Certification Problem
Input (TPDB TRS_Standard/Mixed_TRS/test1)
The rewrite relation of the following TRS is considered.
|
f(s(x),y) |
→ |
f(x,s(s(x))) |
(1) |
|
f(x,s(s(y))) |
→ |
f(y,x) |
(2) |
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#(s(x),y) |
→ |
f#(x,s(s(x))) |
(3) |
|
f#(x,s(s(y))) |
→ |
f#(y,x) |
(4) |
1.1 Reduction Pair Processor with Usable Rules
Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals
| [f#(x1, x2)] |
= |
+ · x1 + · x2
|
| [s(x1)] |
= |
+ · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
|
f#(s(x),y) |
→ |
f#(x,s(s(x))) |
(3) |
could be deleted.
1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [s(x1)] |
= |
1 · x1
|
| [f#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
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#(x,s(s(y))) |
→ |
f#(y,x) |
(4) |
|
|
| 2 |
> |
1 |
| 1 |
≥ |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.