Certification Problem
Input (TPDB SRS_Standard/Waldmann_19/random-421)
The rewrite relation of the following TRS is considered.
|
a(b(a(b(x1)))) |
→ |
b(b(a(a(x1)))) |
(1) |
|
a(a(a(b(x1)))) |
→ |
a(a(b(a(x1)))) |
(2) |
|
b(b(a(b(x1)))) |
→ |
b(a(b(b(x1)))) |
(3) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by ttt2 @ termCOMP 2023)
1 String Reversal
Since only unary symbols occur, one can reverse all terms and obtains the TRS
|
b(a(b(a(x1)))) |
→ |
a(a(b(b(x1)))) |
(4) |
|
b(a(a(a(x1)))) |
→ |
a(b(a(a(x1)))) |
(5) |
|
b(a(b(b(x1)))) |
→ |
b(b(a(b(x1)))) |
(6) |
1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
b#(a(b(a(x1)))) |
→ |
b#(x1) |
(7) |
|
b#(a(b(a(x1)))) |
→ |
b#(b(x1)) |
(8) |
|
b#(a(a(a(x1)))) |
→ |
b#(a(a(x1))) |
(9) |
|
b#(a(b(b(x1)))) |
→ |
b#(a(b(x1))) |
(10) |
|
b#(a(b(b(x1)))) |
→ |
b#(b(a(b(x1)))) |
(11) |
1.1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.
-
The
1st
component contains the
pair
|
b#(a(b(a(x1)))) |
→ |
b#(b(x1)) |
(8) |
|
b#(a(b(b(x1)))) |
→ |
b#(b(a(b(x1)))) |
(11) |
|
b#(a(b(b(x1)))) |
→ |
b#(a(b(x1))) |
(10) |
|
b#(a(b(a(x1)))) |
→ |
b#(x1) |
(7) |
1.1.1.1 Subterm Criterion Processor
We use the projection to multisets
| π(b#)
|
= |
{
1, 1
}
|
| π(a)
|
= |
{
1
}
|
| π(b)
|
= |
{
1, 1
}
|
to remove the pairs:
|
b#(a(b(b(x1)))) |
→ |
b#(a(b(x1))) |
(10) |
|
b#(a(b(a(x1)))) |
→ |
b#(x1) |
(7) |
1.1.1.1.1 Subterm Criterion Processor
We use the projection to multisets
| π(b#)
|
= |
{
1, 1
}
|
| π(a)
|
= |
{
1, 1
}
|
| π(b)
|
= |
{
1, 1
}
|
to remove the pairs:
|
b#(a(b(a(x1)))) |
→ |
b#(b(x1)) |
(8) |
1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1
over the naturals
| [a(x1)] |
= |
· x1 +
|
| [b(x1)] |
= |
· x1 +
|
| [b#(x1)] |
= |
· x1 +
|
together with the usable
rules
|
b(a(b(a(x1)))) |
→ |
a(a(b(b(x1)))) |
(4) |
|
b(a(a(a(x1)))) |
→ |
a(b(a(a(x1)))) |
(5) |
|
b(a(b(b(x1)))) |
→ |
b(b(a(b(x1)))) |
(6) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
b#(a(b(b(x1)))) |
→ |
b#(b(a(b(x1)))) |
(11) |
could be deleted.
1.1.1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
|
b#(a(a(a(x1)))) |
→ |
b#(a(a(x1))) |
(9) |
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.
|
b#(a(a(a(x1)))) |
→ |
b#(a(a(x1))) |
(9) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.