(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(b(x)) → b(r(x))
r(a(x)) → d(r(x))
r(x) → d(x)
d(a(x)) → a(a(d(x)))
d(x) → a(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(b(x)) → R(x)
R(a(x)) → D(r(x))
R(a(x)) → R(x)
R(x) → D(x)
D(a(x)) → A(a(d(x)))
D(a(x)) → A(d(x))
D(a(x)) → D(x)
D(x) → A(x)
The TRS R consists of the following rules:
a(b(x)) → b(r(x))
r(a(x)) → d(r(x))
r(x) → d(x)
d(a(x)) → a(a(d(x)))
d(x) → a(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
A(b(x)) → R(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(A(x1)) = x1
POL(D(x1)) = x1
POL(R(x1)) = x1
POL(a(x1)) = x1
POL(b(x1)) = 1 + x1
POL(d(x1)) = x1
POL(r(x1)) = x1
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
r(a(x)) → d(r(x))
r(x) → d(x)
d(a(x)) → a(a(d(x)))
d(x) → a(x)
a(b(x)) → b(r(x))
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
R(a(x)) → D(r(x))
R(a(x)) → R(x)
R(x) → D(x)
D(a(x)) → A(a(d(x)))
D(a(x)) → A(d(x))
D(a(x)) → D(x)
D(x) → A(x)
The TRS R consists of the following rules:
a(b(x)) → b(r(x))
r(a(x)) → d(r(x))
r(x) → d(x)
d(a(x)) → a(a(d(x)))
d(x) → a(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 5 less nodes.
(6) Complex Obligation (AND)
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
D(a(x)) → D(x)
The TRS R consists of the following rules:
a(b(x)) → b(r(x))
r(a(x)) → d(r(x))
r(x) → d(x)
d(a(x)) → a(a(d(x)))
d(x) → a(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(8) 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.
(9) Obligation:
Q DP problem:
The TRS P consists of the following rules:
D(a(x)) → D(x)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(10) 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:
- D(a(x)) → D(x)
The graph contains the following edges 1 > 1
(11) YES
(12) Obligation:
Q DP problem:
The TRS P consists of the following rules:
R(a(x)) → R(x)
The TRS R consists of the following rules:
a(b(x)) → b(r(x))
r(a(x)) → d(r(x))
r(x) → d(x)
d(a(x)) → a(a(d(x)))
d(x) → a(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(13) 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.
(14) Obligation:
Q DP problem:
The TRS P consists of the following rules:
R(a(x)) → R(x)
R is empty.
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:
- R(a(x)) → R(x)
The graph contains the following edges 1 > 1
(16) YES