Certification Problem
Input (TPDB SRS_Standard/Zantema_04/z121)
The rewrite relation of the following TRS is considered.
|
f(f(x1)) |
→ |
b(b(b(x1))) |
(1) |
|
a(f(x1)) |
→ |
f(a(a(x1))) |
(2) |
|
b(b(x1)) |
→ |
c(c(a(c(x1)))) |
(3) |
|
d(b(x1)) |
→ |
d(a(b(x1))) |
(4) |
|
c(c(x1)) |
→ |
d(d(d(x1))) |
(5) |
|
b(d(x1)) |
→ |
d(b(x1)) |
(6) |
|
c(d(d(x1))) |
→ |
f(x1) |
(7) |
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
|
f(f(x1)) |
→ |
b(b(b(x1))) |
(1) |
|
f(a(x1)) |
→ |
a(a(f(x1))) |
(8) |
|
b(b(x1)) |
→ |
c(a(c(c(x1)))) |
(9) |
|
b(d(x1)) |
→ |
b(a(d(x1))) |
(10) |
|
c(c(x1)) |
→ |
d(d(d(x1))) |
(5) |
|
d(b(x1)) |
→ |
b(d(x1)) |
(11) |
|
d(d(c(x1))) |
→ |
f(x1) |
(12) |
1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [f(x1)] |
= |
1 · x1 + 80 |
| [b(x1)] |
= |
1 · x1 + 53 |
| [a(x1)] |
= |
1 · x1
|
| [c(x1)] |
= |
1 · x1 + 35 |
| [d(x1)] |
= |
1 · x1 + 23 |
all of the following rules can be deleted.
|
f(f(x1)) |
→ |
b(b(b(x1))) |
(1) |
|
b(b(x1)) |
→ |
c(a(c(c(x1)))) |
(9) |
|
c(c(x1)) |
→ |
d(d(d(x1))) |
(5) |
|
d(d(c(x1))) |
→ |
f(x1) |
(12) |
1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
f#(a(x1)) |
→ |
f#(x1) |
(13) |
|
b#(d(x1)) |
→ |
b#(a(d(x1))) |
(14) |
|
d#(b(x1)) |
→ |
b#(d(x1)) |
(15) |
|
d#(b(x1)) |
→ |
d#(x1) |
(16) |
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.
-
The
1st
component contains the
pair
1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [b(x1)] |
= |
1 · x1
|
| [d#(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 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
d#(b(x1)) |
→ |
d#(x1) |
(16) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [a(x1)] |
= |
1 · x1
|
| [f#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
f#(a(x1)) |
→ |
f#(x1) |
(13) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.