The rewrite relation of the following TRS is considered.
The dependency pairs are split into 3
components.
-
The
1st
component contains the
pair
|
plus#(s(x),y) |
→ |
plus#(x,s(y)) |
(13) |
|
plus#(s(x),y) |
→ |
plus#(x,y) |
(12) |
|
plus#(s(x),y) |
→ |
plus#(minus(x,y),double(y)) |
(14) |
|
plus#(s(plus(x,y)),z) |
→ |
plus#(plus(x,y),z) |
(17) |
1.1.1 Reduction Pair Processor
Using the
| prec(plus#) |
= |
0 |
|
stat(plus#) |
= |
lex
|
| prec(s) |
= |
0 |
|
stat(s) |
= |
lex
|
| prec(double) |
= |
1 |
|
stat(double) |
= |
lex
|
| prec(plus) |
= |
2 |
|
stat(plus) |
= |
lex
|
| prec(0) |
= |
1 |
|
stat(0) |
= |
lex
|
| π(plus#) |
= |
[1] |
| π(s) |
= |
[1] |
| π(minus) |
= |
1 |
| π(double) |
= |
[1] |
| π(plus) |
= |
[1,2] |
| π(0) |
= |
[] |
the
pairs
|
plus#(s(x),y) |
→ |
plus#(x,s(y)) |
(13) |
|
plus#(s(x),y) |
→ |
plus#(x,y) |
(12) |
|
plus#(s(x),y) |
→ |
plus#(minus(x,y),double(y)) |
(14) |
|
plus#(s(plus(x,y)),z) |
→ |
plus#(plus(x,y),z) |
(17) |
could be deleted.
1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
|
minus#(s(x),s(y)) |
→ |
minus#(x,y) |
(10) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [s(x1)] |
= |
1 · x1
|
| [minus#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
having no usable rules (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.
|
minus#(s(x),s(y)) |
→ |
minus#(x,y) |
(10) |
|
|
| 1 |
> |
1 |
| 2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
|
double#(s(x)) |
→ |
double#(x) |
(11) |
1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [s(x1)] |
= |
1 · x1
|
| [double#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.3.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
double#(s(x)) |
→ |
double#(x) |
(11) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.