Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex4_4_Luc96b_iGM)
The rewrite relation of the following TRS is considered.
active(f(g(X),Y)) |
→ |
mark(f(X,f(g(X),Y))) |
(1) |
mark(f(X1,X2)) |
→ |
active(f(mark(X1),X2)) |
(2) |
mark(g(X)) |
→ |
active(g(mark(X))) |
(3) |
f(mark(X1),X2) |
→ |
f(X1,X2) |
(4) |
f(X1,mark(X2)) |
→ |
f(X1,X2) |
(5) |
f(active(X1),X2) |
→ |
f(X1,X2) |
(6) |
f(X1,active(X2)) |
→ |
f(X1,X2) |
(7) |
g(mark(X)) |
→ |
g(X) |
(8) |
g(active(X)) |
→ |
g(X) |
(9) |
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
[f(x1, x2)] |
= |
· x1 + · x2 +
|
[mark(x1)] |
= |
· x1 +
|
[g(x1)] |
= |
· x1 +
|
[active(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
active(f(g(X),Y)) |
→ |
mark(f(X,f(g(X),Y))) |
(1) |
1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
[f(x1, x2)] |
= |
· x1 + · x2 +
|
[mark(x1)] |
= |
· x1 +
|
[g(x1)] |
= |
· x1 +
|
[active(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
f(mark(X1),X2) |
→ |
f(X1,X2) |
(4) |
f(X1,mark(X2)) |
→ |
f(X1,X2) |
(5) |
g(mark(X)) |
→ |
g(X) |
(8) |
1.1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
[f(x1, x2)] |
= |
· x1 + · x2 +
|
[mark(x1)] |
= |
· x1 +
|
[g(x1)] |
= |
· x1 +
|
[active(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
f(X1,active(X2)) |
→ |
f(X1,X2) |
(7) |
1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
[f(x1, x2)] |
= |
26 · x1 + 10 · x2 + 4 |
[mark(x1)] |
= |
3 · x1 + 0 |
[g(x1)] |
= |
2 · x1 + 2 |
[active(x1)] |
= |
1 · x1 + 4 |
all of the following rules can be deleted.
mark(f(X1,X2)) |
→ |
active(f(mark(X1),X2)) |
(2) |
f(active(X1),X2) |
→ |
f(X1,X2) |
(6) |
g(active(X)) |
→ |
g(X) |
(9) |
1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
[mark(x1)] |
= |
· x1 +
|
[g(x1)] |
= |
· x1 +
|
[active(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
mark(g(X)) |
→ |
active(g(mark(X))) |
(3) |
1.1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.