Certification Problem
Input (TPDB TRS_Standard/HirokawaMiddeldorp_04/t012)
The rewrite relation of the following TRS is considered.
|
minus(minus(x)) |
→ |
x |
(1) |
|
minus(+(x,y)) |
→ |
*(minus(minus(minus(x))),minus(minus(minus(y)))) |
(2) |
|
minus(*(x,y)) |
→ |
+(minus(minus(minus(x))),minus(minus(minus(y)))) |
(3) |
|
f(minus(x)) |
→ |
minus(minus(minus(f(x)))) |
(4) |
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.
|
minus#(+(x,y)) |
→ |
minus#(minus(minus(x))) |
(5) |
|
minus#(+(x,y)) |
→ |
minus#(minus(x)) |
(6) |
|
minus#(+(x,y)) |
→ |
minus#(x) |
(7) |
|
minus#(+(x,y)) |
→ |
minus#(minus(minus(y))) |
(8) |
|
minus#(+(x,y)) |
→ |
minus#(minus(y)) |
(9) |
|
minus#(+(x,y)) |
→ |
minus#(y) |
(10) |
|
minus#(*(x,y)) |
→ |
minus#(minus(minus(x))) |
(11) |
|
minus#(*(x,y)) |
→ |
minus#(minus(x)) |
(12) |
|
minus#(*(x,y)) |
→ |
minus#(x) |
(13) |
|
minus#(*(x,y)) |
→ |
minus#(minus(minus(y))) |
(14) |
|
minus#(*(x,y)) |
→ |
minus#(minus(y)) |
(15) |
|
minus#(*(x,y)) |
→ |
minus#(y) |
(16) |
|
f#(minus(x)) |
→ |
minus#(minus(minus(f(x)))) |
(17) |
|
f#(minus(x)) |
→ |
minus#(minus(f(x))) |
(18) |
|
f#(minus(x)) |
→ |
minus#(f(x)) |
(19) |
|
f#(minus(x)) |
→ |
f#(x) |
(20) |
1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.
-
The
1st
component contains the
pair
|
f#(minus(x)) |
→ |
f#(x) |
(20) |
1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [minus(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 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
f#(minus(x)) |
→ |
f#(x) |
(20) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
|
minus#(+(x,y)) |
→ |
minus#(minus(x)) |
(6) |
|
minus#(+(x,y)) |
→ |
minus#(minus(minus(x))) |
(5) |
|
minus#(+(x,y)) |
→ |
minus#(x) |
(7) |
|
minus#(+(x,y)) |
→ |
minus#(minus(minus(y))) |
(8) |
|
minus#(+(x,y)) |
→ |
minus#(minus(y)) |
(9) |
|
minus#(+(x,y)) |
→ |
minus#(y) |
(10) |
|
minus#(*(x,y)) |
→ |
minus#(minus(minus(x))) |
(11) |
|
minus#(*(x,y)) |
→ |
minus#(minus(x)) |
(12) |
|
minus#(*(x,y)) |
→ |
minus#(x) |
(13) |
|
minus#(*(x,y)) |
→ |
minus#(minus(minus(y))) |
(14) |
|
minus#(*(x,y)) |
→ |
minus#(minus(y)) |
(15) |
|
minus#(*(x,y)) |
→ |
minus#(y) |
(16) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [minus(x1)] |
= |
1 · x1
|
| [+(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [*(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [minus#(x1)] |
= |
1 · x1
|
together with the usable
rules
|
minus(minus(x)) |
→ |
x |
(1) |
|
minus(+(x,y)) |
→ |
*(minus(minus(minus(x))),minus(minus(minus(y)))) |
(2) |
|
minus(*(x,y)) |
→ |
+(minus(minus(minus(x))),minus(minus(minus(y)))) |
(3) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.2.1 Monotonic Reduction Pair Processor
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
| prec(minus) |
= |
3 |
|
weight(minus) |
= |
0 |
|
|
|
| prec(minus#) |
= |
1 |
|
weight(minus#) |
= |
1 |
|
|
|
| prec(+) |
= |
0 |
|
weight(+) |
= |
0 |
|
|
|
| prec(*) |
= |
2 |
|
weight(*) |
= |
0 |
|
|
|
the
pairs
|
minus#(+(x,y)) |
→ |
minus#(minus(x)) |
(6) |
|
minus#(+(x,y)) |
→ |
minus#(minus(minus(x))) |
(5) |
|
minus#(+(x,y)) |
→ |
minus#(x) |
(7) |
|
minus#(+(x,y)) |
→ |
minus#(minus(minus(y))) |
(8) |
|
minus#(+(x,y)) |
→ |
minus#(minus(y)) |
(9) |
|
minus#(+(x,y)) |
→ |
minus#(y) |
(10) |
|
minus#(*(x,y)) |
→ |
minus#(minus(minus(x))) |
(11) |
|
minus#(*(x,y)) |
→ |
minus#(minus(x)) |
(12) |
|
minus#(*(x,y)) |
→ |
minus#(x) |
(13) |
|
minus#(*(x,y)) |
→ |
minus#(minus(minus(y))) |
(14) |
|
minus#(*(x,y)) |
→ |
minus#(minus(y)) |
(15) |
|
minus#(*(x,y)) |
→ |
minus#(y) |
(16) |
and
the
rules
|
minus(minus(x)) |
→ |
x |
(1) |
|
minus(+(x,y)) |
→ |
*(minus(minus(minus(x))),minus(minus(minus(y)))) |
(2) |
|
minus(*(x,y)) |
→ |
+(minus(minus(minus(x))),minus(minus(minus(y)))) |
(3) |
could be deleted.
1.1.2.1.1 P is empty
There are no pairs anymore.