Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/MYNAT_nosorts_FR)
The rewrite relation of the following TRS is considered.
|
and(tt,X) |
→ |
activate(X) |
(1) |
|
plus(N,0) |
→ |
N |
(2) |
|
plus(N,s(M)) |
→ |
s(plus(N,M)) |
(3) |
|
x(N,0) |
→ |
0 |
(4) |
|
x(N,s(M)) |
→ |
plus(x(N,M),N) |
(5) |
|
activate(X) |
→ |
X |
(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(x) |
= |
5 |
|
status(x) |
= |
[1, 2] |
|
list-extension(x) |
= |
Lex |
| prec(s) |
= |
0 |
|
status(s) |
= |
[1] |
|
list-extension(s) |
= |
Lex |
| prec(plus) |
= |
1 |
|
status(plus) |
= |
[1, 2] |
|
list-extension(plus) |
= |
Lex |
| prec(0) |
= |
0 |
|
status(0) |
= |
[] |
|
list-extension(0) |
= |
Lex |
| prec(activate) |
= |
0 |
|
status(activate) |
= |
[1] |
|
list-extension(activate) |
= |
Lex |
| prec(and) |
= |
1 |
|
status(and) |
= |
[2, 1] |
|
list-extension(and) |
= |
Lex |
| prec(tt) |
= |
0 |
|
status(tt) |
= |
[] |
|
list-extension(tt) |
= |
Lex |
and the following
Max-polynomial interpretation
| [x(x1, x2)] |
=
|
max(0, 0 + 1 · x1, 4 + 1 · x2) |
| [s(x1)] |
=
|
max(0, 0 + 1 · x1) |
| [plus(x1, x2)] |
=
|
max(0, 0 + 1 · x1, 0 + 1 · x2) |
| [0] |
=
|
max(1) |
| [activate(x1)] |
=
|
max(3, 2 + 1 · x1) |
| [and(x1, x2)] |
=
|
3 + 1 · x1 + 1 · x2
|
| [tt] |
=
|
max(0) |
all of the following rules can be deleted.
|
and(tt,X) |
→ |
activate(X) |
(1) |
|
plus(N,0) |
→ |
N |
(2) |
|
plus(N,s(M)) |
→ |
s(plus(N,M)) |
(3) |
|
x(N,0) |
→ |
0 |
(4) |
|
x(N,s(M)) |
→ |
plus(x(N,M),N) |
(5) |
|
activate(X) |
→ |
X |
(6) |
1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.