Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/ExConc_Zan97_FR)
The rewrite relation of the following TRS is considered.
|
f(X) |
→ |
g(n__h(n__f(X))) |
(1) |
|
h(X) |
→ |
n__h(X) |
(2) |
|
f(X) |
→ |
n__f(X) |
(3) |
|
activate(n__h(X)) |
→ |
h(activate(X)) |
(4) |
|
activate(n__f(X)) |
→ |
f(activate(X)) |
(5) |
|
activate(X) |
→ |
X |
(6) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [f(x1)] |
= |
1 · x1
|
| [g(x1)] |
= |
1 · x1
|
| [n__h(x1)] |
= |
1 · x1
|
| [n__f(x1)] |
= |
1 · x1
|
| [h(x1)] |
= |
1 · x1
|
| [activate(x1)] |
= |
1 · x1 + 1 |
all of the following rules can be deleted.
1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
activate#(n__h(X)) |
→ |
h#(activate(X)) |
(7) |
|
activate#(n__h(X)) |
→ |
activate#(X) |
(8) |
|
activate#(n__f(X)) |
→ |
f#(activate(X)) |
(9) |
|
activate#(n__f(X)) |
→ |
activate#(X) |
(10) |
1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
1st
component contains the
pair
|
activate#(n__f(X)) |
→ |
activate#(X) |
(10) |
|
activate#(n__h(X)) |
→ |
activate#(X) |
(8) |
1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [n__f(x1)] |
= |
1 · x1
|
| [n__h(x1)] |
= |
1 · x1
|
| [activate#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
activate#(n__f(X)) |
→ |
activate#(X) |
(10) |
|
| 1 |
> |
1 |
|
activate#(n__h(X)) |
→ |
activate#(X) |
(8) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.