Redefine Subnet
#48
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Redefine Subnet
#48
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The previous definition of `Subnet` seems to require (certain forms of)
functional extensionality to prove simple facts like "every net is a
subnet of itself" or the following statement that being a `Subnet` is an
extensional property:
```coq
Lemma Subnet_compat {I J : DirectedSet} {X : Type}
(x1 x2 : Net I X) (y1 y2 : Net J X) :
(forall i, x1 i = x2 i) ->
(forall j, y1 j = y2 j) ->
Subnet x1 y1 -> Subnet x2 y2.
```
The new definition avoids such difficulties.
The coercion from `DirectedSets` to `Sortclass` makes some code easier to write.
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The current definition of
Subnetrequires (certain forms of) functional extensionality even for simple facts like "every net is a subnet of itself". By restating the definition in an extensional way, such problems/axioms can be avoided.Also, this PR changes
DS_setto be a coercionDirectedSet >-> Sortclass, to simplify various terms. This is analogous to the existing coercionpoint_set : TopologicalSpace >-> Sortclass.