Certification Problem
Input (TPDB SRS_Standard/Zantema_04/z068)
The rewrite relation of the following TRS is considered.
|
C(x1) |
→ |
c(x1) |
(1) |
|
c(c(x1)) |
→ |
x1 |
(2) |
|
b(b(x1)) |
→ |
B(x1) |
(3) |
|
B(B(x1)) |
→ |
b(x1) |
(4) |
|
c(B(c(b(c(x1))))) |
→ |
B(c(b(c(B(c(b(x1))))))) |
(5) |
|
b(B(x1)) |
→ |
x1 |
(6) |
|
B(b(x1)) |
→ |
x1 |
(7) |
|
c(C(x1)) |
→ |
x1 |
(8) |
|
C(c(x1)) |
→ |
x1 |
(9) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by ttt2 @ termCOMP 2023)
1 Rule Removal
Using the
linear polynomial interpretation over the arctic semiring over the integers
| [c(x1)] |
= |
2 · x1 +
-∞ |
| [B(x1)] |
= |
0 · x1 +
-∞ |
| [C(x1)] |
= |
4 · x1 +
-∞ |
| [b(x1)] |
= |
0 · x1 +
-∞ |
all of the following rules can be deleted.
|
C(x1) |
→ |
c(x1) |
(1) |
|
c(c(x1)) |
→ |
x1 |
(2) |
|
c(C(x1)) |
→ |
x1 |
(8) |
|
C(c(x1)) |
→ |
x1 |
(9) |
1.1 Rule Removal
Using the
linear polynomial interpretation over (4 x 4)-matrices with strict dimension 1
over the naturals
| [c(x1)] |
= |
|
| 1 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
1 |
0 |
| 0 |
0 |
0 |
1 |
|
|
· x1 +
|
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
|
|
|
| [B(x1)] |
= |
|
| 1 |
1 |
0 |
0 |
| 0 |
1 |
1 |
1 |
| 0 |
1 |
0 |
0 |
| 0 |
1 |
0 |
0 |
|
|
· x1 +
|
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 1 |
0 |
0 |
0 |
|
|
|
| [b(x1)] |
= |
|
| 1 |
0 |
1 |
0 |
| 0 |
1 |
1 |
1 |
| 0 |
1 |
0 |
0 |
| 0 |
1 |
0 |
0 |
|
|
· x1 +
|
| 0 |
0 |
0 |
0 |
| 1 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
0 |
0 |
|
|
|
all of the following rules can be deleted.
1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
b#(b(x1)) |
→ |
B#(x1) |
(10) |
|
B#(B(x1)) |
→ |
b#(x1) |
(11) |
|
c#(B(c(b(c(x1))))) |
→ |
b#(x1) |
(12) |
|
c#(B(c(b(c(x1))))) |
→ |
c#(b(x1)) |
(13) |
|
c#(B(c(b(c(x1))))) |
→ |
B#(c(b(x1))) |
(14) |
|
c#(B(c(b(c(x1))))) |
→ |
c#(B(c(b(x1)))) |
(15) |
|
c#(B(c(b(c(x1))))) |
→ |
b#(c(B(c(b(x1))))) |
(16) |
|
c#(B(c(b(c(x1))))) |
→ |
c#(b(c(B(c(b(x1)))))) |
(17) |
|
c#(B(c(b(c(x1))))) |
→ |
B#(c(b(c(B(c(b(x1))))))) |
(18) |
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.
-
The
1st
component contains the
pair
|
c#(B(c(b(c(x1))))) |
→ |
c#(B(c(b(x1)))) |
(15) |
|
c#(B(c(b(c(x1))))) |
→ |
c#(b(x1)) |
(13) |
|
c#(B(c(b(c(x1))))) |
→ |
c#(b(c(B(c(b(x1)))))) |
(17) |
1.1.1.1.1 Subterm Criterion Processor
We use the projection to multisets
| π(c#)
|
= |
{
1, 1
}
|
| π(B)
|
= |
{
1
}
|
| π(b)
|
= |
{
1
}
|
| π(c)
|
= |
{
1, 1
}
|
to remove the pairs:
|
c#(B(c(b(c(x1))))) |
→ |
c#(B(c(b(x1)))) |
(15) |
|
c#(B(c(b(c(x1))))) |
→ |
c#(b(x1)) |
(13) |
1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1
over the naturals
| [c(x1)] |
= |
· x1 +
|
| [B(x1)] |
= |
· x1 +
|
| [c#(x1)] |
= |
· x1 +
|
| [b(x1)] |
= |
· x1 +
|
together with the usable
rules
|
b(b(x1)) |
→ |
B(x1) |
(3) |
|
B(B(x1)) |
→ |
b(x1) |
(4) |
|
c(B(c(b(c(x1))))) |
→ |
B(c(b(c(B(c(b(x1))))))) |
(5) |
|
b(B(x1)) |
→ |
x1 |
(6) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
c#(B(c(b(c(x1))))) |
→ |
c#(b(c(B(c(b(x1)))))) |
(17) |
could be deleted.
1.1.1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
|
b#(b(x1)) |
→ |
B#(x1) |
(10) |
|
B#(B(x1)) |
→ |
b#(x1) |
(11) |
1.1.1.1.2 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
b#(b(x1)) |
→ |
B#(x1) |
(10) |
|
| 1 |
> |
1 |
|
B#(B(x1)) |
→ |
b#(x1) |
(11) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.