Certification Problem
Input (TPDB SRS_Standard/Secret_06_SRS/aprove09)
The rewrite relation of the following TRS is considered.
|
q(0(x1)) |
→ |
p(p(s(s(0(s(s(s(s(x1))))))))) |
(1) |
|
q(s(x1)) |
→ |
p(p(s(s(s(s(s(s(r(p(p(s(s(x1))))))))))))) |
(2) |
|
r(0(x1)) |
→ |
p(s(p(s(0(p(p(p(s(s(s(x1))))))))))) |
(3) |
|
r(s(x1)) |
→ |
p(s(p(s(s(q(p(s(p(s(x1)))))))))) |
(4) |
|
p(p(s(x1))) |
→ |
p(x1) |
(5) |
|
p(s(x1)) |
→ |
x1 |
(6) |
|
p(0(x1)) |
→ |
0(s(s(s(x1)))) |
(7) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by ttt2 @ termCOMP 2023)
1 Rule Removal
Using the
linear polynomial interpretation over the arctic semiring over the integers
| [r(x1)] |
= |
1 · x1 +
-∞ |
| [q(x1)] |
= |
1 · x1 +
-∞ |
| [p(x1)] |
= |
0 · x1 +
-∞ |
| [0(x1)] |
= |
0 · x1 +
-∞ |
| [s(x1)] |
= |
0 · x1 +
-∞ |
all of the following rules can be deleted.
|
q(0(x1)) |
→ |
p(p(s(s(0(s(s(s(s(x1))))))))) |
(1) |
|
r(0(x1)) |
→ |
p(s(p(s(0(p(p(p(s(s(s(x1))))))))))) |
(3) |
1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
q#(s(x1)) |
→ |
p#(s(s(x1))) |
(8) |
|
q#(s(x1)) |
→ |
p#(p(s(s(x1)))) |
(9) |
|
q#(s(x1)) |
→ |
r#(p(p(s(s(x1))))) |
(10) |
|
q#(s(x1)) |
→ |
p#(s(s(s(s(s(s(r(p(p(s(s(x1)))))))))))) |
(11) |
|
q#(s(x1)) |
→ |
p#(p(s(s(s(s(s(s(r(p(p(s(s(x1))))))))))))) |
(12) |
|
r#(s(x1)) |
→ |
p#(s(x1)) |
(13) |
|
r#(s(x1)) |
→ |
p#(s(p(s(x1)))) |
(14) |
|
r#(s(x1)) |
→ |
q#(p(s(p(s(x1))))) |
(15) |
|
r#(s(x1)) |
→ |
p#(s(s(q(p(s(p(s(x1)))))))) |
(16) |
|
r#(s(x1)) |
→ |
p#(s(p(s(s(q(p(s(p(s(x1)))))))))) |
(17) |
|
p#(p(s(x1))) |
→ |
p#(x1) |
(18) |
1.1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.
-
The
1st
component contains the
pair
|
r#(s(x1)) |
→ |
q#(p(s(p(s(x1))))) |
(15) |
|
q#(s(x1)) |
→ |
r#(p(p(s(s(x1))))) |
(10) |
1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the arctic semiring over the integers
| [p(x1)] |
= |
-1 · x1 + 0 |
| [q#(x1)] |
= |
1 · x1 + -16 |
| [0(x1)] |
= |
-∞ · x1 + 0 |
| [r#(x1)] |
= |
0 · x1 + -16 |
| [s(x1)] |
= |
1 · x1 + 3 |
together with the usable
rules
|
p(s(x1)) |
→ |
x1 |
(6) |
|
p(p(s(x1))) |
→ |
p(x1) |
(5) |
|
p(0(x1)) |
→ |
0(s(s(s(x1)))) |
(7) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
q#(s(x1)) |
→ |
r#(p(p(s(s(x1))))) |
(10) |
could be deleted.
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) |
(18) |
1.1.1.2 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) |
(18) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.