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.