Certification Problem
Input (TPDB SRS_Standard/Zantema_06/13)
The rewrite relation of the following TRS is considered.
|
a(a(a(b(b(x1))))) |
→ |
b(b(b(x1))) |
(1) |
|
b(a(a(a(b(x1))))) |
→ |
a(a(a(b(a(a(a(x1))))))) |
(2) |
|
a(a(a(x1))) |
→ |
a(a(x1)) |
(3) |
|
b(b(x1)) |
→ |
a(b(a(x1))) |
(4) |
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
| [a(x1)] |
= |
1 · x1 + 1 |
| [b(x1)] |
= |
1 · x1 + 3 |
all of the following rules can be deleted.
|
a(a(a(x1))) |
→ |
a(a(x1)) |
(3) |
|
b(b(x1)) |
→ |
a(b(a(x1))) |
(4) |
1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
a#(a(a(b(b(x1))))) |
→ |
b#(b(b(x1))) |
(5) |
|
b#(a(a(a(b(x1))))) |
→ |
a#(a(a(b(a(a(a(x1))))))) |
(6) |
|
b#(a(a(a(b(x1))))) |
→ |
a#(a(b(a(a(a(x1)))))) |
(7) |
|
b#(a(a(a(b(x1))))) |
→ |
a#(b(a(a(a(x1))))) |
(8) |
|
b#(a(a(a(b(x1))))) |
→ |
b#(a(a(a(x1)))) |
(9) |
|
b#(a(a(a(b(x1))))) |
→ |
a#(a(a(x1))) |
(10) |
|
b#(a(a(a(b(x1))))) |
→ |
a#(a(x1)) |
(11) |
|
b#(a(a(a(b(x1))))) |
→ |
a#(x1) |
(12) |
1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [a(x1)] |
= |
1 + 1 · x1
|
| [b(x1)] |
= |
3 + 1 · x1
|
| [a#(x1)] |
= |
1 · x1
|
| [b#(x1)] |
= |
2 + 1 · x1
|
the
pairs
|
b#(a(a(a(b(x1))))) |
→ |
a#(a(b(a(a(a(x1)))))) |
(7) |
|
b#(a(a(a(b(x1))))) |
→ |
a#(b(a(a(a(x1))))) |
(8) |
|
b#(a(a(a(b(x1))))) |
→ |
b#(a(a(a(x1)))) |
(9) |
|
b#(a(a(a(b(x1))))) |
→ |
a#(a(a(x1))) |
(10) |
|
b#(a(a(a(b(x1))))) |
→ |
a#(a(x1)) |
(11) |
|
b#(a(a(a(b(x1))))) |
→ |
a#(x1) |
(12) |
and
no rules
could be deleted.
1.1.1.1 Reduction Pair Processor
Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals
| [a#(x1)] |
= |
+ · x1
|
| [a(x1)] |
= |
+ · x1
|
| [b(x1)] |
= |
+ · x1
|
| [b#(x1)] |
= |
+
|
| 0 |
-∞ |
-∞ |
|
-∞ |
-∞ |
-∞ |
|
-∞ |
-∞ |
-∞ |
|
|
· x1
|
the
pair
|
a#(a(a(b(b(x1))))) |
→ |
b#(b(b(x1))) |
(5) |
could be deleted.
1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the non-linear polynomial interpretation over the naturals
| [b#(x1)] |
= |
1 |
| [a(x1)] |
= |
0 |
| [b(x1)] |
= |
0 |
| [a#(x1)] |
= |
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
|
b#(a(a(a(b(x1))))) |
→ |
a#(a(a(b(a(a(a(x1))))))) |
(6) |
could be deleted.
1.1.1.1.1.1 P is empty
There are no pairs anymore.