Certification Problem
Input (TPDB SRS_Standard/Zantema_06/01)
The rewrite relation of the following TRS is considered.
|
a(b(x1)) |
→ |
b(r(x1)) |
(1) |
|
r(a(x1)) |
→ |
d(r(x1)) |
(2) |
|
r(x1) |
→ |
d(x1) |
(3) |
|
d(a(x1)) |
→ |
a(a(d(x1))) |
(4) |
|
d(x1) |
→ |
a(x1) |
(5) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by matchbox @ termCOMP 2023)
1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
d#(x1) |
→ |
a#(x1) |
(6) |
|
d#(a(x1)) |
→ |
d#(x1) |
(7) |
|
d#(a(x1)) |
→ |
a#(d(x1)) |
(8) |
|
d#(a(x1)) |
→ |
a#(a(d(x1))) |
(9) |
|
a#(b(x1)) |
→ |
r#(x1) |
(10) |
|
r#(x1) |
→ |
d#(x1) |
(11) |
|
r#(a(x1)) |
→ |
d#(r(x1)) |
(12) |
|
r#(a(x1)) |
→ |
r#(x1) |
(13) |
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
| [d(x1)] |
= |
x1 +
|
| [b(x1)] |
= |
x1 +
|
| [a(x1)] |
= |
x1 +
|
| [r(x1)] |
= |
x1 +
|
| [d#(x1)] |
= |
x1 +
|
| [a#(x1)] |
= |
x1 +
|
| [r#(x1)] |
= |
x1 +
|
together with the usable
rules
|
a(b(x1)) |
→ |
b(r(x1)) |
(1) |
|
r(a(x1)) |
→ |
d(r(x1)) |
(2) |
|
r(x1) |
→ |
d(x1) |
(3) |
|
d(a(x1)) |
→ |
a(a(d(x1))) |
(4) |
|
d(x1) |
→ |
a(x1) |
(5) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
d#(x1) |
→ |
a#(x1) |
(6) |
|
d#(a(x1)) |
→ |
a#(d(x1)) |
(8) |
|
d#(a(x1)) |
→ |
a#(a(d(x1))) |
(9) |
|
a#(b(x1)) |
→ |
r#(x1) |
(10) |
|
r#(x1) |
→ |
d#(x1) |
(11) |
|
r#(a(x1)) |
→ |
d#(r(x1)) |
(12) |
and
no rules
could be deleted.
1.1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.