Certification Problem
Input (TPDB TRS_Standard/AG01/#3.19)
The rewrite relation of the following TRS is considered.
|
minus(x,0) |
→ |
x |
(1) |
|
minus(s(x),s(y)) |
→ |
minus(x,y) |
(2) |
|
double(0) |
→ |
0 |
(3) |
|
double(s(x)) |
→ |
s(s(double(x))) |
(4) |
|
plus(0,y) |
→ |
y |
(5) |
|
plus(s(x),y) |
→ |
s(plus(x,y)) |
(6) |
|
plus(s(x),y) |
→ |
plus(x,s(y)) |
(7) |
|
plus(s(x),y) |
→ |
s(plus(minus(x,y),double(y))) |
(8) |
|
plus(s(plus(x,y)),z) |
→ |
s(plus(plus(x,y),z)) |
(9) |
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.
|
plus#(s(x),y) |
→ |
plus#(x,y) |
(10) |
|
plus#(s(x),y) |
→ |
minus#(x,y) |
(11) |
|
plus#(s(x),y) |
→ |
double#(y) |
(12) |
|
plus#(s(x),y) |
→ |
plus#(minus(x,y),double(y)) |
(13) |
|
plus#(s(plus(x,y)),z) |
→ |
plus#(plus(x,y),z) |
(14) |
|
minus#(s(x),s(y)) |
→ |
minus#(x,y) |
(15) |
|
double#(s(x)) |
→ |
double#(x) |
(16) |
|
plus#(s(x),y) |
→ |
plus#(x,s(y)) |
(17) |
1.1 Dependency Graph Processor
The dependency pairs are split into 3
components.
-
The
1st
component contains the
pair
|
plus#(s(x),y) |
→ |
plus#(x,s(y)) |
(17) |
|
plus#(s(plus(x,y)),z) |
→ |
plus#(plus(x,y),z) |
(14) |
|
plus#(s(x),y) |
→ |
plus#(minus(x,y),double(y)) |
(13) |
|
plus#(s(x),y) |
→ |
plus#(x,y) |
(10) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the argument filter in combination with the following Weighted Path Order with the following precedence and status
| prec(s) |
= |
1 |
|
status(s) |
= |
[1] |
|
list-extension(s) |
= |
Lex |
| prec(minus) |
= |
0 |
|
status(minus) |
= |
[1] |
|
list-extension(minus) |
= |
Lex |
| prec(plus#) |
= |
0 |
|
status(plus#) |
= |
[1] |
|
list-extension(plus#) |
= |
Lex |
| prec(0) |
= |
2 |
|
status(0) |
= |
[] |
|
list-extension(0) |
= |
Lex |
| prec(double#) |
= |
0 |
|
status(double#) |
= |
[] |
|
list-extension(double#) |
= |
Lex |
| prec(double) |
= |
2 |
|
status(double) |
= |
[1] |
|
list-extension(double) |
= |
Lex |
| prec(plus) |
= |
3 |
|
status(plus) |
= |
[1, 2] |
|
list-extension(plus) |
= |
Lex |
and the following
Max-polynomial interpretation
| [s(x1)] |
=
|
x1 + 0 |
| [minus(x1, x2)] |
=
|
max(x1 + 0, 0) |
| [plus#(x1, x2)] |
=
|
max(x1 + 0, 0) |
| [0] |
=
|
0 |
| [double#(x1)] |
=
|
0 |
| [double(x1)] |
=
|
x1 + 0 |
| [plus(x1, x2)] |
=
|
max(x1 + 0, x2 + 0, 0) |
together with the usable
rules
|
double(s(x)) |
→ |
s(s(double(x))) |
(4) |
|
plus(s(x),y) |
→ |
s(plus(minus(x,y),double(y))) |
(8) |
|
minus(x,0) |
→ |
x |
(1) |
|
double(0) |
→ |
0 |
(3) |
|
plus(0,y) |
→ |
y |
(5) |
|
plus(s(x),y) |
→ |
plus(x,s(y)) |
(7) |
|
plus(s(plus(x,y)),z) |
→ |
s(plus(plus(x,y),z)) |
(9) |
|
plus(s(x),y) |
→ |
s(plus(x,y)) |
(6) |
|
minus(s(x),s(y)) |
→ |
minus(x,y) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
plus#(s(x),y) |
→ |
plus#(x,s(y)) |
(17) |
|
plus#(s(plus(x,y)),z) |
→ |
plus#(plus(x,y),z) |
(14) |
|
plus#(s(x),y) |
→ |
plus#(minus(x,y),double(y)) |
(13) |
|
plus#(s(x),y) |
→ |
plus#(x,y) |
(10) |
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#(s(x),s(y)) |
→ |
minus#(x,y) |
(15) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [s(x1)] |
=
|
x1 + 1 |
| [minus(x1, x2)] |
=
|
x1 + x2 + 44022 |
| [plus#(x1, x2)] |
=
|
0 |
| [0] |
=
|
21681 |
| [double#(x1)] |
=
|
0 |
| [double(x1)] |
=
|
21680 |
| [minus#(x1, x2)] |
=
|
x2 + 0 |
| [plus(x1, x2)] |
=
|
x1 + x2 + 9227 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
|
minus#(s(x),s(y)) |
→ |
minus#(x,y) |
(15) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
|
double#(s(x)) |
→ |
double#(x) |
(16) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [s(x1)] |
=
|
x1 + 1 |
| [minus(x1, x2)] |
=
|
x1 + x2 + 96608 |
| [plus#(x1, x2)] |
=
|
0 |
| [0] |
=
|
2 |
| [double#(x1)] |
=
|
x1 + 0 |
| [double(x1)] |
=
|
1 |
| [minus#(x1, x2)] |
=
|
0 |
| [plus(x1, x2)] |
=
|
x1 + x2 + 52868 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
|
double#(s(x)) |
→ |
double#(x) |
(16) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.