Certification Problem
Input (TPDB SRS_Standard/Zantema_04/z115)
The rewrite relation of the following TRS is considered.
|
a(a(x1)) |
→ |
d(c(x1)) |
(1) |
|
a(b(x1)) |
→ |
c(c(c(x1))) |
(2) |
|
b(b(x1)) |
→ |
a(c(c(x1))) |
(3) |
|
c(c(x1)) |
→ |
b(x1) |
(4) |
|
c(d(x1)) |
→ |
a(a(x1)) |
(5) |
|
d(d(x1)) |
→ |
b(a(c(x1))) |
(6) |
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 + 15 |
| [d(x1)] |
= |
1 · x1 + 21 |
| [c(x1)] |
= |
1 · x1 + 9 |
| [b(x1)] |
= |
1 · x1 + 17 |
all of the following rules can be deleted.
|
a(b(x1)) |
→ |
c(c(c(x1))) |
(2) |
|
b(b(x1)) |
→ |
a(c(c(x1))) |
(3) |
|
c(c(x1)) |
→ |
b(x1) |
(4) |
|
d(d(x1)) |
→ |
b(a(c(x1))) |
(6) |
1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
a#(a(x1)) |
→ |
c#(x1) |
(7) |
|
c#(d(x1)) |
→ |
a#(a(x1)) |
(8) |
|
c#(d(x1)) |
→ |
a#(x1) |
(9) |
1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [a(x1)] |
= |
1 + 2 · x1
|
| [d(x1)] |
= |
2 + 2 · x1
|
| [c(x1)] |
= |
2 · x1
|
| [a#(x1)] |
= |
2 · x1
|
| [c#(x1)] |
= |
2 · x1
|
the
pairs
|
a#(a(x1)) |
→ |
c#(x1) |
(7) |
|
c#(d(x1)) |
→ |
a#(a(x1)) |
(8) |
|
c#(d(x1)) |
→ |
a#(x1) |
(9) |
and
the
rules
|
a(a(x1)) |
→ |
d(c(x1)) |
(1) |
|
c(d(x1)) |
→ |
a(a(x1)) |
(5) |
could be deleted.
1.1.1.1 P is empty
There are no pairs anymore.