Certification Problem
Input (TPDB SRS_Standard/Zantema_04/z094)
The rewrite relation of the following TRS is considered.
|
f(x1) |
→ |
n(c(c(x1))) |
(1) |
|
c(f(x1)) |
→ |
f(c(c(x1))) |
(2) |
|
c(c(x1)) |
→ |
c(x1) |
(3) |
|
n(s(x1)) |
→ |
f(s(s(x1))) |
(4) |
|
n(f(x1)) |
→ |
f(n(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
|
f(x1) |
→ |
c(c(n(x1))) |
(6) |
|
f(c(x1)) |
→ |
c(c(f(x1))) |
(7) |
|
c(c(x1)) |
→ |
c(x1) |
(3) |
|
s(n(x1)) |
→ |
s(s(f(x1))) |
(8) |
|
f(n(x1)) |
→ |
n(f(x1)) |
(9) |
1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
f#(x1) |
→ |
c#(c(n(x1))) |
(10) |
|
f#(x1) |
→ |
c#(n(x1)) |
(11) |
|
f#(c(x1)) |
→ |
c#(c(f(x1))) |
(12) |
|
f#(c(x1)) |
→ |
c#(f(x1)) |
(13) |
|
f#(c(x1)) |
→ |
f#(x1) |
(14) |
|
s#(n(x1)) |
→ |
s#(s(f(x1))) |
(15) |
|
s#(n(x1)) |
→ |
s#(f(x1)) |
(16) |
|
s#(n(x1)) |
→ |
f#(x1) |
(17) |
|
f#(n(x1)) |
→ |
f#(x1) |
(18) |
1.1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.
-
The
1st
component contains the
pair
|
s#(n(x1)) |
→ |
s#(f(x1)) |
(16) |
|
s#(n(x1)) |
→ |
s#(s(f(x1))) |
(15) |
1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [s#(x1)] |
= |
1 · x1
|
| [n(x1)] |
= |
1 + 1 · x1
|
| [f(x1)] |
= |
1 + 1 · x1
|
| [s(x1)] |
= |
0 |
| [c(x1)] |
= |
1 · x1
|
the
pair
|
s#(n(x1)) |
→ |
s#(s(f(x1))) |
(15) |
could be deleted.
1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [f(x1)] |
= |
1 · x1
|
| [c(x1)] |
= |
1 · x1
|
| [n(x1)] |
= |
1 · x1
|
| [s#(x1)] |
= |
1 · x1
|
together with the usable
rules
|
f(x1) |
→ |
c(c(n(x1))) |
(6) |
|
f(c(x1)) |
→ |
c(c(f(x1))) |
(7) |
|
f(n(x1)) |
→ |
n(f(x1)) |
(9) |
|
c(c(x1)) |
→ |
c(x1) |
(3) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [s#(x1)] |
= |
-2 + 2 · x1
|
| [f(x1)] |
= |
2 · x1
|
| [c(x1)] |
= |
0 |
| [n(x1)] |
= |
2 + 2 · x1
|
the
pair
|
s#(n(x1)) |
→ |
s#(f(x1)) |
(16) |
could be deleted.
1.1.1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
|
f#(n(x1)) |
→ |
f#(x1) |
(18) |
|
f#(c(x1)) |
→ |
f#(x1) |
(14) |
1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [n(x1)] |
= |
1 · x1
|
| [c(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.
1.1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
f#(n(x1)) |
→ |
f#(x1) |
(18) |
|
| 1 |
> |
1 |
|
f#(c(x1)) |
→ |
f#(x1) |
(14) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.