The rewrite relation of the following TRS is considered.
The dependency pairs are split into 3
components.
-
The
1st
component contains the
pair
|
if_mod#(true,s(x),s(y)) |
→ |
mod#(minus(x,y),s(y)) |
(20) |
|
mod#(s(x),s(y)) |
→ |
if_mod#(le(y,x),s(x),s(y)) |
(18) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the
| prec(if_mod#) |
= |
0 |
|
stat(if_mod#) |
= |
lex
|
| prec(mod#) |
= |
0 |
|
stat(mod#) |
= |
lex
|
| prec(if_minus) |
= |
0 |
|
stat(if_minus) |
= |
lex
|
| prec(minus) |
= |
0 |
|
stat(minus) |
= |
lex
|
| prec(false) |
= |
0 |
|
stat(false) |
= |
lex
|
| prec(s) |
= |
8 |
|
stat(s) |
= |
lex
|
| prec(true) |
= |
0 |
|
stat(true) |
= |
lex
|
| prec(le) |
= |
0 |
|
stat(le) |
= |
lex
|
| prec(0) |
= |
0 |
|
stat(0) |
= |
lex
|
| π(if_mod#) |
= |
2 |
| π(mod#) |
= |
1 |
| π(if_minus) |
= |
2 |
| π(minus) |
= |
1 |
| π(false) |
= |
[] |
| π(s) |
= |
[1] |
| π(true) |
= |
[] |
| π(le) |
= |
[] |
| π(0) |
= |
[] |
together with the usable
rules
|
minus(0,y) |
→ |
0 |
(4) |
|
minus(s(x),y) |
→ |
if_minus(le(s(x),y),s(x),y) |
(5) |
|
if_minus(true,s(x),y) |
→ |
0 |
(6) |
|
if_minus(false,s(x),y) |
→ |
s(minus(x,y)) |
(7) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
if_mod#(true,s(x),s(y)) |
→ |
mod#(minus(x,y),s(y)) |
(20) |
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),y) |
→ |
if_minus#(le(s(x),y),s(x),y) |
(15) |
|
if_minus#(false,s(x),y) |
→ |
minus#(x,y) |
(16) |
1.1.2 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
minus#(s(x),y) |
→ |
if_minus#(le(s(x),y),s(x),y) |
(15) |
|
|
| 2 |
≥ |
3 |
| 1 |
≥ |
2 |
|
if_minus#(false,s(x),y) |
→ |
minus#(x,y) |
(16) |
|
|
| 3 |
≥ |
2 |
| 2 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
|
le#(s(x),s(y)) |
→ |
le#(x,y) |
(13) |
1.1.3 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
le#(s(x),s(y)) |
→ |
le#(x,y) |
(13) |
|
|
| 2 |
> |
2 |
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.