The rewrite relation of the following TRS is considered.
The dependency pairs are split into 5
components.
-
The
1st
component contains the
pair
|
bot'#(rec(x)) |
→ |
bot'#(sent(up(x))) |
(34) |
1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [up(x1)] |
= |
1 · x1
|
| [rec(x1)] |
= |
1 · x1
|
| [sent(x1)] |
= |
1 · x1
|
| [no(x1)] |
= |
1 · x1
|
| [top(x1)] |
= |
1 · x1
|
| [check(x1)] |
= |
1 · x1
|
| [bot'#(x1)] |
= |
1 · x1
|
together with the usable
rules
|
up(rec(x)) |
→ |
rec(up(x)) |
(21) |
|
up(sent(x)) |
→ |
sent(up(x)) |
(22) |
|
up(no(x)) |
→ |
no(up(x)) |
(23) |
|
up(sent(top(x))) |
→ |
rec(check(top(x))) |
(25) |
|
up(check(x)) |
→ |
check(up(x)) |
(27) |
|
sent(rec(x)) |
→ |
rec(sent(x)) |
(18) |
|
sent(check(x)) |
→ |
check(sent(x)) |
(28) |
|
rec(check(x)) |
→ |
check(rec(x)) |
(29) |
|
no(check(x)) |
→ |
check(no(x)) |
(30) |
|
no(check(x)) |
→ |
no(x) |
(31) |
(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
| [bot'#(x1)] |
= |
1 · x1
|
| [rec(x1)] |
= |
1 + 1 · x1
|
| [sent(x1)] |
= |
1 · x1
|
| [up(x1)] |
= |
1 · x1
|
| [no(x1)] |
= |
1 |
| [top(x1)] |
= |
1 + 1 · x1
|
| [check(x1)] |
= |
0 |
the
pair
|
bot'#(rec(x)) |
→ |
bot'#(sent(up(x))) |
(34) |
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
|
up#(sent(x)) |
→ |
up#(x) |
(40) |
|
up#(rec(x)) |
→ |
up#(x) |
(38) |
|
up#(no(x)) |
→ |
up#(x) |
(42) |
|
up#(check(x)) |
→ |
up#(x) |
(44) |
1.1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [sent(x1)] |
= |
1 · x1
|
| [rec(x1)] |
= |
1 · x1
|
| [no(x1)] |
= |
1 · x1
|
| [check(x1)] |
= |
1 · x1
|
| [up#(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.
|
up#(sent(x)) |
→ |
up#(x) |
(40) |
|
| 1 |
> |
1 |
|
up#(rec(x)) |
→ |
up#(x) |
(38) |
|
| 1 |
> |
1 |
|
up#(no(x)) |
→ |
up#(x) |
(42) |
|
| 1 |
> |
1 |
|
up#(check(x)) |
→ |
up#(x) |
(44) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
|
sent#(check(x)) |
→ |
sent#(x) |
(45) |
|
sent#(rec(x)) |
→ |
sent#(x) |
(33) |
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
|
| [rec(x1)] |
= |
1 · x1
|
| [sent#(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.
|
sent#(check(x)) |
→ |
sent#(x) |
(45) |
|
| 1 |
> |
1 |
|
sent#(rec(x)) |
→ |
sent#(x) |
(33) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
|
rec#(check(x)) |
→ |
rec#(x) |
(46) |
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
|
| [rec#(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.
|
rec#(check(x)) |
→ |
rec#(x) |
(46) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
|
no#(check(x)) |
→ |
no#(x) |
(47) |
1.1.1.1.1.5 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [check(x1)] |
= |
1 · x1
|
| [no#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.5.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
no#(check(x)) |
→ |
no#(x) |
(47) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.