The rewrite relation of the following TRS is considered.
The dependency pairs are split into 5
components.
-
The
1st
component contains the
pair
mark#(g(X)) |
→ |
mark#(X) |
(23) |
mark#(f(X)) |
→ |
mark#(X) |
(18) |
1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[g(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 · x1
|
[mark#(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 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
mark#(g(X)) |
→ |
mark#(X) |
(23) |
|
1 |
> |
1 |
mark#(f(X)) |
→ |
mark#(X) |
(18) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
mark#(c(X)) |
→ |
active#(c(X)) |
(20) |
active#(f(f(a))) |
→ |
mark#(c(f(g(f(a))))) |
(12) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[c(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 · x1
|
[a] |
= |
0 |
[g(x1)] |
= |
1 · x1
|
[active#(x1)] |
= |
1 · x1
|
[mark#(x1)] |
= |
1 · x1
|
together with the usable
rules
c(active(X)) |
→ |
c(X) |
(9) |
c(mark(X)) |
→ |
c(X) |
(8) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
f#(active(X)) |
→ |
f#(X) |
(25) |
f#(mark(X)) |
→ |
f#(X) |
(24) |
1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(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.
1.1.3.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
f#(active(X)) |
→ |
f#(X) |
(25) |
|
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
c#(active(X)) |
→ |
c#(X) |
(27) |
c#(mark(X)) |
→ |
c#(X) |
(26) |
1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(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.
1.1.4.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
c#(active(X)) |
→ |
c#(X) |
(27) |
|
1 |
> |
1 |
c#(mark(X)) |
→ |
c#(X) |
(26) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
g#(active(X)) |
→ |
g#(X) |
(29) |
g#(mark(X)) |
→ |
g#(X) |
(28) |
1.1.5 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(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.
1.1.5.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
g#(active(X)) |
→ |
g#(X) |
(29) |
|
1 |
> |
1 |
g#(mark(X)) |
→ |
g#(X) |
(28) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.