The rewrite relation of the following TRS is considered.
The dependency pairs are split into 4
components.
-
The
1st
component contains the
pair
|
s#(n(x1)) |
→ |
s#(f(x1)) |
(24) |
1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [f(x1)] |
= |
1 · x1
|
| [t(x1)] |
= |
1 · x1
|
| [n(x1)] |
= |
1 · x1
|
| [c(x1)] |
= |
1 · x1
|
| [o(x1)] |
= |
1 · x1
|
| [s#(x1)] |
= |
1 · x1
|
together with the usable
rules
|
f(t(x1)) |
→ |
n(c(t(x1))) |
(10) |
|
f(n(x1)) |
→ |
n(f(x1)) |
(11) |
|
f(o(x1)) |
→ |
o(f(x1)) |
(12) |
|
f(c(x1)) |
→ |
c(f(x1)) |
(15) |
|
o(c(x1)) |
→ |
c(o(x1)) |
(17) |
|
o(c(x1)) |
→ |
o(x1) |
(18) |
|
n(c(x1)) |
→ |
c(n(x1)) |
(16) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [f(x1)] |
= |
2 · x1
|
| [t(x1)] |
= |
1 · x1
|
| [n(x1)] |
= |
2 · x1
|
| [c(x1)] |
= |
1 · x1
|
| [o(x1)] |
= |
1 + 2 · x1
|
| [s#(x1)] |
= |
2 · x1
|
the
rule
could be deleted.
1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [f(x1)] |
= |
1 · x1
|
| [t(x1)] |
= |
1 · x1
|
| [n(x1)] |
= |
1 · x1
|
| [c(x1)] |
= |
1 · x1
|
| [s#(x1)] |
= |
1 · x1
|
together with the usable
rules
|
f(t(x1)) |
→ |
n(c(t(x1))) |
(10) |
|
f(n(x1)) |
→ |
n(f(x1)) |
(11) |
|
f(c(x1)) |
→ |
c(f(x1)) |
(15) |
|
n(c(x1)) |
→ |
c(n(x1)) |
(16) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [s#(x1)] |
= |
1 · x1
|
| [n(x1)] |
= |
1 + 1 · x1
|
| [f(x1)] |
= |
1 · x1
|
| [t(x1)] |
= |
1 + 1 · x1
|
| [c(x1)] |
= |
0 |
the
pair
|
s#(n(x1)) |
→ |
s#(f(x1)) |
(24) |
could be deleted.
1.1.1.1.1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
|
f#(o(x1)) |
→ |
f#(x1) |
(23) |
|
f#(n(x1)) |
→ |
f#(x1) |
(21) |
|
f#(c(x1)) |
→ |
f#(x1) |
(26) |
1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [o(x1)] |
= |
1 · x1
|
| [n(x1)] |
= |
1 · x1
|
| [c(x1)] |
= |
1 · x1
|
| [f#(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.
|
f#(o(x1)) |
→ |
f#(x1) |
(23) |
|
| 1 |
> |
1 |
|
f#(n(x1)) |
→ |
f#(x1) |
(21) |
|
| 1 |
> |
1 |
|
f#(c(x1)) |
→ |
f#(x1) |
(26) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
1.1.1.1.3 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.
1.1.1.1.3.1 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) |
(27) |
|
| 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
|
| [o#(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.
|
o#(c(x1)) |
→ |
o#(x1) |
(28) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.