Certification Problem
Input (TPDB TRS_Standard/SK90/4.14)
The rewrite relation of the following TRS is considered.
|
p(s(x)) |
→ |
x |
(1) |
|
s(p(x)) |
→ |
x |
(2) |
|
+(0,y) |
→ |
y |
(3) |
|
+(s(x),y) |
→ |
s(+(x,y)) |
(4) |
|
+(p(x),y) |
→ |
p(+(x,y)) |
(5) |
|
minus(0) |
→ |
0 |
(6) |
|
minus(s(x)) |
→ |
p(minus(x)) |
(7) |
|
minus(p(x)) |
→ |
s(minus(x)) |
(8) |
|
*(0,y) |
→ |
0 |
(9) |
|
*(s(x),y) |
→ |
+(*(x,y),y) |
(10) |
|
*(p(x),y) |
→ |
+(*(x,y),minus(y)) |
(11) |
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.
|
*#(p(x),y) |
→ |
*#(x,y) |
(12) |
|
*#(s(x),y) |
→ |
+#(*(x,y),y) |
(13) |
|
+#(s(x),y) |
→ |
+#(x,y) |
(14) |
|
+#(p(x),y) |
→ |
+#(x,y) |
(15) |
|
minus#(p(x)) |
→ |
minus#(x) |
(16) |
|
+#(s(x),y) |
→ |
s#(+(x,y)) |
(17) |
|
*#(s(x),y) |
→ |
*#(x,y) |
(18) |
|
minus#(p(x)) |
→ |
s#(minus(x)) |
(19) |
|
minus#(s(x)) |
→ |
minus#(x) |
(20) |
|
*#(p(x),y) |
→ |
minus#(y) |
(21) |
|
*#(p(x),y) |
→ |
+#(*(x,y),minus(y)) |
(22) |
|
+#(p(x),y) |
→ |
p#(+(x,y)) |
(23) |
|
minus#(s(x)) |
→ |
p#(minus(x)) |
(24) |
1.1 Dependency Graph Processor
The dependency pairs are split into 3
components.
-
The
1st
component contains the
pair
|
*#(s(x),y) |
→ |
*#(x,y) |
(18) |
|
*#(p(x),y) |
→ |
*#(x,y) |
(12) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [s(x1)] |
=
|
x1 + 1 |
| [minus(x1)] |
=
|
0 |
| [*#(x1, x2)] |
=
|
x1 + 0 |
| [p#(x1)] |
=
|
0 |
| [p(x1)] |
=
|
x1 + 1 |
| [0] |
=
|
0 |
| [s#(x1)] |
=
|
0 |
| [minus#(x1)] |
=
|
0 |
| [+(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
pairs
|
*#(s(x),y) |
→ |
*#(x,y) |
(18) |
|
*#(p(x),y) |
→ |
*#(x,y) |
(12) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
2nd
component contains the
pair
|
minus#(p(x)) |
→ |
minus#(x) |
(16) |
|
minus#(s(x)) |
→ |
minus#(x) |
(20) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [s(x1)] |
=
|
x1 + 1 |
| [minus(x1)] |
=
|
0 |
| [*#(x1, x2)] |
=
|
0 |
| [p#(x1)] |
=
|
0 |
| [p(x1)] |
=
|
x1 + 1 |
| [0] |
=
|
0 |
| [s#(x1)] |
=
|
0 |
| [minus#(x1)] |
=
|
x1 + 0 |
| [+(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
pairs
|
minus#(p(x)) |
→ |
minus#(x) |
(16) |
|
minus#(s(x)) |
→ |
minus#(x) |
(20) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
|
+#(p(x),y) |
→ |
+#(x,y) |
(15) |
|
+#(s(x),y) |
→ |
+#(x,y) |
(14) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [s(x1)] |
=
|
x1 + 1 |
| [minus(x1)] |
=
|
0 |
| [*#(x1, x2)] |
=
|
0 |
| [p#(x1)] |
=
|
0 |
| [p(x1)] |
=
|
x1 + 1 |
| [0] |
=
|
0 |
| [s#(x1)] |
=
|
0 |
| [minus#(x1)] |
=
|
0 |
| [+(x1, x2)] |
=
|
0 |
| [+#(x1, x2)] |
=
|
x1 + 0 |
| [*(x1, x2)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pairs
|
+#(p(x),y) |
→ |
+#(x,y) |
(15) |
|
+#(s(x),y) |
→ |
+#(x,y) |
(14) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.