Certification Problem
Input (TPDB TRS_Standard/Der95/04)
The rewrite relation of the following TRS is considered.
|
f(f(x)) |
→ |
g(f(x)) |
(1) |
|
g(g(x)) |
→ |
f(x) |
(2) |
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
|
f(f(x)) |
→ |
f(g(x)) |
(3) |
|
g(g(x)) |
→ |
f(x) |
(2) |
1.1 Rule Removal
Using the
Weighted Path Order with the following precedence and status
| prec(g) |
= |
1 |
|
status(g) |
= |
[1] |
|
list-extension(g) |
= |
Lex |
| prec(f) |
= |
0 |
|
status(f) |
= |
[1] |
|
list-extension(f) |
= |
Lex |
and the following
Max-polynomial interpretation
| [g(x1)] |
=
|
max(0, 1 + 1 · x1) |
| [f(x1)] |
=
|
max(0, 2 + 1 · x1) |
all of the following rules can be deleted.
|
f(f(x)) |
→ |
f(g(x)) |
(3) |
|
g(g(x)) |
→ |
f(x) |
(2) |
1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.