Certification Problem
Input (TPDB SRS_Standard/Zantema_04/z067)
The rewrite relation of the following TRS is considered.
|
P(x1) |
→ |
Q(Q(p(x1))) |
(1) |
|
p(p(x1)) |
→ |
q(q(x1)) |
(2) |
|
p(Q(Q(x1))) |
→ |
Q(Q(p(x1))) |
(3) |
|
Q(p(q(x1))) |
→ |
q(p(Q(x1))) |
(4) |
|
q(q(p(x1))) |
→ |
p(q(q(x1))) |
(5) |
|
q(Q(x1)) |
→ |
x1 |
(6) |
|
Q(q(x1)) |
→ |
x1 |
(7) |
|
p(P(x1)) |
→ |
x1 |
(8) |
|
P(p(x1)) |
→ |
x1 |
(9) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 String Reversal
Since only unary symbols occur, one can reverse all terms and obtains the TRS
|
P(x1) |
→ |
p(Q(Q(x1))) |
(10) |
|
p(p(x1)) |
→ |
q(q(x1)) |
(2) |
|
Q(Q(p(x1))) |
→ |
p(Q(Q(x1))) |
(11) |
|
q(p(Q(x1))) |
→ |
Q(p(q(x1))) |
(12) |
|
p(q(q(x1))) |
→ |
q(q(p(x1))) |
(13) |
|
Q(q(x1)) |
→ |
x1 |
(7) |
|
q(Q(x1)) |
→ |
x1 |
(6) |
|
P(p(x1)) |
→ |
x1 |
(9) |
|
p(P(x1)) |
→ |
x1 |
(8) |
1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [P(x1)] |
= |
1 · x1 + 2 |
| [p(x1)] |
= |
1 · x1
|
| [Q(x1)] |
= |
1 · x1 + 1 |
| [q(x1)] |
= |
1 · x1
|
all of the following rules can be deleted.
|
Q(q(x1)) |
→ |
x1 |
(7) |
|
q(Q(x1)) |
→ |
x1 |
(6) |
|
P(p(x1)) |
→ |
x1 |
(9) |
|
p(P(x1)) |
→ |
x1 |
(8) |
1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [P(x1)] |
= |
1 · x1 + 2 |
| [p(x1)] |
= |
1 · x1 + 1 |
| [Q(x1)] |
= |
1 · x1
|
| [q(x1)] |
= |
1 · x1
|
all of the following rules can be deleted.
|
P(x1) |
→ |
p(Q(Q(x1))) |
(10) |
|
p(p(x1)) |
→ |
q(q(x1)) |
(2) |
1.1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
Q#(Q(p(x1))) |
→ |
p#(Q(Q(x1))) |
(14) |
|
Q#(Q(p(x1))) |
→ |
Q#(Q(x1)) |
(15) |
|
Q#(Q(p(x1))) |
→ |
Q#(x1) |
(16) |
|
q#(p(Q(x1))) |
→ |
Q#(p(q(x1))) |
(17) |
|
q#(p(Q(x1))) |
→ |
p#(q(x1)) |
(18) |
|
q#(p(Q(x1))) |
→ |
q#(x1) |
(19) |
|
p#(q(q(x1))) |
→ |
q#(q(p(x1))) |
(20) |
|
p#(q(q(x1))) |
→ |
q#(p(x1)) |
(21) |
|
p#(q(q(x1))) |
→ |
p#(x1) |
(22) |
1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [Q#(x1)] |
= |
1 + 1 · x1
|
| [Q(x1)] |
= |
1 + 1 · x1
|
| [p(x1)] |
= |
1 · x1
|
| [p#(x1)] |
= |
1 · x1
|
| [q#(x1)] |
= |
1 · x1
|
| [q(x1)] |
= |
1 · x1
|
the
pairs
|
Q#(Q(p(x1))) |
→ |
Q#(x1) |
(16) |
|
q#(p(Q(x1))) |
→ |
p#(q(x1)) |
(18) |
|
q#(p(Q(x1))) |
→ |
q#(x1) |
(19) |
could be deleted.
1.1.1.1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [Q(x1)] |
= |
2 + 2 · x1
|
| [p(x1)] |
= |
1 · x1
|
| [q(x1)] |
= |
2 · x1
|
| [Q#(x1)] |
= |
2 + 2 · x1
|
| [p#(x1)] |
= |
1 · x1
|
| [q#(x1)] |
= |
2 · x1
|
the
pair
|
q#(p(Q(x1))) |
→ |
Q#(p(q(x1))) |
(17) |
and
the
rule
|
q(p(Q(x1))) |
→ |
Q(p(q(x1))) |
(12) |
could be deleted.
1.1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.
-
The
1st
component contains the
pair
|
Q#(Q(p(x1))) |
→ |
Q#(Q(x1)) |
(15) |
1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [Q(x1)] |
= |
2 · x1
|
| [p(x1)] |
= |
2 · x1
|
| [q(x1)] |
= |
1 + 1 · x1
|
| [Q#(x1)] |
= |
1 · x1
|
the
rule
|
p(q(q(x1))) |
→ |
q(q(p(x1))) |
(13) |
could be deleted.
1.1.1.1.1.1.1.1.1 Switch to Innermost Termination
The TRS does not have overlaps with the pairs and is locally confluent:
20
Hence, it suffices to show innermost termination in the following.
1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [Q#(x1)] |
= |
1 · x1
|
| [Q(x1)] |
= |
1 · x1
|
| [p(x1)] |
= |
1 + 1 · x1
|
the
pair
|
Q#(Q(p(x1))) |
→ |
Q#(Q(x1)) |
(15) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
|
p#(q(q(x1))) |
→ |
p#(x1) |
(22) |
1.1.1.1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [q(x1)] |
= |
1 · x1
|
| [p#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
p#(q(q(x1))) |
→ |
p#(x1) |
(22) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.