) → B(a(x1))
The TRS R consists of the following rules:
a(b(b(x1))) → P(a(b(x1)))
a(P(x1)) → P(a(x(x1)))
a(x(x1)) → x(a(x1))
b(P(x1)) → b(Q(x1))
Q(x(x1)) → a(Q(x1))
Q(a(x1)) → b(b(a(x1)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(15) 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:
- Q1(x(x1)) → Q1(x1)
The graph contains the following edges 1 > 1
- Q1(a(x1)) → B(a(x1))
The graph contains the following edges 1 >= 1
- B(P(x1)) → Q1(x1)
The graph contains the following edges 1 > 1
(16) YES