Certification Problem
Input (TPDB SRS_Standard/Secret_06_SRS/7)
The rewrite relation of the following TRS is considered.
|
a(b(b(x1))) |
→ |
b(a(a(a(x1)))) |
(1) |
|
a(a(b(x1))) |
→ |
b(a(b(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(b(b(x1)))) |
→ |
b(b(a(a(a(x1))))) |
(3) |
|
b(a(a(b(x1)))) |
→ |
b(b(a(b(x1)))) |
(4) |
|
a(a(b(b(x1)))) |
→ |
a(b(a(a(a(x1))))) |
(5) |
|
a(a(a(b(x1)))) |
→ |
a(b(a(b(x1)))) |
(6) |
1.1 Semantic Labeling
The following interpretations form a
model
of the rules.
As carrier we take the set
{0,1}.
Symbols are labeled by the interpretation of their arguments using the interpretations
(modulo 2):
| [b(x1)] |
= |
2x1 + 0 |
| [a(x1)] |
= |
2x1 + 1 |
We obtain the labeled TRS
|
a1(a0(b0(b1(x1)))) |
→ |
a0(b1(a1(a1(a1(x1))))) |
(7) |
|
a1(a0(b0(b0(x1)))) |
→ |
a0(b1(a1(a1(a0(x1))))) |
(8) |
|
b1(a0(b0(b1(x1)))) |
→ |
b0(b1(a1(a1(a1(x1))))) |
(9) |
|
b1(a0(b0(b0(x1)))) |
→ |
b0(b1(a1(a1(a0(x1))))) |
(10) |
|
a1(a1(a0(b1(x1)))) |
→ |
a0(b1(a0(b1(x1)))) |
(11) |
|
a1(a1(a0(b0(x1)))) |
→ |
a0(b1(a0(b0(x1)))) |
(12) |
|
b1(a1(a0(b1(x1)))) |
→ |
b0(b1(a0(b1(x1)))) |
(13) |
|
b1(a1(a0(b0(x1)))) |
→ |
b0(b1(a0(b0(x1)))) |
(14) |
1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
b1#(a0(b0(b0(x1)))) |
→ |
b1#(a1(a1(a0(x1)))) |
(15) |
|
b1#(a0(b0(b0(x1)))) |
→ |
a1#(a0(x1)) |
(16) |
|
b1#(a0(b0(b0(x1)))) |
→ |
a1#(a1(a0(x1))) |
(17) |
|
b1#(a0(b0(b1(x1)))) |
→ |
b1#(a1(a1(a1(x1)))) |
(18) |
|
b1#(a0(b0(b1(x1)))) |
→ |
a1#(x1) |
(19) |
|
b1#(a0(b0(b1(x1)))) |
→ |
a1#(a1(x1)) |
(20) |
|
b1#(a0(b0(b1(x1)))) |
→ |
a1#(a1(a1(x1))) |
(21) |
|
b1#(a1(a0(b0(x1)))) |
→ |
b1#(a0(b0(x1))) |
(22) |
|
b1#(a1(a0(b1(x1)))) |
→ |
b1#(a0(b1(x1))) |
(23) |
|
a1#(a0(b0(b0(x1)))) |
→ |
b1#(a1(a1(a0(x1)))) |
(24) |
|
a1#(a0(b0(b0(x1)))) |
→ |
a1#(a0(x1)) |
(25) |
|
a1#(a0(b0(b0(x1)))) |
→ |
a1#(a1(a0(x1))) |
(26) |
|
a1#(a0(b0(b1(x1)))) |
→ |
b1#(a1(a1(a1(x1)))) |
(27) |
|
a1#(a0(b0(b1(x1)))) |
→ |
a1#(x1) |
(28) |
|
a1#(a0(b0(b1(x1)))) |
→ |
a1#(a1(x1)) |
(29) |
|
a1#(a0(b0(b1(x1)))) |
→ |
a1#(a1(a1(x1))) |
(30) |
|
a1#(a1(a0(b0(x1)))) |
→ |
b1#(a0(b0(x1))) |
(31) |
|
a1#(a1(a0(b1(x1)))) |
→ |
b1#(a0(b1(x1))) |
(32) |
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.