Certification Problem
Input (TPDB SRS_Standard/Waldmann_19/random-106)
The rewrite relation of the following TRS is considered.
|
b(a(b(a(x1)))) |
→ |
b(a(b(b(x1)))) |
(1) |
|
b(b(b(b(x1)))) |
→ |
b(a(a(b(x1)))) |
(2) |
|
a(a(a(b(x1)))) |
→ |
b(a(b(a(x1)))) |
(3) |
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.
|
b#(a(b(a(x1)))) |
→ |
b#(x1) |
(4) |
|
b#(a(b(a(x1)))) |
→ |
b#(b(x1)) |
(5) |
|
b#(a(b(a(x1)))) |
→ |
a#(b(b(x1))) |
(6) |
|
b#(a(b(a(x1)))) |
→ |
b#(a(b(b(x1)))) |
(7) |
|
b#(b(b(b(x1)))) |
→ |
a#(b(x1)) |
(8) |
|
b#(b(b(b(x1)))) |
→ |
a#(a(b(x1))) |
(9) |
|
b#(b(b(b(x1)))) |
→ |
b#(a(a(b(x1)))) |
(10) |
|
a#(a(a(b(x1)))) |
→ |
a#(x1) |
(11) |
|
a#(a(a(b(x1)))) |
→ |
b#(a(x1)) |
(12) |
|
a#(a(a(b(x1)))) |
→ |
a#(b(a(x1))) |
(13) |
|
a#(a(a(b(x1)))) |
→ |
b#(a(b(a(x1)))) |
(14) |
1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.
-
The
1st
component contains the
pair
|
a#(a(a(b(x1)))) |
→ |
a#(x1) |
(11) |
1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
a#(a(a(b(x1)))) |
→ |
a#(x1) |
(11) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
|
b#(b(b(b(x1)))) |
→ |
b#(a(a(b(x1)))) |
(10) |
|
b#(a(b(a(x1)))) |
→ |
b#(a(b(b(x1)))) |
(7) |
|
b#(a(b(a(x1)))) |
→ |
b#(b(x1)) |
(5) |
|
b#(a(b(a(x1)))) |
→ |
b#(x1) |
(4) |
1.1.2 Subterm Criterion Processor
We use the projection to multisets
| π(b#)
|
= |
{
1
}
|
| π(b)
|
= |
{
1, 1
}
|
| π(a)
|
= |
{
1, 1
}
|
to remove the pairs:
|
b#(a(b(a(x1)))) |
→ |
b#(b(x1)) |
(5) |
|
b#(a(b(a(x1)))) |
→ |
b#(x1) |
(4) |
1.1.2.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1
over the naturals
| [b(x1)] |
= |
· x1 +
|
| [a(x1)] |
= |
· x1 +
|
| [b#(x1)] |
= |
· x1 +
|
together with the usable
rules
|
b(a(b(a(x1)))) |
→ |
b(a(b(b(x1)))) |
(1) |
|
b(b(b(b(x1)))) |
→ |
b(a(a(b(x1)))) |
(2) |
|
a(a(a(b(x1)))) |
→ |
b(a(b(a(x1)))) |
(3) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
b#(b(b(b(x1)))) |
→ |
b#(a(a(b(x1)))) |
(10) |
could be deleted.
1.1.2.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over (4 x 4)-matrices with strict dimension 1
over the naturals
| [b(x1)] |
= |
|
| 0 |
0 |
1 |
1 |
| 0 |
0 |
0 |
0 |
| 0 |
1 |
0 |
0 |
| 0 |
0 |
0 |
0 |
|
|
· x1 +
|
| 0 |
0 |
0 |
0 |
| 1 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
|
|
|
| [a(x1)] |
= |
|
| 1 |
0 |
0 |
1 |
| 0 |
0 |
0 |
1 |
| 0 |
1 |
0 |
0 |
| 0 |
0 |
0 |
0 |
|
|
· x1 +
|
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 1 |
0 |
0 |
0 |
|
|
|
| [b#(x1)] |
= |
|
| 1 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
|
|
· x1 +
|
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
|
|
|
together with the usable
rules
|
b(a(b(a(x1)))) |
→ |
b(a(b(b(x1)))) |
(1) |
|
b(b(b(b(x1)))) |
→ |
b(a(a(b(x1)))) |
(2) |
|
a(a(a(b(x1)))) |
→ |
b(a(b(a(x1)))) |
(3) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
b#(a(b(a(x1)))) |
→ |
b#(a(b(b(x1)))) |
(7) |
could be deleted.
1.1.2.1.1.1 P is empty
There are no pairs anymore.