Certification Problem
Input (TPDB SRS_Standard/Mixed_SRS/2)
The rewrite relation of the following TRS is considered.
|
a(a(a(x1))) |
→ |
a(a(b(x1))) |
(1) |
|
b(a(b(x1))) |
→ |
a(b(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
{b(☐), a(☐)}
We obtain the transformed TRS
|
b(a(a(a(x1)))) |
→ |
b(a(a(b(x1)))) |
(3) |
|
b(b(a(b(x1)))) |
→ |
b(a(b(a(x1)))) |
(4) |
|
a(a(a(a(x1)))) |
→ |
a(a(a(b(x1)))) |
(5) |
|
a(b(a(b(x1)))) |
→ |
a(a(b(a(x1)))) |
(6) |
1.1 Closure Under Flat Contexts
Using the flat contexts
{b(☐), a(☐)}
We obtain the transformed TRS
|
b(b(a(a(a(x1))))) |
→ |
b(b(a(a(b(x1))))) |
(7) |
|
b(b(b(a(b(x1))))) |
→ |
b(b(a(b(a(x1))))) |
(8) |
|
b(a(a(a(a(x1))))) |
→ |
b(a(a(a(b(x1))))) |
(9) |
|
b(a(b(a(b(x1))))) |
→ |
b(a(a(b(a(x1))))) |
(10) |
|
a(b(a(a(a(x1))))) |
→ |
a(b(a(a(b(x1))))) |
(11) |
|
a(b(b(a(b(x1))))) |
→ |
a(b(a(b(a(x1))))) |
(12) |
|
a(a(a(a(a(x1))))) |
→ |
a(a(a(a(b(x1))))) |
(13) |
|
a(a(b(a(b(x1))))) |
→ |
a(a(a(b(a(x1))))) |
(14) |
1.1.1 Semantic Labeling
The following interpretations form a
model
of the rules.
As carrier we take the set
{0,...,3}.
Symbols are labeled by the interpretation of their arguments using the interpretations
(modulo 4):
| [b(x1)] |
= |
2x1 + 0 |
| [a(x1)] |
= |
2x1 + 1 |
We obtain the labeled TRS
|
a3(a3(a3(a3(a3(x1))))) |
→ |
a3(a3(a1(a2(b3(x1))))) |
(15) |
|
a3(a3(a3(a3(a1(x1))))) |
→ |
a3(a3(a1(a2(b1(x1))))) |
(16) |
|
a3(a3(a3(a1(a2(x1))))) |
→ |
a3(a3(a1(a0(b2(x1))))) |
(17) |
|
a3(a3(a3(a1(a0(x1))))) |
→ |
a3(a3(a1(a0(b0(x1))))) |
(18) |
|
a2(b3(a3(a3(a3(x1))))) |
→ |
a2(b3(a1(a2(b3(x1))))) |
(19) |
|
a2(b3(a3(a3(a1(x1))))) |
→ |
a2(b3(a1(a2(b1(x1))))) |
(20) |
|
a2(b3(a3(a1(a2(x1))))) |
→ |
a2(b3(a1(a0(b2(x1))))) |
(21) |
|
a2(b3(a3(a1(a0(x1))))) |
→ |
a2(b3(a1(a0(b0(x1))))) |
(22) |
|
b3(a3(a3(a3(a3(x1))))) |
→ |
b3(a3(a1(a2(b3(x1))))) |
(23) |
|
b3(a3(a3(a3(a1(x1))))) |
→ |
b3(a3(a1(a2(b1(x1))))) |
(24) |
|
b3(a3(a3(a1(a2(x1))))) |
→ |
b3(a3(a1(a0(b2(x1))))) |
(25) |
|
b3(a3(a3(a1(a0(x1))))) |
→ |
b3(a3(a1(a0(b0(x1))))) |
(26) |
|
b2(b3(a3(a3(a3(x1))))) |
→ |
b2(b3(a1(a2(b3(x1))))) |
(27) |
|
b2(b3(a3(a3(a1(x1))))) |
→ |
b2(b3(a1(a2(b1(x1))))) |
(28) |
|
b2(b3(a3(a1(a2(x1))))) |
→ |
b2(b3(a1(a0(b2(x1))))) |
(29) |
|
b2(b3(a3(a1(a0(x1))))) |
→ |
b2(b3(a1(a0(b0(x1))))) |
(30) |
|
a1(a2(b1(a2(b3(x1))))) |
→ |
a3(a1(a2(b3(a3(x1))))) |
(31) |
|
a1(a2(b1(a2(b1(x1))))) |
→ |
a3(a1(a2(b3(a1(x1))))) |
(32) |
|
a1(a2(b1(a0(b2(x1))))) |
→ |
a3(a1(a2(b1(a2(x1))))) |
(33) |
|
a1(a2(b1(a0(b0(x1))))) |
→ |
a3(a1(a2(b1(a0(x1))))) |
(34) |
|
a0(b2(b1(a2(b3(x1))))) |
→ |
a2(b1(a2(b3(a3(x1))))) |
(35) |
|
a0(b2(b1(a2(b1(x1))))) |
→ |
a2(b1(a2(b3(a1(x1))))) |
(36) |
|
a0(b2(b1(a0(b2(x1))))) |
→ |
a2(b1(a2(b1(a2(x1))))) |
(37) |
|
a0(b2(b1(a0(b0(x1))))) |
→ |
a2(b1(a2(b1(a0(x1))))) |
(38) |
|
b1(a2(b1(a2(b3(x1))))) |
→ |
b3(a1(a2(b3(a3(x1))))) |
(39) |
|
b1(a2(b1(a2(b1(x1))))) |
→ |
b3(a1(a2(b3(a1(x1))))) |
(40) |
|
b1(a2(b1(a0(b2(x1))))) |
→ |
b3(a1(a2(b1(a2(x1))))) |
(41) |
|
b1(a2(b1(a0(b0(x1))))) |
→ |
b3(a1(a2(b1(a0(x1))))) |
(42) |
|
b0(b2(b1(a2(b3(x1))))) |
→ |
b2(b1(a2(b3(a3(x1))))) |
(43) |
|
b0(b2(b1(a2(b1(x1))))) |
→ |
b2(b1(a2(b3(a1(x1))))) |
(44) |
|
b0(b2(b1(a0(b2(x1))))) |
→ |
b2(b1(a2(b1(a2(x1))))) |
(45) |
|
b0(b2(b1(a0(b0(x1))))) |
→ |
b2(b1(a2(b1(a0(x1))))) |
(46) |
1.1.1.1 Rule Removal
Using the
matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
| [b0(x1)] |
= |
x1 +
|
| [b2(x1)] |
= |
x1 +
|
| [b1(x1)] |
= |
x1 +
|
| [b3(x1)] |
= |
x1 +
|
| [a0(x1)] |
= |
x1 +
|
| [a2(x1)] |
= |
x1 +
|
| [a1(x1)] |
= |
x1 +
|
| [a3(x1)] |
= |
x1 +
|
all of the following rules can be deleted.
|
a3(a3(a3(a3(a3(x1))))) |
→ |
a3(a3(a1(a2(b3(x1))))) |
(15) |
|
a2(b3(a3(a3(a3(x1))))) |
→ |
a2(b3(a1(a2(b3(x1))))) |
(19) |
|
b3(a3(a3(a3(a3(x1))))) |
→ |
b3(a3(a1(a2(b3(x1))))) |
(23) |
|
b2(b3(a3(a3(a3(x1))))) |
→ |
b2(b3(a1(a2(b3(x1))))) |
(27) |
|
a1(a2(b1(a2(b1(x1))))) |
→ |
a3(a1(a2(b3(a1(x1))))) |
(32) |
|
a0(b2(b1(a2(b1(x1))))) |
→ |
a2(b1(a2(b3(a1(x1))))) |
(36) |
|
b1(a2(b1(a2(b3(x1))))) |
→ |
b3(a1(a2(b3(a3(x1))))) |
(39) |
|
b1(a2(b1(a2(b1(x1))))) |
→ |
b3(a1(a2(b3(a1(x1))))) |
(40) |
|
b1(a2(b1(a0(b2(x1))))) |
→ |
b3(a1(a2(b1(a2(x1))))) |
(41) |
|
b1(a2(b1(a0(b0(x1))))) |
→ |
b3(a1(a2(b1(a0(x1))))) |
(42) |
|
b0(b2(b1(a2(b1(x1))))) |
→ |
b2(b1(a2(b3(a1(x1))))) |
(44) |
1.1.1.1.1 Rule Removal
Using the
matrix interpretations of dimension 1 with strict dimension 1 over the naturals
| [b0(x1)] |
= |
· x1 +
|
| [b2(x1)] |
= |
· x1 +
|
| [b1(x1)] |
= |
· x1 +
|
| [b3(x1)] |
= |
· x1 +
|
| [a0(x1)] |
= |
· x1 +
|
| [a2(x1)] |
= |
· x1 +
|
| [a1(x1)] |
= |
· x1 +
|
| [a3(x1)] |
= |
· x1 +
|
all of the following rules can be deleted.
|
a3(a3(a3(a3(a1(x1))))) |
→ |
a3(a3(a1(a2(b1(x1))))) |
(16) |
|
a2(b3(a3(a3(a1(x1))))) |
→ |
a2(b3(a1(a2(b1(x1))))) |
(20) |
|
b3(a3(a3(a3(a1(x1))))) |
→ |
b3(a3(a1(a2(b1(x1))))) |
(24) |
|
b2(b3(a3(a3(a1(x1))))) |
→ |
b2(b3(a1(a2(b1(x1))))) |
(28) |
|
a1(a2(b1(a0(b2(x1))))) |
→ |
a3(a1(a2(b1(a2(x1))))) |
(33) |
|
a1(a2(b1(a0(b0(x1))))) |
→ |
a3(a1(a2(b1(a0(x1))))) |
(34) |
|
a0(b2(b1(a2(b3(x1))))) |
→ |
a2(b1(a2(b3(a3(x1))))) |
(35) |
|
a0(b2(b1(a0(b2(x1))))) |
→ |
a2(b1(a2(b1(a2(x1))))) |
(37) |
|
a0(b2(b1(a0(b0(x1))))) |
→ |
a2(b1(a2(b1(a0(x1))))) |
(38) |
|
b0(b2(b1(a2(b3(x1))))) |
→ |
b2(b1(a2(b3(a3(x1))))) |
(43) |
|
b0(b2(b1(a0(b2(x1))))) |
→ |
b2(b1(a2(b1(a2(x1))))) |
(45) |
|
b0(b2(b1(a0(b0(x1))))) |
→ |
b2(b1(a2(b1(a0(x1))))) |
(46) |
1.1.1.1.1.1 Rule Removal
Using the
matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
| [b0(x1)] |
= |
x1 +
|
| [b2(x1)] |
= |
x1 +
|
| [b1(x1)] |
= |
x1 +
|
| [b3(x1)] |
= |
x1 +
|
| [a0(x1)] |
= |
x1 +
|
| [a2(x1)] |
= |
x1 +
|
| [a1(x1)] |
= |
x1 +
|
| [a3(x1)] |
= |
x1 +
|
all of the following rules can be deleted.
|
a3(a3(a3(a1(a2(x1))))) |
→ |
a3(a3(a1(a0(b2(x1))))) |
(17) |
|
a3(a3(a3(a1(a0(x1))))) |
→ |
a3(a3(a1(a0(b0(x1))))) |
(18) |
|
a2(b3(a3(a1(a2(x1))))) |
→ |
a2(b3(a1(a0(b2(x1))))) |
(21) |
|
a2(b3(a3(a1(a0(x1))))) |
→ |
a2(b3(a1(a0(b0(x1))))) |
(22) |
|
b3(a3(a3(a1(a2(x1))))) |
→ |
b3(a3(a1(a0(b2(x1))))) |
(25) |
|
b3(a3(a3(a1(a0(x1))))) |
→ |
b3(a3(a1(a0(b0(x1))))) |
(26) |
|
b2(b3(a3(a1(a2(x1))))) |
→ |
b2(b3(a1(a0(b2(x1))))) |
(29) |
|
b2(b3(a3(a1(a0(x1))))) |
→ |
b2(b3(a1(a0(b0(x1))))) |
(30) |
|
a1(a2(b1(a2(b3(x1))))) |
→ |
a3(a1(a2(b3(a3(x1))))) |
(31) |
1.1.1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.