Certification Problem
Input (TPDB SRS_Standard/Zantema_04/z013)
The rewrite relation of the following TRS is considered.
|
g(c(x1)) |
→ |
g(f(c(x1))) |
(1) |
|
g(f(c(x1))) |
→ |
g(f(f(c(x1)))) |
(2) |
|
g(g(x1)) |
→ |
g(f(g(x1))) |
(3) |
|
f(f(g(x1))) |
→ |
g(f(x1)) |
(4) |
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
|
c(g(x1)) |
→ |
c(f(g(x1))) |
(5) |
|
c(f(g(x1))) |
→ |
c(f(f(g(x1)))) |
(6) |
|
g(g(x1)) |
→ |
g(f(g(x1))) |
(3) |
|
g(f(f(x1))) |
→ |
f(g(x1)) |
(7) |
1.1 Rule Removal
Using the
linear polynomial interpretation over (5 x 5)-matrices with strict dimension 1
over the naturals
| [g(x1)] |
= |
|
| 1 |
0 |
0 |
1 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
1 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
|
|
· x1 +
|
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 1 |
0 |
0 |
0 |
0 |
| 1 |
0 |
0 |
0 |
0 |
|
|
|
| [c(x1)] |
= |
|
| 1 |
0 |
0 |
0 |
1 |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
1 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
|
|
· x1 +
|
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 1 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
|
|
|
| [f(x1)] |
= |
|
| 1 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
1 |
0 |
| 0 |
0 |
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 |
|
|
|
all of the following rules can be deleted.
|
c(g(x1)) |
→ |
c(f(g(x1))) |
(5) |
1.1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
| [g(x1)] |
= |
· x1 +
|
| [c(x1)] |
= |
· x1 +
|
| [f(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
|
g(f(f(x1))) |
→ |
f(g(x1)) |
(7) |
1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
| [g(x1)] |
= |
· x1 +
|
| [c(x1)] |
= |
· x1 +
|
| [f(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
|
g(g(x1)) |
→ |
g(f(g(x1))) |
(3) |
1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
| [g(x1)] |
= |
· x1 +
|
| [c(x1)] |
= |
· x1 +
|
| [f(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
|
c(f(g(x1))) |
→ |
c(f(f(g(x1)))) |
(6) |
1.1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.