Certification Problem
Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-2-num-22)
The rewrite relation of the following TRS is considered.
|
a(x1) |
→ |
x1 |
(1) |
|
a(a(b(a(x1)))) |
→ |
b(b(a(a(a(x1))))) |
(2) |
|
b(x1) |
→ |
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.
|
a#(a(b(a(x1)))) |
→ |
a#(a(x1)) |
(4) |
|
a#(a(b(a(x1)))) |
→ |
a#(a(a(x1))) |
(5) |
|
a#(a(b(a(x1)))) |
→ |
b#(a(a(a(x1)))) |
(6) |
|
a#(a(b(a(x1)))) |
→ |
b#(b(a(a(a(x1))))) |
(7) |
|
b#(x1) |
→ |
a#(x1) |
(8) |
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 +
|
| [a#(x1)] |
= |
· x1 +
|
together with the usable
rules
|
a(x1) |
→ |
x1 |
(1) |
|
a(a(b(a(x1)))) |
→ |
b(b(a(a(a(x1))))) |
(2) |
|
b(x1) |
→ |
a(x1) |
(3) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
a#(a(b(a(x1)))) |
→ |
a#(a(x1)) |
(4) |
|
a#(a(b(a(x1)))) |
→ |
a#(a(a(x1))) |
(5) |
|
a#(a(b(a(x1)))) |
→ |
b#(a(a(a(x1)))) |
(6) |
could be deleted.
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 +
|
| [a#(x1)] |
= |
· x1 +
|
together with the usable
rules
|
a(x1) |
→ |
x1 |
(1) |
|
a(a(b(a(x1)))) |
→ |
b(b(a(a(a(x1))))) |
(2) |
|
b(x1) |
→ |
a(x1) |
(3) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.