Certification Problem
Input (TPDB SRS_Standard/Trafo_06/dup14)
The rewrite relation of the following TRS is considered.
|
a(a(b(b(b(b(a(a(x1)))))))) |
→ |
a(a(c(c(a(a(b(b(x1)))))))) |
(1) |
|
a(a(c(c(x1)))) |
→ |
c(c(c(c(a(a(x1)))))) |
(2) |
|
c(c(c(c(c(c(x1)))))) |
→ |
b(b(c(c(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
|
a(a(b(b(b(b(a(a(x1)))))))) |
→ |
b(b(a(a(c(c(a(a(x1)))))))) |
(4) |
|
c(c(a(a(x1)))) |
→ |
a(a(c(c(c(c(x1)))))) |
(5) |
|
c(c(c(c(c(c(x1)))))) |
→ |
b(b(c(c(b(b(x1)))))) |
(3) |
1.1 Rule Removal
Using the
linear polynomial interpretation over (5 x 5)-matrices with strict dimension 1
over the naturals
| [b(x1)] |
= |
|
| 1 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
1 |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
1 |
0 |
|
|
· x1 +
|
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 1 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
|
|
|
| [a(x1)] |
= |
|
| 1 |
0 |
1 |
0 |
0 |
| 0 |
0 |
1 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
1 |
0 |
0 |
0 |
| 0 |
1 |
0 |
0 |
0 |
|
|
· x1 +
|
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
|
|
|
| [c(x1)] |
= |
|
| 1 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
1 |
| 0 |
1 |
0 |
0 |
0 |
|
|
· x1 +
|
| 0 |
0 |
0 |
0 |
0 |
| 1 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
|
|
|
all of the following rules can be deleted.
|
a(a(b(b(b(b(a(a(x1)))))))) |
→ |
b(b(a(a(c(c(a(a(x1)))))))) |
(4) |
1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
c#(c(a(a(x1)))) |
→ |
c#(x1) |
(6) |
|
c#(c(a(a(x1)))) |
→ |
c#(c(x1)) |
(7) |
|
c#(c(a(a(x1)))) |
→ |
c#(c(c(x1))) |
(8) |
|
c#(c(a(a(x1)))) |
→ |
c#(c(c(c(x1)))) |
(9) |
|
c#(c(c(c(c(c(x1)))))) |
→ |
c#(b(b(x1))) |
(10) |
|
c#(c(c(c(c(c(x1)))))) |
→ |
c#(c(b(b(x1)))) |
(11) |
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.