The rewrite relation of the following TRS is considered.
The dependency pairs are split into 4
components.
-
The
1st
component contains the
pair
|
serve'#(new(x)) |
→ |
serve'#(free(x)) |
(26) |
1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [free(x1)] |
= |
1 · x1
|
| [top(x1)] |
= |
1 · x1
|
| [new(x1)] |
= |
1 · x1
|
| [check(x1)] |
= |
1 · x1
|
| [old(x1)] |
= |
1 · x1
|
| [serve'#(x1)] |
= |
1 · x1
|
together with the usable
rules
|
free(top(x)) |
→ |
new(check(top(x))) |
(12) |
|
free(new(x)) |
→ |
new(free(x)) |
(13) |
|
free(old(x)) |
→ |
old(free(x)) |
(14) |
|
free(check(x)) |
→ |
check(free(x)) |
(17) |
|
old(check(x)) |
→ |
check(old(x)) |
(19) |
|
old(check(x)) |
→ |
old(x) |
(20) |
|
new(check(x)) |
→ |
check(new(x)) |
(18) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [serve'#(x1)] |
= |
1 · x1
|
| [new(x1)] |
= |
1 + 1 · x1
|
| [free(x1)] |
= |
1 · x1
|
| [top(x1)] |
= |
1 + 1 · x1
|
| [check(x1)] |
= |
0 |
| [old(x1)] |
= |
1 |
the
pair
|
serve'#(new(x)) |
→ |
serve'#(free(x)) |
(26) |
could be deleted.
1.1.1.1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
|
free#(old(x)) |
→ |
free#(x) |
(25) |
|
free#(new(x)) |
→ |
free#(x) |
(23) |
|
free#(check(x)) |
→ |
free#(x) |
(28) |
1.1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [old(x1)] |
= |
1 · x1
|
| [new(x1)] |
= |
1 · x1
|
| [check(x1)] |
= |
1 · x1
|
| [free#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
free#(old(x)) |
→ |
free#(x) |
(25) |
|
| 1 |
> |
1 |
|
free#(new(x)) |
→ |
free#(x) |
(23) |
|
| 1 |
> |
1 |
|
free#(check(x)) |
→ |
free#(x) |
(28) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
|
new#(check(x)) |
→ |
new#(x) |
(29) |
1.1.1.1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [check(x1)] |
= |
1 · x1
|
| [new#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.3.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
new#(check(x)) |
→ |
new#(x) |
(29) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
|
old#(check(x)) |
→ |
old#(x) |
(30) |
1.1.1.1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [check(x1)] |
= |
1 · x1
|
| [old#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.4.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
old#(check(x)) |
→ |
old#(x) |
(30) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.