(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(y, f(x, y)) → f(f(a, y), f(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:
F(y, f(x, y)) → F(f(a, y), f(a, x))
F(y, f(x, y)) → F(a, y)
F(y, f(x, y)) → F(a, x)
The TRS R consists of the following rules:
f(y, f(x, y)) → f(f(a, y), f(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].
The following pairs can be oriented strictly and are deleted.
F(y, f(x, y)) → F(a, y)
F(y, f(x, y)) → F(a, x)
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(x, y)) → f(f(a, y), f(a, x))
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(y, f(x, y)) → F(f(a, y), f(a, x))
The TRS R consists of the following rules:
f(y, f(x, y)) → f(f(a, y), f(a, x))
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}(x, y)) → F_{f,f}(f_{a,f}(a, y), f_{a,f}(a, x))
F_{f,f}(y, f_{a,f}(x, y)) → F_{f,f}(f_{a,f}(a, y), f_{a,a}(a, x))
F_{a,f}(y, f_{f,a}(x, y)) → F_{f,f}(f_{a,a}(a, y), f_{a,f}(a, x))
F_{a,f}(y, f_{a,a}(x, y)) → F_{f,f}(f_{a,a}(a, y), f_{a,a}(a, x))
The TRS R consists of the following rules:
f_{f,f}(y, f_{f,f}(x, y)) → f_{f,f}(f_{a,f}(a, y), f_{a,f}(a, x))
f_{f,f}(y, f_{a,f}(x, y)) → f_{f,f}(f_{a,f}(a, y), f_{a,a}(a, x))
f_{a,f}(y, f_{f,a}(x, y)) → f_{f,f}(f_{a,a}(a, y), f_{a,f}(a, x))
f_{a,f}(y, f_{a,a}(x, y)) → f_{f,f}(f_{a,a}(a, y), f_{a,a}(a, 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 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}(x, y)) → F_{f,f}(f_{a,f}(a, y), f_{a,f}(a, x))
The TRS R consists of the following rules:
f_{f,f}(y, f_{f,f}(x, y)) → f_{f,f}(f_{a,f}(a, y), f_{a,f}(a, x))
f_{f,f}(y, f_{a,f}(x, y)) → f_{f,f}(f_{a,f}(a, y), f_{a,a}(a, x))
f_{a,f}(y, f_{f,a}(x, y)) → f_{f,f}(f_{a,a}(a, y), f_{a,f}(a, x))
f_{a,f}(y, f_{a,a}(x, y)) → f_{f,f}(f_{a,a}(a, y), f_{a,a}(a, x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) UsableRulesReductionPairsProof (EQUIVALENT transformation)
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
No dependency pairs are removed.
The following rules are removed from R:
f_{a,f}(y, f_{f,a}(x, y)) → f_{f,f}(f_{a,a}(a, y), f_{a,f}(a, x))
Used ordering: POLO with Polynomial interpretation [POLO]:
POL(F_{f,f}(x1, x2)) = 2·x1 + x2
POL(a) = 0
POL(f_{a,a}(x1, x2)) = 2·x1 + x2
POL(f_{a,f}(x1, x2)) = x1 + x2
POL(f_{f,a}(x1, x2)) = 2·x1 + x2
POL(f_{f,f}(x1, x2)) = 2·x1 + x2
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F_{f,f}(y, f_{f,f}(x, y)) → F_{f,f}(f_{a,f}(a, y), f_{a,f}(a, x))
The TRS R consists of the following rules:
f_{a,f}(y, f_{a,a}(x, y)) → f_{f,f}(f_{a,a}(a, y), f_{a,a}(a, x))
f_{f,f}(y, f_{f,f}(x, y)) → f_{f,f}(f_{a,f}(a, y), f_{a,f}(a, x))
f_{f,f}(y, f_{a,f}(x, y)) → f_{f,f}(f_{a,f}(a, y), f_{a,a}(a, x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(11) 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.
(12) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F_{f,f}(y, f_{f,f}(x, y)) → F_{f,f}(f_{a,f}(a, y), f_{a,f}(a, x))
The TRS R consists of the following rules:
f_{a,f}(y, f_{a,a}(x, y)) → f_{f,f}(f_{a,a}(a, y), f_{a,a}(a, x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(13) MNOCProof (EQUIVALENT transformation)
We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
(14) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F_{f,f}(y, f_{f,f}(x, y)) → F_{f,f}(f_{a,f}(a, y), f_{a,f}(a, x))
The TRS R consists of the following rules:
f_{a,f}(y, f_{a,a}(x, y)) → f_{f,f}(f_{a,a}(a, y), f_{a,a}(a, x))
The set Q consists of the following terms:
f_{a,f}(x0, f_{a,a}(x1, x0))
We have to consider all minimal (P,Q,R)-chains.
(15) Narrowing (EQUIVALENT transformation)
By narrowing [LPAR04] the rule
F_{f,f}(
y,
f_{f,f}(
x,
y)) →
F_{f,f}(
f_{a,f}(
a,
y),
f_{a,f}(
a,
x)) at position [1] we obtained the following new rules [LPAR04]:
F_{f,f}(y0, f_{f,f}(f_{a,a}(x1, a), y0)) → F_{f,f}(f_{a,f}(a, y0), f_{f,f}(f_{a,a}(a, a), f_{a,a}(a, x1)))
(16) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F_{f,f}(y0, f_{f,f}(f_{a,a}(x1, a), y0)) → F_{f,f}(f_{a,f}(a, y0), f_{f,f}(f_{a,a}(a, a), f_{a,a}(a, x1)))
The TRS R consists of the following rules:
f_{a,f}(y, f_{a,a}(x, y)) → f_{f,f}(f_{a,a}(a, y), f_{a,a}(a, x))
The set Q consists of the following terms:
f_{a,f}(x0, f_{a,a}(x1, x0))
We have to consider all minimal (P,Q,R)-chains.
(17) Narrowing (EQUIVALENT transformation)
By narrowing [LPAR04] the rule
F_{f,f}(
y0,
f_{f,f}(
f_{a,a}(
x1,
a),
y0)) →
F_{f,f}(
f_{a,f}(
a,
y0),
f_{f,f}(
f_{a,a}(
a,
a),
f_{a,a}(
a,
x1))) at position [] we obtained the following new rules [LPAR04]:
F_{f,f}(f_{a,a}(x1, a), f_{f,f}(f_{a,a}(y1, a), f_{a,a}(x1, a))) → F_{f,f}(f_{f,f}(f_{a,a}(a, a), f_{a,a}(a, x1)), f_{f,f}(f_{a,a}(a, a), f_{a,a}(a, y1)))
(18) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F_{f,f}(f_{a,a}(x1, a), f_{f,f}(f_{a,a}(y1, a), f_{a,a}(x1, a))) → F_{f,f}(f_{f,f}(f_{a,a}(a, a), f_{a,a}(a, x1)), f_{f,f}(f_{a,a}(a, a), f_{a,a}(a, y1)))
The TRS R consists of the following rules:
f_{a,f}(y, f_{a,a}(x, y)) → f_{f,f}(f_{a,a}(a, y), f_{a,a}(a, x))
The set Q consists of the following terms:
f_{a,f}(x0, f_{a,a}(x1, x0))
We have to consider all minimal (P,Q,R)-chains.
(19) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
(20) TRUE