Certification Problem
Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-2-num-3)
The rewrite relation of the following TRS is considered.
a(a(x1)) |
→ |
x1 |
(1) |
a(b(x1)) |
→ |
x1 |
(2) |
b(b(a(x1))) |
→ |
a(b(a(b(b(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#(b(a(x1))) |
→ |
b#(x1) |
(4) |
b#(b(a(x1))) |
→ |
b#(b(x1)) |
(5) |
b#(b(a(x1))) |
→ |
a#(b(b(x1))) |
(6) |
b#(b(a(x1))) |
→ |
b#(a(b(b(x1)))) |
(7) |
b#(b(a(x1))) |
→ |
a#(b(a(b(b(x1))))) |
(8) |
1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
1st
component contains the
pair
b#(b(a(x1))) |
→ |
b#(b(x1)) |
(5) |
b#(b(a(x1))) |
→ |
b#(x1) |
(4) |
b#(b(a(x1))) |
→ |
b#(a(b(b(x1)))) |
(7) |
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
[b(x1)] |
= |
· x1 +
|
[b#(x1)] |
= |
· x1 +
|
[a(x1)] |
= |
· x1 +
|
together with the usable
rules
a(a(x1)) |
→ |
x1 |
(1) |
a(b(x1)) |
→ |
x1 |
(2) |
b(b(a(x1))) |
→ |
a(b(a(b(b(x1))))) |
(3) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
b#(b(a(x1))) |
→ |
b#(b(x1)) |
(5) |
b#(b(a(x1))) |
→ |
b#(a(b(b(x1)))) |
(7) |
could be deleted.
1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
b#(b(a(x1))) |
→ |
b#(x1) |
(4) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.