Certification Problem
Input (TPDB TRS_Standard/Der95/13)
The rewrite relation of the following TRS is considered.
|
*(x,+(y,z)) |
→ |
+(*(x,y),*(x,z)) |
(1) |
|
*(+(y,z),x) |
→ |
+(*(x,y),*(x,z)) |
(2) |
|
*(*(x,y),z) |
→ |
*(x,*(y,z)) |
(3) |
|
+(+(x,y),z) |
→ |
+(x,+(y,z)) |
(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.
|
*#(x,+(y,z)) |
→ |
+#(*(x,y),*(x,z)) |
(5) |
|
*#(x,+(y,z)) |
→ |
*#(x,y) |
(6) |
|
*#(x,+(y,z)) |
→ |
*#(x,z) |
(7) |
|
*#(+(y,z),x) |
→ |
+#(*(x,y),*(x,z)) |
(8) |
|
*#(+(y,z),x) |
→ |
*#(x,y) |
(9) |
|
*#(+(y,z),x) |
→ |
*#(x,z) |
(10) |
|
*#(*(x,y),z) |
→ |
*#(x,*(y,z)) |
(11) |
|
*#(*(x,y),z) |
→ |
*#(y,z) |
(12) |
|
+#(+(x,y),z) |
→ |
+#(x,+(y,z)) |
(13) |
|
+#(+(x,y),z) |
→ |
+#(y,z) |
(14) |
1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.
-
The
1st
component contains the
pair
|
*#(x,+(y,z)) |
→ |
*#(x,z) |
(7) |
|
*#(x,+(y,z)) |
→ |
*#(x,y) |
(6) |
|
*#(+(y,z),x) |
→ |
*#(x,y) |
(9) |
|
*#(+(y,z),x) |
→ |
*#(x,z) |
(10) |
|
*#(*(x,y),z) |
→ |
*#(x,*(y,z)) |
(11) |
|
*#(*(x,y),z) |
→ |
*#(y,z) |
(12) |
1.1.1 Reduction Pair Processor
Using the non-linear polynomial interpretation over the naturals
| [*#(x1, x2)] |
= |
1 · x2 + 1 · x1 + 1 · x1 · x2
|
| [+(x1, x2)] |
= |
1 + 1 · x2 + 1 · x1
|
| [*(x1, x2)] |
= |
1 · x2 + 1 · x1 + 1 · x1 · x2
|
the
pairs
|
*#(x,+(y,z)) |
→ |
*#(x,z) |
(7) |
|
*#(x,+(y,z)) |
→ |
*#(x,y) |
(6) |
|
*#(+(y,z),x) |
→ |
*#(x,y) |
(9) |
|
*#(+(y,z),x) |
→ |
*#(x,z) |
(10) |
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.
|
*#(*(x,y),z) |
→ |
*#(x,*(y,z)) |
(11) |
|
| 1 |
> |
1 |
|
*#(*(x,y),z) |
→ |
*#(y,z) |
(12) |
|
|
| 1 |
> |
1 |
| 2 |
≥ |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
|
+#(+(x,y),z) |
→ |
+#(y,z) |
(14) |
|
+#(+(x,y),z) |
→ |
+#(x,+(y,z)) |
(13) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [+(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [+#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
together with the usable
rule
|
+(+(x,y),z) |
→ |
+(x,+(y,z)) |
(4) |
(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.
|
+#(+(x,y),z) |
→ |
+#(y,z) |
(14) |
|
|
| 1 |
> |
1 |
| 2 |
≥ |
2 |
|
+#(+(x,y),z) |
→ |
+#(x,+(y,z)) |
(13) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.