Certification Problem
Input (TPDB TRS_Standard/AG01/#3.16)
The rewrite relation of the following TRS is considered.
|
times(x,0) |
→ |
0 |
(1) |
|
times(x,s(y)) |
→ |
plus(times(x,y),x) |
(2) |
|
plus(x,0) |
→ |
x |
(3) |
|
plus(0,x) |
→ |
x |
(4) |
|
plus(x,s(y)) |
→ |
s(plus(x,y)) |
(5) |
|
plus(s(x),y) |
→ |
s(plus(x,y)) |
(6) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by ttt2 @ termCOMP 2023)
1 Rule Removal
Using the
Weighted Path Order with the following precedence and status
| prec(plus) |
= |
2 |
|
status(plus) |
= |
[1, 2] |
|
list-extension(plus) |
= |
Lex |
| prec(s) |
= |
0 |
|
status(s) |
= |
[1] |
|
list-extension(s) |
= |
Lex |
| prec(times) |
= |
3 |
|
status(times) |
= |
[1, 2] |
|
list-extension(times) |
= |
Lex |
| prec(0) |
= |
0 |
|
status(0) |
= |
[] |
|
list-extension(0) |
= |
Lex |
and the following
Max-polynomial interpretation
| [plus(x1, x2)] |
=
|
max(5, 0 + 1 · x1, 5 + 1 · x2) |
| [s(x1)] |
=
|
max(0, 0 + 1 · x1) |
| [times(x1, x2)] |
=
|
max(0, 5 + 1 · x1, 0 + 1 · x2) |
| [0] |
=
|
max(3) |
all of the following rules can be deleted.
|
times(x,0) |
→ |
0 |
(1) |
|
times(x,s(y)) |
→ |
plus(times(x,y),x) |
(2) |
|
plus(x,0) |
→ |
x |
(3) |
|
plus(0,x) |
→ |
x |
(4) |
|
plus(x,s(y)) |
→ |
s(plus(x,y)) |
(5) |
|
plus(s(x),y) |
→ |
s(plus(x,y)) |
(6) |
1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.