(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(y, f(y, x)) → f(f(a, y), f(a, y))
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:
F(y, f(y, x)) → F(f(a, y), f(a, y))
F(y, f(y, x)) → F(a, y)
The TRS R consists of the following rules:
f(y, f(y, x)) → f(f(a, y), f(a, y))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
F(y, f(y, x)) → F(a, y)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
| POL(F(x1, x2)) = | 0A | + | 1A | · | x1 | + | 1A | · | x2 |
| POL(f(x1, x2)) = | 5A | + | 1A | · | x1 | + | 1A | · | x2 |
The following usable rules [FROCOS05] were oriented:
f(y, f(y, x)) → f(f(a, y), f(a, y))
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(y, f(y, x)) → F(f(a, y), f(a, y))
The TRS R consists of the following rules:
f(y, f(y, x)) → f(f(a, y), f(a, y))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) RootLabelingFC2Proof (EQUIVALENT transformation)
We used root labeling (second transformation) [ROOTLAB] with the following heuristic:
LabelAll: All function symbols get labeled
As Q is empty the root labeling was sound AND complete.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F_{f,f}(y, f_{f,f}(y, x)) → F_{f,f}(f_{a,f}(a, y), f_{a,f}(a, y))
F_{f,f}(y, f_{f,a}(y, x)) → F_{f,f}(f_{a,f}(a, y), f_{a,f}(a, y))
F_{a,f}(y, f_{a,f}(y, x)) → F_{f,f}(f_{a,a}(a, y), f_{a,a}(a, y))
F_{a,f}(y, f_{a,a}(y, x)) → F_{f,f}(f_{a,a}(a, y), f_{a,a}(a, y))
The TRS R consists of the following rules:
f_{f,f}(y, f_{f,f}(y, x)) → f_{f,f}(f_{a,f}(a, y), f_{a,f}(a, y))
f_{f,f}(y, f_{f,a}(y, x)) → f_{f,f}(f_{a,f}(a, y), f_{a,f}(a, y))
f_{a,f}(y, f_{a,f}(y, x)) → f_{f,f}(f_{a,a}(a, y), f_{a,a}(a, y))
f_{a,f}(y, f_{a,a}(y, x)) → f_{f,f}(f_{a,a}(a, y), f_{a,a}(a, y))
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 1 SCC with 3 less nodes.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F_{f,f}(y, f_{f,f}(y, x)) → F_{f,f}(f_{a,f}(a, y), f_{a,f}(a, y))
The TRS R consists of the following rules:
f_{f,f}(y, f_{f,f}(y, x)) → f_{f,f}(f_{a,f}(a, y), f_{a,f}(a, y))
f_{f,f}(y, f_{f,a}(y, x)) → f_{f,f}(f_{a,f}(a, y), f_{a,f}(a, y))
f_{a,f}(y, f_{a,f}(y, x)) → f_{f,f}(f_{a,a}(a, y), f_{a,a}(a, y))
f_{a,f}(y, f_{a,a}(y, x)) → f_{f,f}(f_{a,a}(a, y), f_{a,a}(a, y))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) 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.
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F_{f,f}(y, f_{f,f}(y, x)) → F_{f,f}(f_{a,f}(a, y), f_{a,f}(a, y))
The TRS R consists of the following rules:
f_{a,f}(y, f_{a,f}(y, x)) → f_{f,f}(f_{a,a}(a, y), f_{a,a}(a, y))
f_{a,f}(y, f_{a,a}(y, x)) → f_{f,f}(f_{a,a}(a, y), f_{a,a}(a, y))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(11) SemLabProof (SOUND transformation)
We found the following model for the rules of the TRS R.
Interpretation over the domain with elements from 0 to 1.a: 0
f_{f,f}: 0
f_{a,a}: 1
f_{a,f}: 0
F_{f,f}: 0
By semantic labelling [SEMLAB] we obtain the following labelled TRS.
(12) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F_{f,f}.0-0(y, f_{f,f}.0-0(y, x)) → F_{f,f}.0-0(f_{a,f}.0-0(a., y), f_{a,f}.0-0(a., y))
F_{f,f}.0-0(y, f_{f,f}.0-1(y, x)) → F_{f,f}.0-0(f_{a,f}.0-0(a., y), f_{a,f}.0-0(a., y))
F_{f,f}.1-0(y, f_{f,f}.1-0(y, x)) → F_{f,f}.0-0(f_{a,f}.0-1(a., y), f_{a,f}.0-1(a., y))
F_{f,f}.1-0(y, f_{f,f}.1-1(y, x)) → F_{f,f}.0-0(f_{a,f}.0-1(a., y), f_{a,f}.0-1(a., y))
The TRS R consists of the following rules:
f_{a,f}.0-0(y, f_{a,f}.0-1(y, x)) → f_{f,f}.1-1(f_{a,a}.0-0(a., y), f_{a,a}.0-0(a., y))
f_{a,f}.1-0(y, f_{a,f}.1-1(y, x)) → f_{f,f}.1-1(f_{a,a}.0-1(a., y), f_{a,a}.0-1(a., y))
f_{a,f}.1-1(y, f_{a,a}.1-0(y, x)) → f_{f,f}.1-1(f_{a,a}.0-1(a., y), f_{a,a}.0-1(a., y))
f_{a,f}.1-0(y, f_{a,f}.1-0(y, x)) → f_{f,f}.1-1(f_{a,a}.0-1(a., y), f_{a,a}.0-1(a., y))
f_{a,f}.1-1(y, f_{a,a}.1-1(y, x)) → f_{f,f}.1-1(f_{a,a}.0-1(a., y), f_{a,a}.0-1(a., y))
f_{a,f}.0-1(y, f_{a,a}.0-0(y, x)) → f_{f,f}.1-1(f_{a,a}.0-0(a., y), f_{a,a}.0-0(a., y))
f_{a,f}.0-1(y, f_{a,a}.0-1(y, x)) → f_{f,f}.1-1(f_{a,a}.0-0(a., y), f_{a,a}.0-0(a., y))
f_{a,f}.0-0(y, f_{a,f}.0-0(y, x)) → f_{f,f}.1-1(f_{a,a}.0-0(a., y), f_{a,a}.0-0(a., y))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(13) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.
(14) TRUE