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 ttt2 @ termCOMP 2023)
1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
a#(s(x1)) |
→ |
p#(s(s(x1))) |
(6) |
|
a#(s(x1)) |
→ |
p#(p(s(s(x1)))) |
(7) |
|
a#(s(x1)) |
→ |
b#(p(p(s(s(x1))))) |
(8) |
|
a#(s(x1)) |
→ |
p#(s(b(p(p(s(s(x1))))))) |
(9) |
|
b#(s(x1)) |
→ |
p#(s(x1)) |
(10) |
|
b#(s(x1)) |
→ |
p#(s(p(s(x1)))) |
(11) |
|
b#(s(x1)) |
→ |
c#(p(s(p(s(x1))))) |
(12) |
|
b#(s(x1)) |
→ |
p#(s(s(c(p(s(p(s(x1)))))))) |
(13) |
|
b#(s(x1)) |
→ |
p#(p(s(s(c(p(s(p(s(x1))))))))) |
(14) |
|
c#(s(x1)) |
→ |
p#(s(x1)) |
(15) |
|
c#(s(x1)) |
→ |
p#(s(p(s(x1)))) |
(16) |
|
c#(s(x1)) |
→ |
a#(p(s(p(s(x1))))) |
(17) |
|
c#(s(x1)) |
→ |
p#(s(a(p(s(p(s(x1))))))) |
(18) |
|
c#(s(x1)) |
→ |
p#(s(p(s(a(p(s(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
|
c#(s(x1)) |
→ |
a#(p(s(p(s(x1))))) |
(17) |
|
a#(s(x1)) |
→ |
b#(p(p(s(s(x1))))) |
(8) |
|
b#(s(x1)) |
→ |
c#(p(s(p(s(x1))))) |
(12) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the arctic semiring over the integers
| [c#(x1)] |
= |
2 · x1 + 0 |
| [a#(x1)] |
= |
3 · x1 + 0 |
| [s(x1)] |
= |
1 · x1 + 1 |
| [b#(x1)] |
= |
4 · x1 + 0 |
| [p(x1)] |
= |
-1 · x1 + 0 |
together with the usable
rules
|
p(s(x1)) |
→ |
x1 |
(5) |
|
p(p(s(x1))) |
→ |
p(x1) |
(4) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
b#(s(x1)) |
→ |
c#(p(s(p(s(x1))))) |
(12) |
could be deleted.
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 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.