Certification Problem
Input (TPDB SRS_Standard/Zantema_04/z100)
The rewrite relation of the following TRS is considered.
|
r1(a(x1)) |
→ |
a(a(a(r1(x1)))) |
(1) |
|
r2(a(x1)) |
→ |
a(a(a(r2(x1)))) |
(2) |
|
a(l1(x1)) |
→ |
l1(a(a(a(x1)))) |
(3) |
|
a(a(l2(x1))) |
→ |
l2(a(a(x1))) |
(4) |
|
r1(b(x1)) |
→ |
l1(b(x1)) |
(5) |
|
r2(b(x1)) |
→ |
l2(a(b(x1))) |
(6) |
|
b(l1(x1)) |
→ |
b(r2(x1)) |
(7) |
|
b(l2(x1)) |
→ |
b(r1(x1)) |
(8) |
|
a(a(x1)) |
→ |
x1 |
(9) |
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.
|
r2#(a(x1)) |
→ |
a#(a(r2(x1))) |
(10) |
|
b#(l1(x1)) |
→ |
b#(r2(x1)) |
(11) |
|
r1#(a(x1)) |
→ |
a#(r1(x1)) |
(12) |
|
a#(l1(x1)) |
→ |
a#(a(x1)) |
(13) |
|
r1#(a(x1)) |
→ |
a#(a(a(r1(x1)))) |
(14) |
|
r1#(a(x1)) |
→ |
a#(a(r1(x1))) |
(15) |
|
r1#(a(x1)) |
→ |
r1#(x1) |
(16) |
|
b#(l1(x1)) |
→ |
r2#(x1) |
(17) |
|
a#(l1(x1)) |
→ |
a#(x1) |
(18) |
|
r2#(b(x1)) |
→ |
a#(b(x1)) |
(19) |
|
a#(a(l2(x1))) |
→ |
a#(a(x1)) |
(20) |
|
b#(l2(x1)) |
→ |
r1#(x1) |
(21) |
|
r2#(a(x1)) |
→ |
a#(r2(x1)) |
(22) |
|
r2#(a(x1)) |
→ |
a#(a(a(r2(x1)))) |
(23) |
|
a#(l1(x1)) |
→ |
a#(a(a(x1))) |
(24) |
|
b#(l2(x1)) |
→ |
b#(r1(x1)) |
(25) |
|
r2#(a(x1)) |
→ |
r2#(x1) |
(26) |
|
a#(a(l2(x1))) |
→ |
a#(x1) |
(27) |
1.1 Dependency Graph Processor
The dependency pairs are split into 4
components.
-
The
1st
component contains the
pair
|
b#(l2(x1)) |
→ |
b#(r1(x1)) |
(25) |
|
b#(l1(x1)) |
→ |
b#(r2(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 +
|
| [l1(x1)] |
= |
· x1 +
|
| [l2(x1)] |
= |
· x1 +
|
| [r1#(x1)] |
= |
|
| [r1(x1)] |
= |
· x1 +
|
| [r2#(x1)] |
= |
|
| [a#(x1)] |
= |
|
| [r2(x1)] |
= |
· x1 +
|
| [b#(x1)] |
= |
· x1 +
|
together with the usable
rules
|
a(a(l2(x1))) |
→ |
l2(a(a(x1))) |
(4) |
|
b(l2(x1)) |
→ |
b(r1(x1)) |
(8) |
|
r1(a(x1)) |
→ |
a(a(a(r1(x1)))) |
(1) |
|
a(l1(x1)) |
→ |
l1(a(a(a(x1)))) |
(3) |
|
r1(b(x1)) |
→ |
l1(b(x1)) |
(5) |
|
b(l1(x1)) |
→ |
b(r2(x1)) |
(7) |
|
a(a(x1)) |
→ |
x1 |
(9) |
|
r2(b(x1)) |
→ |
l2(a(b(x1))) |
(6) |
|
r2(a(x1)) |
→ |
a(a(a(r2(x1)))) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
b#(l2(x1)) |
→ |
b#(r1(x1)) |
(25) |
|
b#(l1(x1)) |
→ |
b#(r2(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
|
r1#(a(x1)) |
→ |
r1#(x1) |
(16) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [a(x1)] |
=
|
x1 + 1 |
| [b(x1)] |
=
|
x1 + 1 |
| [l1(x1)] |
=
|
20166 |
| [l2(x1)] |
=
|
20163 |
| [r1#(x1)] |
=
|
x1 + 0 |
| [r1(x1)] |
=
|
x1 + 20164 |
| [r2#(x1)] |
=
|
0 |
| [a#(x1)] |
=
|
0 |
| [r2(x1)] |
=
|
20167 |
| [b#(x1)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
|
r1#(a(x1)) |
→ |
r1#(x1) |
(16) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
|
r2#(a(x1)) |
→ |
r2#(x1) |
(26) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [a(x1)] |
=
|
x1 + 1 |
| [b(x1)] |
=
|
x1 + 1 |
| [l1(x1)] |
=
|
4 |
| [l2(x1)] |
=
|
1 |
| [r1#(x1)] |
=
|
0 |
| [r1(x1)] |
=
|
x1 + 2 |
| [r2#(x1)] |
=
|
x1 + 0 |
| [a#(x1)] |
=
|
0 |
| [r2(x1)] |
=
|
20167 |
| [b#(x1)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
|
r2#(a(x1)) |
→ |
r2#(x1) |
(26) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
4th
component contains the
pair
|
a#(a(l2(x1))) |
→ |
a#(x1) |
(27) |
|
a#(l1(x1)) |
→ |
a#(a(a(x1))) |
(24) |
|
a#(l1(x1)) |
→ |
a#(a(x1)) |
(13) |
|
a#(a(l2(x1))) |
→ |
a#(a(x1)) |
(20) |
|
a#(l1(x1)) |
→ |
a#(x1) |
(18) |
1.1.4 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [a(x1)] |
=
|
x1 + 0 |
| [b(x1)] |
=
|
x1 + 1143 |
| [l1(x1)] |
=
|
x1 + 4 |
| [l2(x1)] |
=
|
x1 + 1 |
| [r1#(x1)] |
=
|
0 |
| [r1(x1)] |
=
|
x1 + 3 |
| [r2#(x1)] |
=
|
0 |
| [a#(x1)] |
=
|
x1 + 0 |
| [r2(x1)] |
=
|
1147 |
| [b#(x1)] |
=
|
0 |
together with the usable
rules
|
a(a(l2(x1))) |
→ |
l2(a(a(x1))) |
(4) |
|
a(l1(x1)) |
→ |
l1(a(a(a(x1)))) |
(3) |
|
a(a(x1)) |
→ |
x1 |
(9) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
a#(a(l2(x1))) |
→ |
a#(x1) |
(27) |
|
a#(l1(x1)) |
→ |
a#(a(a(x1))) |
(24) |
|
a#(l1(x1)) |
→ |
a#(a(x1)) |
(13) |
|
a#(a(l2(x1))) |
→ |
a#(a(x1)) |
(20) |
|
a#(l1(x1)) |
→ |
a#(x1) |
(18) |
could be deleted.
1.1.4.1 Dependency Graph Processor
The dependency pairs are split into 0
components.