Certification Problem
Input (TPDB SRS_Standard/Zantema_04/z072)
The rewrite relation of the following TRS is considered.
|
a(l(x1)) |
→ |
l(a(x1)) |
(1) |
|
r(a(a(x1))) |
→ |
a(a(r(x1))) |
(2) |
|
b(l(x1)) |
→ |
b(a(r(x1))) |
(3) |
|
r(b(x1)) |
→ |
l(b(x1)) |
(4) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by NaTT @ termCOMP 2023)
1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
r#(a(a(x1))) |
→ |
a#(r(x1)) |
(5) |
|
b#(l(x1)) |
→ |
r#(x1) |
(6) |
|
a#(l(x1)) |
→ |
a#(x1) |
(7) |
|
b#(l(x1)) |
→ |
a#(r(x1)) |
(8) |
|
r#(a(a(x1))) |
→ |
r#(x1) |
(9) |
|
r#(a(a(x1))) |
→ |
a#(a(r(x1))) |
(10) |
|
b#(l(x1)) |
→ |
b#(a(r(x1))) |
(11) |
1.1 Dependency Graph Processor
The dependency pairs are split into 3
components.
-
The
1st
component contains the
pair
|
b#(l(x1)) |
→ |
b#(a(r(x1))) |
(11) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the matrix interpretations of dimension 2 with strict dimension 1 over the naturals
| [a(x1)] |
= |
· x1 +
|
| [b(x1)] |
= |
· x1 +
|
| [r(x1)] |
= |
· x1 +
|
| [l(x1)] |
= |
x1 +
|
| [r#(x1)] |
= |
|
| [a#(x1)] |
= |
|
| [b#(x1)] |
= |
· x1 +
|
together with the usable
rules
|
r(b(x1)) |
→ |
l(b(x1)) |
(4) |
|
a(l(x1)) |
→ |
l(a(x1)) |
(1) |
|
b(l(x1)) |
→ |
b(a(r(x1))) |
(3) |
|
r(a(a(x1))) |
→ |
a(a(r(x1))) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
b#(l(x1)) |
→ |
b#(a(r(x1))) |
(11) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
2nd
component contains the
pair
|
r#(a(a(x1))) |
→ |
r#(x1) |
(9) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [a(x1)] |
=
|
x1 + 2 |
| [b(x1)] |
=
|
x1 + 1 |
| [r(x1)] |
=
|
5854 |
| [l(x1)] |
=
|
5855 |
| [r#(x1)] |
=
|
x1 + 0 |
| [a#(x1)] |
=
|
0 |
| [b#(x1)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
|
r#(a(a(x1))) |
→ |
r#(x1) |
(9) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [a(x1)] |
=
|
x1 + 0 |
| [b(x1)] |
=
|
x1 + 1144 |
| [r(x1)] |
=
|
1144 |
| [l(x1)] |
=
|
x1 + 1 |
| [r#(x1)] |
=
|
0 |
| [a#(x1)] |
=
|
x1 + 0 |
| [b#(x1)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.