Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex4_4_Luc96b_C)
The rewrite relation of the following TRS is considered.
active(f(g(X),Y)) |
→ |
mark(f(X,f(g(X),Y))) |
(1) |
active(f(X1,X2)) |
→ |
f(active(X1),X2) |
(2) |
active(g(X)) |
→ |
g(active(X)) |
(3) |
f(mark(X1),X2) |
→ |
mark(f(X1,X2)) |
(4) |
g(mark(X)) |
→ |
mark(g(X)) |
(5) |
proper(f(X1,X2)) |
→ |
f(proper(X1),proper(X2)) |
(6) |
proper(g(X)) |
→ |
g(proper(X)) |
(7) |
f(ok(X1),ok(X2)) |
→ |
ok(f(X1,X2)) |
(8) |
g(ok(X)) |
→ |
ok(g(X)) |
(9) |
top(mark(X)) |
→ |
top(proper(X)) |
(10) |
top(ok(X)) |
→ |
top(active(X)) |
(11) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 Rule Removal
Using the
linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 + 2 · x1
|
[f(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[g(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[proper(x1)] |
= |
1 · x1
|
[ok(x1)] |
= |
1 + 2 · x1
|
[top(x1)] |
= |
1 · x1
|
all of the following rules can be deleted.
active(f(g(X),Y)) |
→ |
mark(f(X,f(g(X),Y))) |
(1) |
f(ok(X1),ok(X2)) |
→ |
ok(f(X1,X2)) |
(8) |
1.1 Rule Removal
Using the
Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(active) |
= |
4 |
|
weight(active) |
= |
2 |
|
|
|
prec(g) |
= |
3 |
|
weight(g) |
= |
4 |
|
|
|
prec(mark) |
= |
0 |
|
weight(mark) |
= |
1 |
|
|
|
prec(proper) |
= |
6 |
|
weight(proper) |
= |
0 |
|
|
|
prec(ok) |
= |
2 |
|
weight(ok) |
= |
3 |
|
|
|
prec(top) |
= |
5 |
|
weight(top) |
= |
1 |
|
|
|
prec(f) |
= |
1 |
|
weight(f) |
= |
0 |
|
|
|
all of the following rules can be deleted.
active(f(X1,X2)) |
→ |
f(active(X1),X2) |
(2) |
active(g(X)) |
→ |
g(active(X)) |
(3) |
f(mark(X1),X2) |
→ |
mark(f(X1,X2)) |
(4) |
g(mark(X)) |
→ |
mark(g(X)) |
(5) |
proper(f(X1,X2)) |
→ |
f(proper(X1),proper(X2)) |
(6) |
proper(g(X)) |
→ |
g(proper(X)) |
(7) |
g(ok(X)) |
→ |
ok(g(X)) |
(9) |
top(mark(X)) |
→ |
top(proper(X)) |
(10) |
top(ok(X)) |
→ |
top(active(X)) |
(11) |
1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.