Certification Problem
Input (TPDB SRS_Standard/Mixed_SRS/04)
The rewrite relation of the following TRS is considered.
|
a(a(a(x1))) |
→ |
a(b(b(b(x1)))) |
(1) |
|
b(b(a(b(x1)))) |
→ |
a(a(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(b(b(b(x1))))) |
(3) |
|
b(b(b(a(b(x1))))) |
→ |
b(a(a(a(x1)))) |
(4) |
|
a(a(a(a(x1)))) |
→ |
a(a(b(b(b(x1))))) |
(5) |
|
a(b(b(a(b(x1))))) |
→ |
a(a(a(a(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(a1(a1(a1(x1)))) |
→ |
a1(a0(b0(b0(b1(x1))))) |
(7) |
|
a1(a1(a1(a0(x1)))) |
→ |
a1(a0(b0(b0(b0(x1))))) |
(8) |
|
b1(a1(a1(a1(x1)))) |
→ |
b1(a0(b0(b0(b1(x1))))) |
(9) |
|
b1(a1(a1(a0(x1)))) |
→ |
b1(a0(b0(b0(b0(x1))))) |
(10) |
|
a0(b0(b1(a0(b1(x1))))) |
→ |
a1(a1(a1(a1(x1)))) |
(11) |
|
a0(b0(b1(a0(b0(x1))))) |
→ |
a1(a1(a1(a0(x1)))) |
(12) |
|
b0(b0(b1(a0(b1(x1))))) |
→ |
b1(a1(a1(a1(x1)))) |
(13) |
|
b0(b0(b1(a0(b0(x1))))) |
→ |
b1(a1(a1(a0(x1)))) |
(14) |
1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
b0#(b0(b1(a0(b0(x1))))) |
→ |
b1#(a1(a1(a0(x1)))) |
(15) |
|
b0#(b0(b1(a0(b0(x1))))) |
→ |
a0#(x1) |
(16) |
|
b0#(b0(b1(a0(b0(x1))))) |
→ |
a1#(a0(x1)) |
(17) |
|
b0#(b0(b1(a0(b0(x1))))) |
→ |
a1#(a1(a0(x1))) |
(18) |
|
b0#(b0(b1(a0(b1(x1))))) |
→ |
b1#(a1(a1(a1(x1)))) |
(19) |
|
b0#(b0(b1(a0(b1(x1))))) |
→ |
a1#(x1) |
(20) |
|
b0#(b0(b1(a0(b1(x1))))) |
→ |
a1#(a1(x1)) |
(21) |
|
b0#(b0(b1(a0(b1(x1))))) |
→ |
a1#(a1(a1(x1))) |
(22) |
|
b1#(a1(a1(a0(x1)))) |
→ |
b0#(x1) |
(23) |
|
b1#(a1(a1(a0(x1)))) |
→ |
b0#(b0(x1)) |
(24) |
|
b1#(a1(a1(a0(x1)))) |
→ |
b0#(b0(b0(x1))) |
(25) |
|
b1#(a1(a1(a0(x1)))) |
→ |
b1#(a0(b0(b0(b0(x1))))) |
(26) |
|
b1#(a1(a1(a0(x1)))) |
→ |
a0#(b0(b0(b0(x1)))) |
(27) |
|
b1#(a1(a1(a1(x1)))) |
→ |
b0#(b0(b1(x1))) |
(28) |
|
b1#(a1(a1(a1(x1)))) |
→ |
b0#(b1(x1)) |
(29) |
|
b1#(a1(a1(a1(x1)))) |
→ |
b1#(x1) |
(30) |
|
b1#(a1(a1(a1(x1)))) |
→ |
b1#(a0(b0(b0(b1(x1))))) |
(31) |
|
b1#(a1(a1(a1(x1)))) |
→ |
a0#(b0(b0(b1(x1)))) |
(32) |
|
a0#(b0(b1(a0(b0(x1))))) |
→ |
a0#(x1) |
(33) |
|
a0#(b0(b1(a0(b0(x1))))) |
→ |
a1#(a0(x1)) |
(34) |
|
a0#(b0(b1(a0(b0(x1))))) |
→ |
a1#(a1(a0(x1))) |
(35) |
|
a0#(b0(b1(a0(b0(x1))))) |
→ |
a1#(a1(a1(a0(x1)))) |
(36) |
|
a0#(b0(b1(a0(b1(x1))))) |
→ |
a1#(x1) |
(37) |
|
a0#(b0(b1(a0(b1(x1))))) |
→ |
a1#(a1(x1)) |
(38) |
|
a0#(b0(b1(a0(b1(x1))))) |
→ |
a1#(a1(a1(x1))) |
(39) |
|
a0#(b0(b1(a0(b1(x1))))) |
→ |
a1#(a1(a1(a1(x1)))) |
(40) |
|
a1#(a1(a1(a0(x1)))) |
→ |
b0#(x1) |
(41) |
|
a1#(a1(a1(a0(x1)))) |
→ |
b0#(b0(x1)) |
(42) |
|
a1#(a1(a1(a0(x1)))) |
→ |
b0#(b0(b0(x1))) |
(43) |
|
a1#(a1(a1(a0(x1)))) |
→ |
a0#(b0(b0(b0(x1)))) |
(44) |
|
a1#(a1(a1(a0(x1)))) |
→ |
a1#(a0(b0(b0(b0(x1))))) |
(45) |
|
a1#(a1(a1(a1(x1)))) |
→ |
b0#(b0(b1(x1))) |
(46) |
|
a1#(a1(a1(a1(x1)))) |
→ |
b0#(b1(x1)) |
(47) |
|
a1#(a1(a1(a1(x1)))) |
→ |
b1#(x1) |
(48) |
|
a1#(a1(a1(a1(x1)))) |
→ |
a0#(b0(b0(b1(x1)))) |
(49) |
|
a1#(a1(a1(a1(x1)))) |
→ |
a1#(a0(b0(b0(b1(x1))))) |
(50) |
1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
| [b0(x1)] |
= |
x1 +
|
| [b1(x1)] |
= |
x1 +
|
| [a0(x1)] |
= |
x1 +
|
| [a1(x1)] |
= |
x1 +
|
| [b0#(x1)] |
= |
x1 +
|
| [b1#(x1)] |
= |
x1 +
|
| [a0#(x1)] |
= |
x1 +
|
| [a1#(x1)] |
= |
x1 +
|
together with the usable
rules
|
a1(a1(a1(a1(x1)))) |
→ |
a1(a0(b0(b0(b1(x1))))) |
(7) |
|
a1(a1(a1(a0(x1)))) |
→ |
a1(a0(b0(b0(b0(x1))))) |
(8) |
|
b1(a1(a1(a1(x1)))) |
→ |
b1(a0(b0(b0(b1(x1))))) |
(9) |
|
b1(a1(a1(a0(x1)))) |
→ |
b1(a0(b0(b0(b0(x1))))) |
(10) |
|
a0(b0(b1(a0(b1(x1))))) |
→ |
a1(a1(a1(a1(x1)))) |
(11) |
|
a0(b0(b1(a0(b0(x1))))) |
→ |
a1(a1(a1(a0(x1)))) |
(12) |
|
b0(b0(b1(a0(b1(x1))))) |
→ |
b1(a1(a1(a1(x1)))) |
(13) |
|
b0(b0(b1(a0(b0(x1))))) |
→ |
b1(a1(a1(a0(x1)))) |
(14) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
b0#(b0(b1(a0(b0(x1))))) |
→ |
b1#(a1(a1(a0(x1)))) |
(15) |
|
b0#(b0(b1(a0(b0(x1))))) |
→ |
a0#(x1) |
(16) |
|
b0#(b0(b1(a0(b0(x1))))) |
→ |
a1#(a0(x1)) |
(17) |
|
b0#(b0(b1(a0(b0(x1))))) |
→ |
a1#(a1(a0(x1))) |
(18) |
|
b0#(b0(b1(a0(b1(x1))))) |
→ |
b1#(a1(a1(a1(x1)))) |
(19) |
|
b0#(b0(b1(a0(b1(x1))))) |
→ |
a1#(x1) |
(20) |
|
b0#(b0(b1(a0(b1(x1))))) |
→ |
a1#(a1(x1)) |
(21) |
|
b0#(b0(b1(a0(b1(x1))))) |
→ |
a1#(a1(a1(x1))) |
(22) |
|
b1#(a1(a1(a0(x1)))) |
→ |
b0#(x1) |
(23) |
|
b1#(a1(a1(a0(x1)))) |
→ |
b0#(b0(x1)) |
(24) |
|
b1#(a1(a1(a0(x1)))) |
→ |
b0#(b0(b0(x1))) |
(25) |
|
b1#(a1(a1(a0(x1)))) |
→ |
a0#(b0(b0(b0(x1)))) |
(27) |
|
b1#(a1(a1(a1(x1)))) |
→ |
b0#(b0(b1(x1))) |
(28) |
|
b1#(a1(a1(a1(x1)))) |
→ |
b0#(b1(x1)) |
(29) |
|
b1#(a1(a1(a1(x1)))) |
→ |
b1#(x1) |
(30) |
|
b1#(a1(a1(a1(x1)))) |
→ |
a0#(b0(b0(b1(x1)))) |
(32) |
|
a0#(b0(b1(a0(b0(x1))))) |
→ |
a0#(x1) |
(33) |
|
a0#(b0(b1(a0(b0(x1))))) |
→ |
a1#(a0(x1)) |
(34) |
|
a0#(b0(b1(a0(b0(x1))))) |
→ |
a1#(a1(a0(x1))) |
(35) |
|
a0#(b0(b1(a0(b0(x1))))) |
→ |
a1#(a1(a1(a0(x1)))) |
(36) |
|
a0#(b0(b1(a0(b1(x1))))) |
→ |
a1#(x1) |
(37) |
|
a0#(b0(b1(a0(b1(x1))))) |
→ |
a1#(a1(x1)) |
(38) |
|
a0#(b0(b1(a0(b1(x1))))) |
→ |
a1#(a1(a1(x1))) |
(39) |
|
a0#(b0(b1(a0(b1(x1))))) |
→ |
a1#(a1(a1(a1(x1)))) |
(40) |
|
a1#(a1(a1(a0(x1)))) |
→ |
b0#(x1) |
(41) |
|
a1#(a1(a1(a0(x1)))) |
→ |
b0#(b0(x1)) |
(42) |
|
a1#(a1(a1(a0(x1)))) |
→ |
b0#(b0(b0(x1))) |
(43) |
|
a1#(a1(a1(a0(x1)))) |
→ |
a0#(b0(b0(b0(x1)))) |
(44) |
|
a1#(a1(a1(a1(x1)))) |
→ |
b0#(b0(b1(x1))) |
(46) |
|
a1#(a1(a1(a1(x1)))) |
→ |
b0#(b1(x1)) |
(47) |
|
a1#(a1(a1(a1(x1)))) |
→ |
b1#(x1) |
(48) |
|
a1#(a1(a1(a1(x1)))) |
→ |
a0#(b0(b0(b1(x1)))) |
(49) |
and
no rules
could be deleted.
1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.