Certification Problem
Input (TPDB TRS_Standard/AProVE_08/id_inc)
The rewrite relation of the following TRS is considered.
|
f(s(x)) |
→ |
f(id_inc(c(x,x))) |
(1) |
|
f(c(s(x),y)) |
→ |
g(c(x,y)) |
(2) |
|
g(c(s(x),y)) |
→ |
g(c(y,x)) |
(3) |
|
g(c(x,s(y))) |
→ |
g(c(y,x)) |
(4) |
|
g(c(x,x)) |
→ |
f(x) |
(5) |
|
id_inc(c(x,y)) |
→ |
c(id_inc(x),id_inc(y)) |
(6) |
|
id_inc(s(x)) |
→ |
s(id_inc(x)) |
(7) |
|
id_inc(0) |
→ |
0 |
(8) |
|
id_inc(0) |
→ |
s(0) |
(9) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
f#(s(x)) |
→ |
f#(id_inc(c(x,x))) |
(10) |
|
f#(s(x)) |
→ |
id_inc#(c(x,x)) |
(11) |
|
f#(c(s(x),y)) |
→ |
g#(c(x,y)) |
(12) |
|
g#(c(s(x),y)) |
→ |
g#(c(y,x)) |
(13) |
|
g#(c(x,s(y))) |
→ |
g#(c(y,x)) |
(14) |
|
g#(c(x,x)) |
→ |
f#(x) |
(15) |
|
id_inc#(c(x,y)) |
→ |
id_inc#(x) |
(16) |
|
id_inc#(c(x,y)) |
→ |
id_inc#(y) |
(17) |
|
id_inc#(s(x)) |
→ |
id_inc#(x) |
(18) |
1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.
-
The
1st
component contains the
pair
|
f#(c(s(x),y)) |
→ |
g#(c(x,y)) |
(12) |
|
g#(c(s(x),y)) |
→ |
g#(c(y,x)) |
(13) |
|
g#(c(x,s(y))) |
→ |
g#(c(y,x)) |
(14) |
|
g#(c(x,x)) |
→ |
f#(x) |
(15) |
|
f#(s(x)) |
→ |
f#(id_inc(c(x,x))) |
(10) |
1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [id_inc(x1)] |
= |
1 · x1
|
| [c(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [s(x1)] |
= |
2 · x1
|
| [0] |
= |
0 |
| [f#(x1)] |
= |
2 · x1
|
| [g#(x1)] |
= |
1 · x1
|
together with the usable
rules
|
id_inc(c(x,y)) |
→ |
c(id_inc(x),id_inc(y)) |
(6) |
|
id_inc(s(x)) |
→ |
s(id_inc(x)) |
(7) |
|
id_inc(0) |
→ |
0 |
(8) |
|
id_inc(0) |
→ |
s(0) |
(9) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the rationals with delta = 1/4
| [f#(x1)] |
= |
0 + 2 · x1
|
| [c(x1, x2)] |
= |
0 + 1/2 · x1 + 1/2 · x2
|
| [s(x1)] |
= |
1/4 + 1 · x1
|
| [g#(x1)] |
= |
1/4 + 2 · x1
|
| [id_inc(x1)] |
= |
1/4 + 1 · x1
|
| [0] |
= |
0 |
the
pairs
|
g#(c(s(x),y)) |
→ |
g#(c(y,x)) |
(13) |
|
g#(c(x,s(y))) |
→ |
g#(c(y,x)) |
(14) |
|
g#(c(x,x)) |
→ |
f#(x) |
(15) |
could be deleted.
1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
2nd
component contains the
pair
|
id_inc#(c(x,y)) |
→ |
id_inc#(y) |
(17) |
|
id_inc#(c(x,y)) |
→ |
id_inc#(x) |
(16) |
|
id_inc#(s(x)) |
→ |
id_inc#(x) |
(18) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [c(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [s(x1)] |
= |
1 · x1
|
| [id_inc#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
id_inc#(c(x,y)) |
→ |
id_inc#(y) |
(17) |
|
| 1 |
> |
1 |
|
id_inc#(c(x,y)) |
→ |
id_inc#(x) |
(16) |
|
| 1 |
> |
1 |
|
id_inc#(s(x)) |
→ |
id_inc#(x) |
(18) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.