Certification Problem
Input (TPDB TRS_Standard/AProVE_04/improved_usable2)
The rewrite relation of the following TRS is considered.
|
f(a,x) |
→ |
f(g(x),x) |
(1) |
|
h(g(x)) |
→ |
h(a) |
(2) |
|
g(h(x)) |
→ |
g(x) |
(3) |
|
h(h(x)) |
→ |
x |
(4) |
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#(a,x) |
→ |
f#(g(x),x) |
(5) |
|
f#(a,x) |
→ |
g#(x) |
(6) |
|
h#(g(x)) |
→ |
h#(a) |
(7) |
|
g#(h(x)) |
→ |
g#(x) |
(8) |
1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.
-
The
1st
component contains the
pair
1.1.1 Reduction Pair Processor with Usable Rules
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
| prec(a) |
= |
1 |
|
weight(a) |
= |
1 |
|
|
|
| prec(g) |
= |
0 |
|
weight(g) |
= |
1 |
|
|
|
in combination with the following argument filter
| π(f#) |
= |
1 |
| π(a) |
= |
[] |
| π(g) |
= |
[] |
together with the usable
rule
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
could be deleted.
1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [h(x1)] |
= |
1 · x1
|
| [g#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
g#(h(x)) |
→ |
g#(x) |
(8) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.