Certification Problem
Input (TPDB SRS_Standard/Secret_07_SRS/x04)
The rewrite relation of the following TRS is considered.
|
a(a(x1)) |
→ |
a(b(a(b(a(x1))))) |
(1) |
|
c(a(x1)) |
→ |
a(b(a(a(c(x1))))) |
(2) |
|
b(b(b(x1))) |
→ |
a(b(x1)) |
(3) |
|
c(b(x1)) |
→ |
a(a(c(x1))) |
(4) |
|
c(b(x1)) |
→ |
b(a(d(x1))) |
(5) |
|
d(d(x1)) |
→ |
d(b(d(b(d(x1))))) |
(6) |
|
c(c(x1)) |
→ |
c(d(c(x1))) |
(7) |
|
a(a(a(x1))) |
→ |
a(b(b(x1))) |
(8) |
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
| [b(x1)] |
= |
0 · x1 +
-∞ |
| [d(x1)] |
= |
0 · x1 +
-∞ |
| [a(x1)] |
= |
0 · x1 +
-∞ |
| [c(x1)] |
= |
2 · x1 +
-∞ |
all of the following rules can be deleted.
|
c(b(x1)) |
→ |
b(a(d(x1))) |
(5) |
1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
| [b(x1)] |
= |
· x1 +
|
| [d(x1)] |
= |
· x1 +
|
| [a(x1)] |
= |
· x1 +
|
| [c(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
|
d(d(x1)) |
→ |
d(b(d(b(d(x1))))) |
(6) |
1.1.1 Rule Removal
Using the
linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1
over the naturals
| [b(x1)] |
= |
· x1 +
|
| [d(x1)] |
= |
· x1 +
|
| [a(x1)] |
= |
· x1 +
|
| [c(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
|
c(c(x1)) |
→ |
c(d(c(x1))) |
(7) |
1.1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
a#(a(x1)) |
→ |
b#(a(x1)) |
(9) |
|
a#(a(x1)) |
→ |
a#(b(a(x1))) |
(10) |
|
a#(a(x1)) |
→ |
b#(a(b(a(x1)))) |
(11) |
|
a#(a(x1)) |
→ |
a#(b(a(b(a(x1))))) |
(12) |
|
c#(a(x1)) |
→ |
c#(x1) |
(13) |
|
c#(a(x1)) |
→ |
a#(c(x1)) |
(14) |
|
c#(a(x1)) |
→ |
a#(a(c(x1))) |
(15) |
|
c#(a(x1)) |
→ |
b#(a(a(c(x1)))) |
(16) |
|
c#(a(x1)) |
→ |
a#(b(a(a(c(x1))))) |
(17) |
|
b#(b(b(x1))) |
→ |
a#(b(x1)) |
(18) |
|
c#(b(x1)) |
→ |
c#(x1) |
(19) |
|
c#(b(x1)) |
→ |
a#(c(x1)) |
(20) |
|
c#(b(x1)) |
→ |
a#(a(c(x1))) |
(21) |
|
a#(a(a(x1))) |
→ |
b#(x1) |
(22) |
|
a#(a(a(x1))) |
→ |
b#(b(x1)) |
(23) |
|
a#(a(a(x1))) |
→ |
a#(b(b(x1))) |
(24) |
1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.
-
The
1st
component contains the
pair
|
c#(b(x1)) |
→ |
c#(x1) |
(19) |
|
c#(a(x1)) |
→ |
c#(x1) |
(13) |
1.1.1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
c#(b(x1)) |
→ |
c#(x1) |
(19) |
|
| 1 |
> |
1 |
|
c#(a(x1)) |
→ |
c#(x1) |
(13) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
|
a#(a(a(x1))) |
→ |
a#(b(b(x1))) |
(24) |
|
a#(a(a(x1))) |
→ |
b#(b(x1)) |
(23) |
|
b#(b(b(x1))) |
→ |
a#(b(x1)) |
(18) |
|
a#(a(a(x1))) |
→ |
b#(x1) |
(22) |
|
a#(a(x1)) |
→ |
a#(b(a(b(a(x1))))) |
(12) |
|
a#(a(x1)) |
→ |
a#(b(a(x1))) |
(10) |
1.1.1.1.1.2 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1
over the arctic semiring over the integers
| [a#(x1)] |
= |
· x1 +
|
| [b(x1)] |
= |
· x1 +
|
| [b#(x1)] |
= |
· x1 +
|
| [a(x1)] |
= |
· x1 +
|
together with the usable
rules
|
b(b(b(x1))) |
→ |
a(b(x1)) |
(3) |
|
a(a(x1)) |
→ |
a(b(a(b(a(x1))))) |
(1) |
|
a(a(a(x1))) |
→ |
a(b(b(x1))) |
(8) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
a#(a(x1)) |
→ |
a#(b(a(b(a(x1))))) |
(12) |
|
a#(a(x1)) |
→ |
a#(b(a(x1))) |
(10) |
could be deleted.
1.1.1.1.1.2.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1
over the arctic semiring over the integers
| [a#(x1)] |
= |
· x1 +
|
| [b(x1)] |
= |
· x1 +
|
| [b#(x1)] |
= |
· x1 +
|
| [a(x1)] |
= |
· x1 +
|
together with the usable
rules
|
b(b(b(x1))) |
→ |
a(b(x1)) |
(3) |
|
a(a(x1)) |
→ |
a(b(a(b(a(x1))))) |
(1) |
|
a(a(a(x1))) |
→ |
a(b(b(x1))) |
(8) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
b#(b(b(x1))) |
→ |
a#(b(x1)) |
(18) |
|
a#(a(a(x1))) |
→ |
b#(x1) |
(22) |
could be deleted.
1.1.1.1.1.2.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.