The rewrite relation of the following TRS is considered.
The dependency pairs are split into 5
components.
-
The
1st
component contains the
pair
|
-#(s(x),s(y)) |
→ |
-#(x,y) |
(18) |
1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
-#(s(x),s(y)) |
→ |
-#(x,y) |
(18) |
|
|
| 2 |
> |
2 |
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
|
f#(x,s(y),z) |
→ |
f#(*(x,x),half(s(y)),z) |
(25) |
|
f#(x,s(y),z) |
→ |
f#(x,y,*(x,z)) |
(27) |
1.1.2 Subterm Criterion Processor
We use the projection to multisets
| π(f#)
|
= |
{
2, 2
}
|
| π(half)
|
= |
{
1
}
|
| π(+)
|
= |
{
1, 1, 1
}
|
| π(s)
|
= |
{
1, 1
}
|
to remove the pairs:
|
f#(x,s(y),z) |
→ |
f#(x,y,*(x,z)) |
(27) |
1.1.2.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the rationals with delta = 1/64
| [f#(x1, x2, x3)] |
= |
1 · x1 + 5/2 · x2 + 0 · x3 + 1 |
| [+(x1, x2)] |
= |
1 · x1 + 0 · x2 + 0 |
| [*(x1, x2)] |
= |
0 · x1 + 0 · x2 + 1 |
| [0] |
= |
0 |
| [half(x1)] |
= |
1/2 · x1 + 0 |
| [s(x1)] |
= |
2 · x1 + 2 |
together with the usable
rules
|
half(s(0)) |
→ |
0 |
(11) |
|
half(s(s(x))) |
→ |
s(half(x)) |
(12) |
|
half(0) |
→ |
0 |
(10) |
|
*(x,0) |
→ |
0 |
(3) |
|
*(x,s(y)) |
→ |
+(*(x,y),x) |
(4) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
f#(x,s(y),z) |
→ |
f#(*(x,x),half(s(y)),z) |
(25) |
could be deleted.
1.1.2.1.1 P is empty
There are no pairs anymore.
-
The
3rd
component contains the
pair
|
odd#(s(s(x))) |
→ |
odd#(x) |
(20) |
1.1.3 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
odd#(s(s(x))) |
→ |
odd#(x) |
(20) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
|
*#(x,s(y)) |
→ |
*#(x,y) |
(19) |
1.1.4 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
*#(x,s(y)) |
→ |
*#(x,y) |
(19) |
|
|
| 2 |
> |
2 |
| 1 |
≥ |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
|
half#(s(s(x))) |
→ |
half#(x) |
(21) |
1.1.5 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.