Certification Problem
Input (TPDB SRS_Standard/Zantema_04/z112)
The rewrite relation of the following TRS is considered.
|
a(a(x1)) |
→ |
b(c(x1)) |
(1) |
|
b(b(x1)) |
→ |
c(d(x1)) |
(2) |
|
b(x1) |
→ |
a(x1) |
(3) |
|
c(c(x1)) |
→ |
d(f(x1)) |
(4) |
|
d(d(x1)) |
→ |
f(f(f(x1))) |
(5) |
|
d(x1) |
→ |
b(x1) |
(6) |
|
f(f(x1)) |
→ |
g(a(x1)) |
(7) |
|
g(g(x1)) |
→ |
a(x1) |
(8) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [a(x1)] |
= |
1 · x1 + 155 |
| [b(x1)] |
= |
1 · x1 + 162 |
| [c(x1)] |
= |
1 · x1 + 147 |
| [d(x1)] |
= |
1 · x1 + 176 |
| [f(x1)] |
= |
1 · x1 + 117 |
| [g(x1)] |
= |
1 · x1 + 78 |
all of the following rules can be deleted.
|
a(a(x1)) |
→ |
b(c(x1)) |
(1) |
|
b(b(x1)) |
→ |
c(d(x1)) |
(2) |
|
b(x1) |
→ |
a(x1) |
(3) |
|
c(c(x1)) |
→ |
d(f(x1)) |
(4) |
|
d(d(x1)) |
→ |
f(f(f(x1))) |
(5) |
|
d(x1) |
→ |
b(x1) |
(6) |
|
f(f(x1)) |
→ |
g(a(x1)) |
(7) |
|
g(g(x1)) |
→ |
a(x1) |
(8) |
1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.