Certification Problem
Input (TPDB SRS_Relative/Waldmann_19/random-184)
The rewrite relation of the following TRS is considered.
a(a(c(x1))) |
→ |
a(c(c(x1))) |
(1) |
a(a(c(x1))) |
→ |
b(a(c(x1))) |
(2) |
a(a(c(x1))) |
→ |
a(b(c(x1))) |
(3) |
c(c(c(x1))) |
→ |
b(b(b(x1))) |
(4) |
c(a(a(x1))) |
→ |
c(b(b(x1))) |
(5) |
c(a(c(x1))) |
→ |
c(b(c(x1))) |
(6) |
b(b(a(x1))) |
→ |
c(c(a(x1))) |
(7) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by matchbox @ termCOMP 2023)
1 Rule Removal
Using the
matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
[c(x1)] |
= |
x1 +
|
[b(x1)] |
= |
x1 +
|
[a(x1)] |
= |
x1 +
|
all of the following rules can be deleted.
a(a(c(x1))) |
→ |
a(c(c(x1))) |
(1) |
a(a(c(x1))) |
→ |
b(a(c(x1))) |
(2) |
a(a(c(x1))) |
→ |
a(b(c(x1))) |
(3) |
c(a(a(x1))) |
→ |
c(b(b(x1))) |
(5) |
c(a(c(x1))) |
→ |
c(b(c(x1))) |
(6) |
1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
c#(c(c(x1))) |
→ |
b#(x1) |
(8) |
c#(c(c(x1))) |
→ |
b#(b(x1)) |
(9) |
c#(c(c(x1))) |
→ |
b#(b(b(x1))) |
(10) |
b#(b(a(x1))) |
→ |
c#(c(a(x1))) |
(11) |
b#(b(a(x1))) |
→ |
c#(a(x1)) |
(12) |
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
[c(x1)] |
= |
x1 +
|
[b(x1)] |
= |
x1 +
|
[a(x1)] |
= |
x1 +
|
[c#(x1)] |
= |
x1 +
|
[b#(x1)] |
= |
x1 +
|
together with the usable
rules
c(c(c(x1))) |
→ |
b(b(b(x1))) |
(4) |
b(b(a(x1))) |
→ |
c(c(a(x1))) |
(7) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
c#(c(c(x1))) |
→ |
b#(x1) |
(8) |
c#(c(c(x1))) |
→ |
b#(b(x1)) |
(9) |
b#(b(a(x1))) |
→ |
c#(a(x1)) |
(12) |
and
no rules
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.