Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex4_4_Luc96b_GM)
The rewrite relation of the following TRS is considered.
|
a__f(g(X),Y) |
→ |
a__f(mark(X),f(g(X),Y)) |
(1) |
|
mark(f(X1,X2)) |
→ |
a__f(mark(X1),X2) |
(2) |
|
mark(g(X)) |
→ |
g(mark(X)) |
(3) |
|
a__f(X1,X2) |
→ |
f(X1,X2) |
(4) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by ttt2 @ termCOMP 2023)
1 Rule Removal
Using the
Weighted Path Order with the following precedence and status
| prec(f) |
= |
0 |
|
status(f) |
= |
[2, 1] |
|
list-extension(f) |
= |
Lex |
| prec(mark) |
= |
3 |
|
status(mark) |
= |
[1] |
|
list-extension(mark) |
= |
Lex |
| prec(a__f) |
= |
1 |
|
status(a__f) |
= |
[1, 2] |
|
list-extension(a__f) |
= |
Lex |
| prec(g) |
= |
0 |
|
status(g) |
= |
[1] |
|
list-extension(g) |
= |
Lex |
and the following
Max-polynomial interpretation
| [f(x1, x2)] |
=
|
max(0, 1 + 1 · x1, 0 + 1 · x2) |
| [mark(x1)] |
=
|
0 + 1 · x1
|
| [a__f(x1, x2)] |
=
|
max(0, 1 + 1 · x1, 0 + 1 · x2) |
| [g(x1)] |
=
|
max(2, 4 + 1 · x1) |
all of the following rules can be deleted.
|
a__f(g(X),Y) |
→ |
a__f(mark(X),f(g(X),Y)) |
(1) |
|
mark(f(X1,X2)) |
→ |
a__f(mark(X1),X2) |
(2) |
|
mark(g(X)) |
→ |
g(mark(X)) |
(3) |
|
a__f(X1,X2) |
→ |
f(X1,X2) |
(4) |
1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.