Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex23_Luc06_iGM)
The rewrite relation of the following TRS is considered.
active(f(f(a))) |
→ |
mark(c(f(g(f(a))))) |
(1) |
mark(f(X)) |
→ |
active(f(mark(X))) |
(2) |
mark(a) |
→ |
active(a) |
(3) |
mark(c(X)) |
→ |
active(c(X)) |
(4) |
mark(g(X)) |
→ |
active(g(mark(X))) |
(5) |
f(mark(X)) |
→ |
f(X) |
(6) |
f(active(X)) |
→ |
f(X) |
(7) |
c(mark(X)) |
→ |
c(X) |
(8) |
c(active(X)) |
→ |
c(X) |
(9) |
g(mark(X)) |
→ |
g(X) |
(10) |
g(active(X)) |
→ |
g(X) |
(11) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by ttt2 @ termCOMP 2023)
1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
[c(x1)] |
= |
· x1 +
|
[f(x1)] |
= |
· x1 +
|
[g(x1)] |
= |
· x1 +
|
[mark(x1)] |
= |
· x1 +
|
[a] |
= |
|
[active(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
[c(x1)] |
= |
· x1 +
|
[f(x1)] |
= |
· x1 +
|
[g(x1)] |
= |
· x1 +
|
[mark(x1)] |
= |
· x1 +
|
[a] |
= |
|
[active(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
mark(f(X)) |
→ |
active(f(mark(X))) |
(2) |
1.1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
[c(x1)] |
= |
· x1 +
|
[f(x1)] |
= |
· x1 +
|
[g(x1)] |
= |
· x1 +
|
[mark(x1)] |
= |
· x1 +
|
[a] |
= |
|
[active(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
f(mark(X)) |
→ |
f(X) |
(6) |
f(active(X)) |
→ |
f(X) |
(7) |
c(active(X)) |
→ |
c(X) |
(9) |
1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
[c(x1)] |
= |
· x1 +
|
[f(x1)] |
= |
· x1 +
|
[g(x1)] |
= |
· x1 +
|
[mark(x1)] |
= |
· x1 +
|
[a] |
= |
|
[active(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
[c(x1)] |
= |
· x1 +
|
[f(x1)] |
= |
· x1 +
|
[g(x1)] |
= |
· x1 +
|
[mark(x1)] |
= |
· x1 +
|
[a] |
= |
|
[active(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
g(mark(X)) |
→ |
g(X) |
(10) |
g(active(X)) |
→ |
g(X) |
(11) |
1.1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
[c(x1)] |
= |
· x1 +
|
[f(x1)] |
= |
· x1 +
|
[g(x1)] |
= |
· x1 +
|
[mark(x1)] |
= |
· x1 +
|
[a] |
= |
|
[active(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
active(f(f(a))) |
→ |
mark(c(f(g(f(a))))) |
(1) |
1.1.1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the arctic semiring over the integers
[c(x1)] |
= |
3 · x1 +
-∞ |
[g(x1)] |
= |
10 · x1 +
-∞ |
[mark(x1)] |
= |
1 · x1 +
-∞ |
[active(x1)] |
= |
0 · x1 +
-∞ |
all of the following rules can be deleted.
mark(c(X)) |
→ |
active(c(X)) |
(4) |
1.1.1.1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over (5 x 5)-matrices with strict dimension 1
over the naturals
[g(x1)] |
= |
|
1 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
|
|
· x1 +
|
0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
|
|
|
[mark(x1)] |
= |
|
1 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
|
|
· x1 +
|
0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
|
|
[active(x1)] |
= |
|
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
|
· x1 +
|
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
|
|
all of the following rules can be deleted.
mark(g(X)) |
→ |
active(g(mark(X))) |
(5) |
1.1.1.1.1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.