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 AProVE @ termCOMP 2023)
1 String Reversal
Since only unary symbols occur, one can reverse all terms and obtains the TRS
|
b(a(x1)) |
→ |
r(b(x1)) |
(6) |
|
a(r(x1)) |
→ |
r(d(x1)) |
(7) |
|
r(x1) |
→ |
d(x1) |
(3) |
|
a(d(x1)) |
→ |
d(a(a(x1))) |
(8) |
|
d(x1) |
→ |
a(x1) |
(5) |
1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
b#(a(x1)) |
→ |
r#(b(x1)) |
(9) |
|
b#(a(x1)) |
→ |
b#(x1) |
(10) |
|
a#(r(x1)) |
→ |
r#(d(x1)) |
(11) |
|
a#(r(x1)) |
→ |
d#(x1) |
(12) |
|
r#(x1) |
→ |
d#(x1) |
(13) |
|
a#(d(x1)) |
→ |
d#(a(a(x1))) |
(14) |
|
a#(d(x1)) |
→ |
a#(a(x1)) |
(15) |
|
a#(d(x1)) |
→ |
a#(x1) |
(16) |
|
d#(x1) |
→ |
a#(x1) |
(17) |
1.1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.
-
The
1st
component contains the
pair
1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [a(x1)] |
= |
1 · x1
|
| [b#(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 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
b#(a(x1)) |
→ |
b#(x1) |
(10) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
|
r#(x1) |
→ |
d#(x1) |
(13) |
|
d#(x1) |
→ |
a#(x1) |
(17) |
|
a#(r(x1)) |
→ |
r#(d(x1)) |
(11) |
|
a#(r(x1)) |
→ |
d#(x1) |
(12) |
|
a#(d(x1)) |
→ |
d#(a(a(x1))) |
(14) |
|
a#(d(x1)) |
→ |
a#(a(x1)) |
(15) |
|
a#(d(x1)) |
→ |
a#(x1) |
(16) |
1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [r(x1)] |
= |
1 · x1
|
| [d(x1)] |
= |
1 · x1
|
| [a(x1)] |
= |
1 · x1
|
| [d#(x1)] |
= |
1 · x1
|
| [r#(x1)] |
= |
1 · x1
|
| [a#(x1)] |
= |
1 · x1
|
together with the usable
rules
|
r(x1) |
→ |
d(x1) |
(3) |
|
d(x1) |
→ |
a(x1) |
(5) |
|
a(r(x1)) |
→ |
r(d(x1)) |
(7) |
|
a(d(x1)) |
→ |
d(a(a(x1))) |
(8) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.2.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [r(x1)] |
= |
1 + 1 · x1
|
| [d(x1)] |
= |
1 · x1
|
| [a(x1)] |
= |
1 · x1
|
| [r#(x1)] |
= |
1 + 2 · x1
|
| [d#(x1)] |
= |
2 · x1
|
| [a#(x1)] |
= |
2 · x1
|
the
pairs
|
r#(x1) |
→ |
d#(x1) |
(13) |
|
a#(r(x1)) |
→ |
r#(d(x1)) |
(11) |
|
a#(r(x1)) |
→ |
d#(x1) |
(12) |
and
the
rule
could be deleted.
1.1.1.2.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [a#(x1)] |
= |
-2 + 2 · x1
|
| [d#(x1)] |
= |
1 + 2 · x1
|
| [a(x1)] |
= |
x1 |
| [r(x1)] |
= |
-2 |
| [d(x1)] |
= |
2 + x1
|
the
pairs
|
d#(x1) |
→ |
a#(x1) |
(17) |
|
a#(d(x1)) |
→ |
d#(a(a(x1))) |
(14) |
|
a#(d(x1)) |
→ |
a#(a(x1)) |
(15) |
|
a#(d(x1)) |
→ |
a#(x1) |
(16) |
could be deleted.
1.1.1.2.1.1.1 P is empty
There are no pairs anymore.