Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex5_Zan97_iGM)
The rewrite relation of the following TRS is considered.
active(f(X)) |
→ |
mark(if(X,c,f(true))) |
(1) |
active(if(true,X,Y)) |
→ |
mark(X) |
(2) |
active(if(false,X,Y)) |
→ |
mark(Y) |
(3) |
mark(f(X)) |
→ |
active(f(mark(X))) |
(4) |
mark(if(X1,X2,X3)) |
→ |
active(if(mark(X1),mark(X2),X3)) |
(5) |
mark(c) |
→ |
active(c) |
(6) |
mark(true) |
→ |
active(true) |
(7) |
mark(false) |
→ |
active(false) |
(8) |
f(mark(X)) |
→ |
f(X) |
(9) |
f(active(X)) |
→ |
f(X) |
(10) |
if(mark(X1),X2,X3) |
→ |
if(X1,X2,X3) |
(11) |
if(X1,mark(X2),X3) |
→ |
if(X1,X2,X3) |
(12) |
if(X1,X2,mark(X3)) |
→ |
if(X1,X2,X3) |
(13) |
if(active(X1),X2,X3) |
→ |
if(X1,X2,X3) |
(14) |
if(X1,active(X2),X3) |
→ |
if(X1,X2,X3) |
(15) |
if(X1,X2,active(X3)) |
→ |
if(X1,X2,X3) |
(16) |
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
[if(x1, x2, x3)] |
= |
· x1 + · x2 + · x3 +
|
[active(x1)] |
= |
· x1 +
|
[true] |
= |
|
[mark(x1)] |
= |
· x1 +
|
[f(x1)] |
= |
· x1 +
|
[false] |
= |
|
[c] |
= |
|
all of the following rules can be deleted.
active(if(false,X,Y)) |
→ |
mark(Y) |
(3) |
1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
[if(x1, x2, x3)] |
= |
· x1 + · x2 + · x3 +
|
[active(x1)] |
= |
· x1 +
|
[true] |
= |
|
[mark(x1)] |
= |
· x1 +
|
[f(x1)] |
= |
· x1 +
|
[false] |
= |
|
[c] |
= |
|
all of the following rules can be deleted.
active(if(true,X,Y)) |
→ |
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
[if(x1, x2, x3)] |
= |
· x1 + · x2 + · x3 +
|
[active(x1)] |
= |
· x1 +
|
[true] |
= |
|
[mark(x1)] |
= |
· x1 +
|
[f(x1)] |
= |
· x1 +
|
[false] |
= |
|
[c] |
= |
|
all of the following rules can be deleted.
mark(f(X)) |
→ |
active(f(mark(X))) |
(4) |
1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
[if(x1, x2, x3)] |
= |
· x1 + · x2 + · x3 +
|
[active(x1)] |
= |
· x1 +
|
[true] |
= |
|
[mark(x1)] |
= |
· x1 +
|
[f(x1)] |
= |
· x1 +
|
[false] |
= |
|
[c] |
= |
|
all of the following rules can be deleted.
f(mark(X)) |
→ |
f(X) |
(9) |
f(active(X)) |
→ |
f(X) |
(10) |
1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
[if(x1, x2, x3)] |
= |
· x1 + · x2 + · x3 +
|
[active(x1)] |
= |
· x1 +
|
[true] |
= |
|
[mark(x1)] |
= |
· x1 +
|
[f(x1)] |
= |
· x1 +
|
[false] |
= |
|
[c] |
= |
|
all of the following rules can be deleted.
mark(false) |
→ |
active(false) |
(8) |
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
[if(x1, x2, x3)] |
= |
· x1 + · x2 + · x3 +
|
[active(x1)] |
= |
· x1 +
|
[true] |
= |
|
[mark(x1)] |
= |
· x1 +
|
[f(x1)] |
= |
· x1 +
|
[c] |
= |
|
all of the following rules can be deleted.
active(f(X)) |
→ |
mark(if(X,c,f(true))) |
(1) |
1.1.1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
[if(x1, x2, x3)] |
= |
· x1 + · x2 + · x3 +
|
[active(x1)] |
= |
· x1 +
|
[true] |
= |
|
[mark(x1)] |
= |
· x1 +
|
[c] |
= |
|
all of the following rules can be deleted.
if(X1,X2,mark(X3)) |
→ |
if(X1,X2,X3) |
(13) |
1.1.1.1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
[if(x1, x2, x3)] |
= |
· x1 + · x2 + · x3 +
|
[active(x1)] |
= |
· x1 +
|
[true] |
= |
|
[mark(x1)] |
= |
· x1 +
|
[c] |
= |
|
all of the following rules can be deleted.
if(X1,X2,active(X3)) |
→ |
if(X1,X2,X3) |
(16) |
1.1.1.1.1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
[if(x1, x2, x3)] |
= |
3 · x1 + 2 · x2 + 16 · x3 + 4 |
[active(x1)] |
= |
1 · x1 + 0 |
[true] |
= |
0 |
[mark(x1)] |
= |
2 · x1 + 1 |
[c] |
= |
0 |
all of the following rules can be deleted.
mark(c) |
→ |
active(c) |
(6) |
mark(true) |
→ |
active(true) |
(7) |
if(mark(X1),X2,X3) |
→ |
if(X1,X2,X3) |
(11) |
if(X1,mark(X2),X3) |
→ |
if(X1,X2,X3) |
(12) |
1.1.1.1.1.1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
[if(x1, x2, x3)] |
= |
· x1 + · x2 + · x3 +
|
[active(x1)] |
= |
· x1 +
|
[mark(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
mark(if(X1,X2,X3)) |
→ |
active(if(mark(X1),mark(X2),X3)) |
(5) |
1.1.1.1.1.1.1.1.1.1.1 Rule Removal
Using the
Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(if) |
= |
0 |
|
weight(if) |
= |
0 |
|
|
|
prec(active) |
= |
1 |
|
weight(active) |
= |
2 |
|
|
|
all of the following rules can be deleted.
if(active(X1),X2,X3) |
→ |
if(X1,X2,X3) |
(14) |
if(X1,active(X2),X3) |
→ |
if(X1,X2,X3) |
(15) |
1.1.1.1.1.1.1.1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.