The rewrite relation of the following TRS is considered.
The dependency pairs are split into 5
components.
-
The
1st
component contains the
pair
|
b#(r(x1)) |
→ |
b#(s(u(x1))) |
(33) |
1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [u(x1)] |
= |
1 · x1
|
| [r(x1)] |
= |
1 · x1
|
| [s(x1)] |
= |
1 · x1
|
| [n(x1)] |
= |
1 · x1
|
| [t(x1)] |
= |
1 · x1
|
| [c(x1)] |
= |
1 · x1
|
| [b#(x1)] |
= |
1 · x1
|
together with the usable
rules
|
u(r(x1)) |
→ |
r(u(x1)) |
(20) |
|
u(s(x1)) |
→ |
s(u(x1)) |
(21) |
|
u(n(x1)) |
→ |
n(u(x1)) |
(22) |
|
u(s(t(x1))) |
→ |
r(c(t(x1))) |
(24) |
|
u(c(x1)) |
→ |
c(u(x1)) |
(26) |
|
s(r(x1)) |
→ |
r(s(x1)) |
(17) |
|
s(c(x1)) |
→ |
c(s(x1)) |
(27) |
|
r(c(x1)) |
→ |
c(r(x1)) |
(28) |
|
n(c(x1)) |
→ |
c(n(x1)) |
(29) |
|
n(c(x1)) |
→ |
n(x1) |
(30) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [b#(x1)] |
= |
1 · x1
|
| [r(x1)] |
= |
1 + 1 · x1
|
| [s(x1)] |
= |
1 · x1
|
| [u(x1)] |
= |
1 · x1
|
| [n(x1)] |
= |
1 |
| [t(x1)] |
= |
1 + 1 · x1
|
| [c(x1)] |
= |
0 |
the
pair
|
b#(r(x1)) |
→ |
b#(s(u(x1))) |
(33) |
could be deleted.
1.1.1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
|
u#(s(x1)) |
→ |
u#(x1) |
(39) |
|
u#(r(x1)) |
→ |
u#(x1) |
(37) |
|
u#(n(x1)) |
→ |
u#(x1) |
(41) |
|
u#(c(x1)) |
→ |
u#(x1) |
(43) |
1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [s(x1)] |
= |
1 · x1
|
| [r(x1)] |
= |
1 · x1
|
| [n(x1)] |
= |
1 · x1
|
| [c(x1)] |
= |
1 · x1
|
| [u#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
u#(s(x1)) |
→ |
u#(x1) |
(39) |
|
| 1 |
> |
1 |
|
u#(r(x1)) |
→ |
u#(x1) |
(37) |
|
| 1 |
> |
1 |
|
u#(n(x1)) |
→ |
u#(x1) |
(41) |
|
| 1 |
> |
1 |
|
u#(c(x1)) |
→ |
u#(x1) |
(43) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
|
s#(c(x1)) |
→ |
s#(x1) |
(44) |
|
s#(r(x1)) |
→ |
s#(x1) |
(32) |
1.1.1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [c(x1)] |
= |
1 · x1
|
| [r(x1)] |
= |
1 · x1
|
| [s#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.3.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
s#(c(x1)) |
→ |
s#(x1) |
(44) |
|
| 1 |
> |
1 |
|
s#(r(x1)) |
→ |
s#(x1) |
(32) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
1.1.1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [c(x1)] |
= |
1 · x1
|
| [r#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.4.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
r#(c(x1)) |
→ |
r#(x1) |
(45) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
1.1.1.1.5 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [c(x1)] |
= |
1 · x1
|
| [n#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.5.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
n#(c(x1)) |
→ |
n#(x1) |
(46) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.