Certification Problem
Input (TPDB TRS_Standard/SK90/4.16)
The rewrite relation of the following TRS is considered.
|
f(0) |
→ |
s(0) |
(1) |
|
f(s(0)) |
→ |
s(s(0)) |
(2) |
|
f(s(0)) |
→ |
*(s(s(0)),f(0)) |
(3) |
|
f(+(x,s(0))) |
→ |
+(s(s(0)),f(x)) |
(4) |
|
f(+(x,y)) |
→ |
*(f(x),f(y)) |
(5) |
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
| [f(x1)] |
= |
1 + 2 · x1
|
| [0] |
= |
0 |
| [s(x1)] |
= |
2 · x1
|
| [*(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [+(x1, x2)] |
= |
2 + 2 · x1 + 1 · x2
|
all of the following rules can be deleted.
|
f(0) |
→ |
s(0) |
(1) |
|
f(s(0)) |
→ |
s(s(0)) |
(2) |
|
f(+(x,s(0))) |
→ |
+(s(s(0)),f(x)) |
(4) |
|
f(+(x,y)) |
→ |
*(f(x),f(y)) |
(5) |
1.1 Rule Removal
Using the
| prec(f) |
= |
2 |
|
stat(f) |
= |
mul
|
| prec(s) |
= |
0 |
|
stat(s) |
= |
mul
|
| prec(0) |
= |
1 |
|
stat(0) |
= |
mul
|
| prec(*) |
= |
1 |
|
stat(*) |
= |
mul
|
| π(f) |
= |
[1] |
| π(s) |
= |
[1] |
| π(0) |
= |
[] |
| π(*) |
= |
[1,2] |
all of the following rules can be deleted.
|
f(s(0)) |
→ |
*(s(s(0)),f(0)) |
(3) |
1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.