The rewrite relation of the following TRS is considered.
The dependency pairs are split into 7
components.
-
The
1st
component contains the
pair
active#(h(X)) |
→ |
active#(X) |
(23) |
active#(f(X)) |
→ |
active#(X) |
(21) |
1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[h(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 · x1
|
[active#(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 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
active#(h(X)) |
→ |
active#(X) |
(23) |
|
1 |
> |
1 |
active#(f(X)) |
→ |
active#(X) |
(21) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
proper#(c(X)) |
→ |
proper#(X) |
(29) |
proper#(f(X)) |
→ |
proper#(X) |
(27) |
proper#(g(X)) |
→ |
proper#(X) |
(31) |
proper#(d(X)) |
→ |
proper#(X) |
(33) |
proper#(h(X)) |
→ |
proper#(X) |
(35) |
1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[c(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 · x1
|
[g(x1)] |
= |
1 · x1
|
[d(x1)] |
= |
1 · x1
|
[h(x1)] |
= |
1 · x1
|
[proper#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
proper#(c(X)) |
→ |
proper#(X) |
(29) |
|
1 |
> |
1 |
proper#(f(X)) |
→ |
proper#(X) |
(27) |
|
1 |
> |
1 |
proper#(g(X)) |
→ |
proper#(X) |
(31) |
|
1 |
> |
1 |
proper#(d(X)) |
→ |
proper#(X) |
(33) |
|
1 |
> |
1 |
proper#(h(X)) |
→ |
proper#(X) |
(35) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
f#(ok(X)) |
→ |
f#(X) |
(36) |
f#(mark(X)) |
→ |
f#(X) |
(24) |
1.1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(x1)] |
= |
1 · x1
|
[mark(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.
1.1.1.3.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
f#(ok(X)) |
→ |
f#(X) |
(36) |
|
1 |
> |
1 |
f#(mark(X)) |
→ |
f#(X) |
(24) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
h#(ok(X)) |
→ |
h#(X) |
(40) |
h#(mark(X)) |
→ |
h#(X) |
(25) |
1.1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[h#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.4.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
h#(ok(X)) |
→ |
h#(X) |
(40) |
|
1 |
> |
1 |
h#(mark(X)) |
→ |
h#(X) |
(25) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
1.1.1.5 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(x1)] |
= |
1 · x1
|
[c#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.5.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
c#(ok(X)) |
→ |
c#(X) |
(37) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
1.1.1.6 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(x1)] |
= |
1 · x1
|
[g#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.6.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
g#(ok(X)) |
→ |
g#(X) |
(38) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
7th
component contains the
pair
1.1.1.7 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(x1)] |
= |
1 · x1
|
[d#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.7.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
d#(ok(X)) |
→ |
d#(X) |
(39) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.