Certification Problem
Input (TPDB TRS_Standard/Secret_05_TRS/matchbox1)
The rewrite relation of the following TRS is considered.
|
f(f(x,a),y) |
→ |
f(y,f(x,y)) |
(1) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
f#(f(x,a),y) |
→ |
f#(y,f(x,y)) |
(2) |
|
f#(f(x,a),y) |
→ |
f#(x,y) |
(3) |
1.1 Forward Instantiation Processor
We instantiate the pair
to the following set of pairs
|
f#(f(x0,a),f(y_0,a)) |
→ |
f#(f(y_0,a),f(x0,f(y_0,a))) |
(4) |
1.1.1 Forward Instantiation Processor
We instantiate the pair
to the following set of pairs
|
f#(f(f(y_0,a),a),x1) |
→ |
f#(f(y_0,a),x1) |
(5) |
|
f#(f(f(y_0,a),a),f(y_1,a)) |
→ |
f#(f(y_0,a),f(y_1,a)) |
(6) |
1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the rationals with delta = 3/256
| [f#(x1, x2)] |
= |
0 + 0 · x1 + 1/4 · x2
|
| [f(x1, x2)] |
= |
0 + 0 · x1 + 1/4 · x2
|
| [a] |
= |
1/4 |
the
pair
|
f#(f(x0,a),f(y_0,a)) |
→ |
f#(f(y_0,a),f(x0,f(y_0,a))) |
(4) |
could be deleted.
1.1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
f#(f(f(y_0,a),a),x1) |
→ |
f#(f(y_0,a),x1) |
(5) |
|
|
| 1 |
> |
1 |
| 2 |
≥ |
2 |
|
f#(f(f(y_0,a),a),f(y_1,a)) |
→ |
f#(f(y_0,a),f(y_1,a)) |
(6) |
|
|
| 1 |
> |
1 |
| 2 |
≥ |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.