Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex25_Luc06_GM)
The rewrite relation of the following TRS is considered.
a__f(f(X)) |
→ |
a__c(f(g(f(X)))) |
(1) |
a__c(X) |
→ |
d(X) |
(2) |
a__h(X) |
→ |
a__c(d(X)) |
(3) |
mark(f(X)) |
→ |
a__f(mark(X)) |
(4) |
mark(c(X)) |
→ |
a__c(X) |
(5) |
mark(h(X)) |
→ |
a__h(mark(X)) |
(6) |
mark(g(X)) |
→ |
g(X) |
(7) |
mark(d(X)) |
→ |
d(X) |
(8) |
a__f(X) |
→ |
f(X) |
(9) |
a__c(X) |
→ |
c(X) |
(10) |
a__h(X) |
→ |
h(X) |
(11) |
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
[a__f(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 · x1
|
[a__c(x1)] |
= |
1 · x1
|
[g(x1)] |
= |
1 · x1
|
[d(x1)] |
= |
1 · x1
|
[a__h(x1)] |
= |
1 · x1 + 1 |
[mark(x1)] |
= |
1 · x1
|
[c(x1)] |
= |
1 · x1
|
[h(x1)] |
= |
1 · x1 + 1 |
all of the following rules can be deleted.
1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
[a__f(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 · x1
|
[a__c(x1)] |
= |
1 · x1
|
[g(x1)] |
= |
1 · x1
|
[d(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1 + 1 |
[c(x1)] |
= |
1 · x1
|
[h(x1)] |
= |
1 · x1
|
[a__h(x1)] |
= |
1 · x1
|
all of the following rules can be deleted.
mark(c(X)) |
→ |
a__c(X) |
(5) |
mark(g(X)) |
→ |
g(X) |
(7) |
mark(d(X)) |
→ |
d(X) |
(8) |
1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
a__f#(f(X)) |
→ |
a__c#(f(g(f(X)))) |
(12) |
mark#(f(X)) |
→ |
a__f#(mark(X)) |
(13) |
mark#(f(X)) |
→ |
mark#(X) |
(14) |
mark#(h(X)) |
→ |
a__h#(mark(X)) |
(15) |
mark#(h(X)) |
→ |
mark#(X) |
(16) |
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
1st
component contains the
pair
mark#(h(X)) |
→ |
mark#(X) |
(16) |
mark#(f(X)) |
→ |
mark#(X) |
(14) |
1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[h(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 · x1
|
[mark#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
mark#(h(X)) |
→ |
mark#(X) |
(16) |
|
1 |
> |
1 |
mark#(f(X)) |
→ |
mark#(X) |
(14) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.