Certification Problem
Input (TPDB TRS_Standard/SK90/2.22)
The rewrite relation of the following TRS is considered.
|
exp(x,0) |
→ |
s(0) |
(1) |
|
exp(x,s(y)) |
→ |
*(x,exp(x,y)) |
(2) |
|
*(0,y) |
→ |
0 |
(3) |
|
*(s(x),y) |
→ |
+(y,*(x,y)) |
(4) |
|
-(0,y) |
→ |
0 |
(5) |
|
-(x,0) |
→ |
x |
(6) |
|
-(s(x),s(y)) |
→ |
-(x,y) |
(7) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by NaTT @ termCOMP 2023)
1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
exp#(x,s(y)) |
→ |
exp#(x,y) |
(8) |
|
-#(s(x),s(y)) |
→ |
-#(x,y) |
(9) |
|
exp#(x,s(y)) |
→ |
*#(x,exp(x,y)) |
(10) |
|
*#(s(x),y) |
→ |
*#(x,y) |
(11) |
1.1 Dependency Graph Processor
The dependency pairs are split into 3
components.
-
The
1st
component contains the
pair
|
-#(s(x),s(y)) |
→ |
-#(x,y) |
(9) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [s(x1)] |
=
|
x1 + 1 |
| [*#(x1, x2)] |
=
|
0 |
| [exp#(x1, x2)] |
=
|
0 |
| [0] |
=
|
0 |
| [-(x1, x2)] |
=
|
0 |
| [-#(x1, x2)] |
=
|
x1 + x2 + 0 |
| [exp(x1, x2)] |
=
|
0 |
| [+(x1, x2)] |
=
|
0 |
| [*(x1, x2)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
|
-#(s(x),s(y)) |
→ |
-#(x,y) |
(9) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
2nd
component contains the
pair
|
exp#(x,s(y)) |
→ |
exp#(x,y) |
(8) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [s(x1)] |
=
|
x1 + 1 |
| [*#(x1, x2)] |
=
|
0 |
| [exp#(x1, x2)] |
=
|
x2 + 0 |
| [0] |
=
|
0 |
| [-(x1, x2)] |
=
|
0 |
| [-#(x1, x2)] |
=
|
0 |
| [exp(x1, x2)] |
=
|
0 |
| [+(x1, x2)] |
=
|
0 |
| [*(x1, x2)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
|
exp#(x,s(y)) |
→ |
exp#(x,y) |
(8) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
|
*#(s(x),y) |
→ |
*#(x,y) |
(11) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [s(x1)] |
=
|
x1 + 1 |
| [*#(x1, x2)] |
=
|
x1 + 0 |
| [exp#(x1, x2)] |
=
|
0 |
| [0] |
=
|
0 |
| [-(x1, x2)] |
=
|
0 |
| [-#(x1, x2)] |
=
|
0 |
| [exp(x1, x2)] |
=
|
0 |
| [+(x1, x2)] |
=
|
0 |
| [*(x1, x2)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
|
*#(s(x),y) |
→ |
*#(x,y) |
(11) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.