Certification Problem
Input (TPDB SRS_Standard/Secret_06_SRS/aprove00)
The rewrite relation of the following TRS is considered.
|
a(s(x1)) |
→ |
s(s(s(p(s(b(p(p(s(s(x1)))))))))) |
(1) |
|
b(s(x1)) |
→ |
s(s(s(p(p(s(s(c(p(s(p(s(x1)))))))))))) |
(2) |
|
c(s(x1)) |
→ |
p(s(p(s(a(p(s(p(s(x1))))))))) |
(3) |
|
p(p(s(x1))) |
→ |
p(x1) |
(4) |
|
p(s(x1)) |
→ |
x1 |
(5) |
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.
|
a#(s(x1)) |
→ |
p#(s(b(p(p(s(s(x1))))))) |
(6) |
|
a#(s(x1)) |
→ |
b#(p(p(s(s(x1))))) |
(7) |
|
a#(s(x1)) |
→ |
p#(p(s(s(x1)))) |
(8) |
|
a#(s(x1)) |
→ |
p#(s(s(x1))) |
(9) |
|
b#(s(x1)) |
→ |
p#(p(s(s(c(p(s(p(s(x1))))))))) |
(10) |
|
b#(s(x1)) |
→ |
p#(s(s(c(p(s(p(s(x1)))))))) |
(11) |
|
b#(s(x1)) |
→ |
c#(p(s(p(s(x1))))) |
(12) |
|
b#(s(x1)) |
→ |
p#(s(p(s(x1)))) |
(13) |
|
b#(s(x1)) |
→ |
p#(s(x1)) |
(14) |
|
c#(s(x1)) |
→ |
p#(s(p(s(a(p(s(p(s(x1))))))))) |
(15) |
|
c#(s(x1)) |
→ |
p#(s(a(p(s(p(s(x1))))))) |
(16) |
|
c#(s(x1)) |
→ |
a#(p(s(p(s(x1))))) |
(17) |
|
c#(s(x1)) |
→ |
p#(s(p(s(x1)))) |
(18) |
|
c#(s(x1)) |
→ |
p#(s(x1)) |
(19) |
|
p#(p(s(x1))) |
→ |
p#(x1) |
(20) |
1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.
-
The
1st
component contains the
pair
|
a#(s(x1)) |
→ |
b#(p(p(s(s(x1))))) |
(7) |
|
b#(s(x1)) |
→ |
c#(p(s(p(s(x1))))) |
(12) |
|
c#(s(x1)) |
→ |
a#(p(s(p(s(x1))))) |
(17) |
1.1.1 Switch to Innermost Termination
The TRS does not have overlaps with the pairs and is locally confluent:
20
Hence, it suffices to show innermost termination in the following.
1.1.1.1 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
1.1.1.1.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
1.1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [a#(x1)] |
= |
-2 + 2 · x1
|
| [b#(x1)] |
= |
-2 + 2 · x1
|
| [c#(x1)] |
= |
-1 + 2 · x1
|
| [p(x1)] |
= |
-1 + x1
|
| [s(x1)] |
= |
1 + x1
|
the
pair
|
c#(s(x1)) |
→ |
a#(p(s(p(s(x1))))) |
(17) |
could be deleted.
1.1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
2nd
component contains the
pair
|
p#(p(s(x1))) |
→ |
p#(x1) |
(20) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [p(x1)] |
= |
1 · x1
|
| [s(x1)] |
= |
1 · x1
|
| [p#(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.
|
p#(p(s(x1))) |
→ |
p#(x1) |
(20) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.