Certification Problem
Input (TPDB SRS_Standard/Mixed_SRS/3)
The rewrite relation of the following TRS is considered.
|
a(a(x1)) |
→ |
a(b(a(x1))) |
(1) |
|
b(a(b(x1))) |
→ |
a(c(a(x1))) |
(2) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by matchbox @ termCOMP 2023)
1 Closure Under Flat Contexts
Using the flat contexts
{c(☐), b(☐), a(☐)}
We obtain the transformed TRS
|
c(a(a(x1))) |
→ |
c(a(b(a(x1)))) |
(3) |
|
c(b(a(b(x1)))) |
→ |
c(a(c(a(x1)))) |
(4) |
|
b(a(a(x1))) |
→ |
b(a(b(a(x1)))) |
(5) |
|
b(b(a(b(x1)))) |
→ |
b(a(c(a(x1)))) |
(6) |
|
a(a(a(x1))) |
→ |
a(a(b(a(x1)))) |
(7) |
|
a(b(a(b(x1)))) |
→ |
a(a(c(a(x1)))) |
(8) |
1.1 Closure Under Flat Contexts
Using the flat contexts
{c(☐), b(☐), a(☐)}
We obtain the transformed TRS
|
c(c(a(a(x1)))) |
→ |
c(c(a(b(a(x1))))) |
(9) |
|
c(c(b(a(b(x1))))) |
→ |
c(c(a(c(a(x1))))) |
(10) |
|
c(b(a(a(x1)))) |
→ |
c(b(a(b(a(x1))))) |
(11) |
|
c(b(b(a(b(x1))))) |
→ |
c(b(a(c(a(x1))))) |
(12) |
|
c(a(a(a(x1)))) |
→ |
c(a(a(b(a(x1))))) |
(13) |
|
c(a(b(a(b(x1))))) |
→ |
c(a(a(c(a(x1))))) |
(14) |
|
b(c(a(a(x1)))) |
→ |
b(c(a(b(a(x1))))) |
(15) |
|
b(c(b(a(b(x1))))) |
→ |
b(c(a(c(a(x1))))) |
(16) |
|
b(b(a(a(x1)))) |
→ |
b(b(a(b(a(x1))))) |
(17) |
|
b(b(b(a(b(x1))))) |
→ |
b(b(a(c(a(x1))))) |
(18) |
|
b(a(a(a(x1)))) |
→ |
b(a(a(b(a(x1))))) |
(19) |
|
b(a(b(a(b(x1))))) |
→ |
b(a(a(c(a(x1))))) |
(20) |
|
a(c(a(a(x1)))) |
→ |
a(c(a(b(a(x1))))) |
(21) |
|
a(c(b(a(b(x1))))) |
→ |
a(c(a(c(a(x1))))) |
(22) |
|
a(b(a(a(x1)))) |
→ |
a(b(a(b(a(x1))))) |
(23) |
|
a(b(b(a(b(x1))))) |
→ |
a(b(a(c(a(x1))))) |
(24) |
|
a(a(a(a(x1)))) |
→ |
a(a(a(b(a(x1))))) |
(25) |
|
a(a(b(a(b(x1))))) |
→ |
a(a(a(c(a(x1))))) |
(26) |
1.1.1 Semantic Labeling
The following interpretations form a
model
of the rules.
As carrier we take the set
{0,...,8}.
Symbols are labeled by the interpretation of their arguments using the interpretations
(modulo 9):
| [c(x1)] |
= |
3x1 + 0 |
| [b(x1)] |
= |
3x1 + 1 |
| [a(x1)] |
= |
3x1 + 2 |
We obtain the labeled TRS
There are 162 ruless (increase limit for explicit display).
1.1.1.1 Rule Removal
Using the
matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
| [c3(x1)] |
= |
x1 +
|
| [c6(x1)] |
= |
x1 +
|
| [c4(x1)] |
= |
x1 +
|
| [c7(x1)] |
= |
x1 +
|
| [c2(x1)] |
= |
x1 +
|
| [c5(x1)] |
= |
x1 +
|
| [c8(x1)] |
= |
x1 +
|
| [b0(x1)] |
= |
x1 +
|
| [b3(x1)] |
= |
x1 +
|
| [b6(x1)] |
= |
x1 +
|
| [b1(x1)] |
= |
x1 +
|
| [b4(x1)] |
= |
x1 +
|
| [b7(x1)] |
= |
x1 +
|
| [b2(x1)] |
= |
x1 +
|
| [b5(x1)] |
= |
x1 +
|
| [b8(x1)] |
= |
x1 +
|
| [a0(x1)] |
= |
x1 +
|
| [a3(x1)] |
= |
x1 +
|
| [a6(x1)] |
= |
x1 +
|
| [a1(x1)] |
= |
x1 +
|
| [a4(x1)] |
= |
x1 +
|
| [a7(x1)] |
= |
x1 +
|
| [a2(x1)] |
= |
x1 +
|
| [a5(x1)] |
= |
x1 +
|
| [a8(x1)] |
= |
x1 +
|
all of the following rules can be deleted.
There are 162 ruless (increase limit for explicit display).
1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.