The rewrite relation of the following TRS is considered.
The dependency pairs are split into 3
components.
-
The
1st
component contains the
pair
|
ifb#(true,x,y) |
→ |
help#(half(x),s(y)) |
(20) |
|
help#(x,y) |
→ |
ifb#(lt(y,x),x,y) |
(18) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the arctic semiring over the integers
| [false] |
= |
1 |
| [help#(x1, x2)] |
= |
-1 · x1 +
-∞ · x2 + 0 |
| [s(x1)] |
= |
2 · x1 + 2 |
| [true] |
= |
2 |
| [ifb#(x1, x2, x3)] |
= |
-1 · x1 + -1 · x2 +
-∞ · x3 + 0 |
| [0] |
= |
0 |
| [half(x1)] |
= |
-1 · x1 + 0 |
| [lt(x1, x2)] |
= |
-∞ · x1 + 0 · x2 + 1 |
together with the usable
rules
|
half(0) |
→ |
0 |
(10) |
|
half(s(0)) |
→ |
0 |
(11) |
|
half(s(s(x))) |
→ |
s(half(x)) |
(12) |
|
lt(0,s(x)) |
→ |
true |
(1) |
|
lt(x,0) |
→ |
false |
(2) |
|
lt(s(x),s(y)) |
→ |
lt(x,y) |
(3) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
ifb#(true,x,y) |
→ |
help#(half(x),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
|
lt#(s(x),s(y)) |
→ |
lt#(x,y) |
(13) |
1.1.2 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
lt#(s(x),s(y)) |
→ |
lt#(x,y) |
(13) |
|
|
| 2 |
> |
2 |
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
|
half#(s(s(x))) |
→ |
half#(x) |
(21) |
1.1.3 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
half#(s(s(x))) |
→ |
half#(x) |
(21) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.