Certification Problem
Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-2-num-5)
The rewrite relation of the following TRS is considered.
|
a(a(x1)) |
→ |
x1 |
(1) |
|
a(b(b(a(x1)))) |
→ |
a(a(b(a(b(b(x1)))))) |
(2) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by ttt2 @ termCOMP 2023)
1 Closure Under Flat Contexts
Using the flat contexts
{a(☐), b(☐)}
We obtain the transformed TRS
|
a(a(a(x1))) |
→ |
a(x1) |
(3) |
|
b(a(a(x1))) |
→ |
b(x1) |
(4) |
|
a(b(b(a(x1)))) |
→ |
a(a(b(a(b(b(x1)))))) |
(2) |
1.1 Semantic Labeling
Root-labeling is applied.
We obtain the labeled TRS
|
aa(aa(aa(x1))) |
→ |
aa(x1) |
(5) |
|
aa(aa(ab(x1))) |
→ |
ab(x1) |
(6) |
|
ba(aa(aa(x1))) |
→ |
ba(x1) |
(7) |
|
ba(aa(ab(x1))) |
→ |
bb(x1) |
(8) |
|
ab(bb(ba(aa(x1)))) |
→ |
aa(ab(ba(ab(bb(ba(x1)))))) |
(9) |
|
ab(bb(ba(ab(x1)))) |
→ |
aa(ab(ba(ab(bb(bb(x1)))))) |
(10) |
1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
ba#(aa(aa(x1))) |
→ |
ba#(x1) |
(11) |
|
ab#(bb(ba(aa(x1)))) |
→ |
ba#(x1) |
(12) |
|
ab#(bb(ba(aa(x1)))) |
→ |
ab#(bb(ba(x1))) |
(13) |
|
ab#(bb(ba(aa(x1)))) |
→ |
ba#(ab(bb(ba(x1)))) |
(14) |
|
ab#(bb(ba(aa(x1)))) |
→ |
ab#(ba(ab(bb(ba(x1))))) |
(15) |
|
ab#(bb(ba(aa(x1)))) |
→ |
aa#(ab(ba(ab(bb(ba(x1)))))) |
(16) |
|
ab#(bb(ba(ab(x1)))) |
→ |
ab#(bb(bb(x1))) |
(17) |
|
ab#(bb(ba(ab(x1)))) |
→ |
ba#(ab(bb(bb(x1)))) |
(18) |
|
ab#(bb(ba(ab(x1)))) |
→ |
ab#(ba(ab(bb(bb(x1))))) |
(19) |
|
ab#(bb(ba(ab(x1)))) |
→ |
aa#(ab(ba(ab(bb(bb(x1)))))) |
(20) |
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.
-
The
1st
component contains the
pair
|
ab#(bb(ba(aa(x1)))) |
→ |
ab#(bb(ba(x1))) |
(13) |
|
ab#(bb(ba(aa(x1)))) |
→ |
ab#(ba(ab(bb(ba(x1))))) |
(15) |
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 arctic semiring over the integers
| [ba(x1)] |
= |
· x1 +
|
| [ab#(x1)] |
= |
· x1 +
|
| [ab(x1)] |
= |
· x1 +
|
| [bb(x1)] |
= |
· x1 +
|
| [aa(x1)] |
= |
· x1 +
|
together with the usable
rules
|
aa(aa(aa(x1))) |
→ |
aa(x1) |
(5) |
|
aa(aa(ab(x1))) |
→ |
ab(x1) |
(6) |
|
ba(aa(aa(x1))) |
→ |
ba(x1) |
(7) |
|
ba(aa(ab(x1))) |
→ |
bb(x1) |
(8) |
|
ab(bb(ba(aa(x1)))) |
→ |
aa(ab(ba(ab(bb(ba(x1)))))) |
(9) |
|
ab(bb(ba(ab(x1)))) |
→ |
aa(ab(ba(ab(bb(bb(x1)))))) |
(10) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
ab#(bb(ba(aa(x1)))) |
→ |
ab#(bb(ba(x1))) |
(13) |
|
ab#(bb(ba(aa(x1)))) |
→ |
ab#(ba(ab(bb(ba(x1))))) |
(15) |
could be deleted.
1.1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
|
ba#(aa(aa(x1))) |
→ |
ba#(x1) |
(11) |
1.1.1.1.2 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
ba#(aa(aa(x1))) |
→ |
ba#(x1) |
(11) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.