Certification Problem
Input (TPDB SRS_Standard/Zantema_04/z124)
The rewrite relation of the following TRS is considered.
|
q0(a(x1)) |
→ |
x(q1(x1)) |
(1) |
|
q1(a(x1)) |
→ |
a(q1(x1)) |
(2) |
|
q1(y(x1)) |
→ |
y(q1(x1)) |
(3) |
|
a(q1(b(x1))) |
→ |
q2(a(y(x1))) |
(4) |
|
a(q2(a(x1))) |
→ |
q2(a(a(x1))) |
(5) |
|
a(q2(y(x1))) |
→ |
q2(a(y(x1))) |
(6) |
|
y(q1(b(x1))) |
→ |
q2(y(y(x1))) |
(7) |
|
y(q2(a(x1))) |
→ |
q2(y(a(x1))) |
(8) |
|
y(q2(y(x1))) |
→ |
q2(y(y(x1))) |
(9) |
|
q2(x(x1)) |
→ |
x(q0(x1)) |
(10) |
|
q0(y(x1)) |
→ |
y(q3(x1)) |
(11) |
|
q3(y(x1)) |
→ |
y(q3(x1)) |
(12) |
|
q3(bl(x1)) |
→ |
bl(q4(x1)) |
(13) |
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
| [bl(x1)] |
= |
x1 +
|
| [q1(x1)] |
= |
x1 +
|
| [q0(x1)] |
= |
x1 +
|
| [q3(x1)] |
= |
x1 +
|
| [q2(x1)] |
= |
x1 +
|
| [q4(x1)] |
= |
x1 +
|
| [b(x1)] |
= |
x1 +
|
| [a(x1)] |
= |
x1 +
|
| [y(x1)] |
= |
x1 +
|
| [x(x1)] |
= |
x1 +
|
all of the following rules can be deleted.
|
q0(a(x1)) |
→ |
x(q1(x1)) |
(1) |
|
a(q1(b(x1))) |
→ |
q2(a(y(x1))) |
(4) |
|
y(q1(b(x1))) |
→ |
q2(y(y(x1))) |
(7) |
|
q2(x(x1)) |
→ |
x(q0(x1)) |
(10) |
|
q0(y(x1)) |
→ |
y(q3(x1)) |
(11) |
|
q3(bl(x1)) |
→ |
bl(q4(x1)) |
(13) |
1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
q1#(a(x1)) |
→ |
q1#(x1) |
(14) |
|
q1#(a(x1)) |
→ |
a#(q1(x1)) |
(15) |
|
q1#(y(x1)) |
→ |
q1#(x1) |
(16) |
|
q1#(y(x1)) |
→ |
y#(q1(x1)) |
(17) |
|
q3#(y(x1)) |
→ |
q3#(x1) |
(18) |
|
q3#(y(x1)) |
→ |
y#(q3(x1)) |
(19) |
|
a#(q2(a(x1))) |
→ |
a#(a(x1)) |
(20) |
|
a#(q2(y(x1))) |
→ |
a#(y(x1)) |
(21) |
|
y#(q2(a(x1))) |
→ |
y#(a(x1)) |
(22) |
|
y#(q2(y(x1))) |
→ |
y#(y(x1)) |
(23) |
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
| [q1(x1)] |
= |
x1 +
|
| [q3(x1)] |
= |
x1 +
|
| [q2(x1)] |
= |
x1 +
|
| [a(x1)] |
= |
x1 +
|
| [y(x1)] |
= |
x1 +
|
| [q1#(x1)] |
= |
x1 +
|
| [q3#(x1)] |
= |
x1 +
|
| [a#(x1)] |
= |
x1 +
|
| [y#(x1)] |
= |
x1 +
|
together with the usable
rules
|
q1(a(x1)) |
→ |
a(q1(x1)) |
(2) |
|
q1(y(x1)) |
→ |
y(q1(x1)) |
(3) |
|
a(q2(a(x1))) |
→ |
q2(a(a(x1))) |
(5) |
|
a(q2(y(x1))) |
→ |
q2(a(y(x1))) |
(6) |
|
y(q2(a(x1))) |
→ |
q2(y(a(x1))) |
(8) |
|
y(q2(y(x1))) |
→ |
q2(y(y(x1))) |
(9) |
|
q3(y(x1)) |
→ |
y(q3(x1)) |
(12) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
q1#(a(x1)) |
→ |
q1#(x1) |
(14) |
|
q1#(a(x1)) |
→ |
a#(q1(x1)) |
(15) |
|
q1#(y(x1)) |
→ |
q1#(x1) |
(16) |
|
q1#(y(x1)) |
→ |
y#(q1(x1)) |
(17) |
|
q3#(y(x1)) |
→ |
q3#(x1) |
(18) |
|
q3#(y(x1)) |
→ |
y#(q3(x1)) |
(19) |
|
a#(q2(a(x1))) |
→ |
a#(a(x1)) |
(20) |
|
a#(q2(y(x1))) |
→ |
a#(y(x1)) |
(21) |
|
y#(q2(a(x1))) |
→ |
y#(a(x1)) |
(22) |
|
y#(q2(y(x1))) |
→ |
y#(y(x1)) |
(23) |
and
no rules
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.