Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex6_9_Luc02c_FR)
The rewrite relation of the following TRS is considered.
2nd(cons1(X,cons(Y,Z))) |
→ |
Y |
(1) |
2nd(cons(X,X1)) |
→ |
2nd(cons1(X,activate(X1))) |
(2) |
from(X) |
→ |
cons(X,n__from(n__s(X))) |
(3) |
from(X) |
→ |
n__from(X) |
(4) |
s(X) |
→ |
n__s(X) |
(5) |
activate(n__from(X)) |
→ |
from(activate(X)) |
(6) |
activate(n__s(X)) |
→ |
s(activate(X)) |
(7) |
activate(X) |
→ |
X |
(8) |
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.
2nd#(cons(X,X1)) |
→ |
2nd#(cons1(X,activate(X1))) |
(9) |
2nd#(cons(X,X1)) |
→ |
activate#(X1) |
(10) |
activate#(n__from(X)) |
→ |
from#(activate(X)) |
(11) |
activate#(n__from(X)) |
→ |
activate#(X) |
(12) |
activate#(n__s(X)) |
→ |
s#(activate(X)) |
(13) |
activate#(n__s(X)) |
→ |
activate#(X) |
(14) |
1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
1st
component contains the
pair
activate#(n__s(X)) |
→ |
activate#(X) |
(14) |
activate#(n__from(X)) |
→ |
activate#(X) |
(12) |
1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[n__s(x1)] |
= |
1 · x1
|
[n__from(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 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
activate#(n__s(X)) |
→ |
activate#(X) |
(14) |
|
1 |
> |
1 |
activate#(n__from(X)) |
→ |
activate#(X) |
(12) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.