Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex1_Luc02b_Z)
The rewrite relation of the following TRS is considered.
|
from(X) |
→ |
cons(X,n__from(s(X))) |
(1) |
|
first(0,Z) |
→ |
nil |
(2) |
|
first(s(X),cons(Y,Z)) |
→ |
cons(Y,n__first(X,activate(Z))) |
(3) |
|
sel(0,cons(X,Z)) |
→ |
X |
(4) |
|
sel(s(X),cons(Y,Z)) |
→ |
sel(X,activate(Z)) |
(5) |
|
from(X) |
→ |
n__from(X) |
(6) |
|
first(X1,X2) |
→ |
n__first(X1,X2) |
(7) |
|
activate(n__from(X)) |
→ |
from(X) |
(8) |
|
activate(n__first(X1,X2)) |
→ |
first(X1,X2) |
(9) |
|
activate(X) |
→ |
X |
(10) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 Rule Removal
Using the
| prec(from) |
= |
4 |
|
stat(from) |
= |
lex
|
| prec(cons) |
= |
0 |
|
stat(cons) |
= |
mul
|
| prec(s) |
= |
2 |
|
stat(s) |
= |
mul
|
| prec(first) |
= |
4 |
|
stat(first) |
= |
lex
|
| prec(0) |
= |
5 |
|
stat(0) |
= |
mul
|
| prec(nil) |
= |
3 |
|
stat(nil) |
= |
mul
|
| prec(n__first) |
= |
1 |
|
stat(n__first) |
= |
lex
|
| prec(activate) |
= |
4 |
|
stat(activate) |
= |
lex
|
| prec(sel) |
= |
6 |
|
stat(sel) |
= |
lex
|
| π(from) |
= |
[1] |
| π(cons) |
= |
[1,2] |
| π(n__from) |
= |
1 |
| π(s) |
= |
[1] |
| π(first) |
= |
[2,1] |
| π(0) |
= |
[] |
| π(nil) |
= |
[] |
| π(n__first) |
= |
[1,2] |
| π(activate) |
= |
[1] |
| π(sel) |
= |
[1,2] |
all of the following rules can be deleted.
|
from(X) |
→ |
cons(X,n__from(s(X))) |
(1) |
|
first(0,Z) |
→ |
nil |
(2) |
|
first(s(X),cons(Y,Z)) |
→ |
cons(Y,n__first(X,activate(Z))) |
(3) |
|
sel(0,cons(X,Z)) |
→ |
X |
(4) |
|
sel(s(X),cons(Y,Z)) |
→ |
sel(X,activate(Z)) |
(5) |
|
from(X) |
→ |
n__from(X) |
(6) |
|
first(X1,X2) |
→ |
n__first(X1,X2) |
(7) |
|
activate(n__first(X1,X2)) |
→ |
first(X1,X2) |
(9) |
|
activate(X) |
→ |
X |
(10) |
1.1 Rule Removal
Using the
Knuth Bendix order with w0 = 1 and the following precedence and weight functions
| prec(activate) |
= |
2 |
|
weight(activate) |
= |
1 |
|
|
|
| prec(n__from) |
= |
0 |
|
weight(n__from) |
= |
1 |
|
|
|
| prec(from) |
= |
1 |
|
weight(from) |
= |
2 |
|
|
|
all of the following rules can be deleted.
|
activate(n__from(X)) |
→ |
from(X) |
(8) |
1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.