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Cluster1 #850
Cluster1 #850
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theories/topology.v
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Proof. | ||
apply/ predeqP => ?. | ||
by split; [exact: compact_near_covering| exact: near_covering_compact]. | ||
Qed. |
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Proof. | |
apply/ predeqP => ?. | |
by split; [exact: compact_near_covering| exact: near_covering_compact]. | |
Qed. | |
Proof. by rewrite eqEsubset; exact: compact_near_coveringP. Qed. |
theories/topology.v
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have cptVU : compact (V `\` U). | ||
apply: (subclosed_compact _ cptV) => //. | ||
by apply: closedI; [exact: compact_closed | exact: open_closedC]. | ||
rewrite compact_near_coveringE in cptVU. | ||
have [] := cptVU _ (powerset_filter_from F) (fun W x => ~ W x). |
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This is where compact_near_coveringE
is used. Wouldn't
Lemma blah A : compact A <-> near_covering A.
be more useful here? We can /blah
at line 3045, line 3048 is not need anymore, and we do not need to name the fact compact (V \ U)
. But maybe there is some combinator to from compact_near_coveringP
to blah
that I miss...
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I think it's cleaner to just change compact_near_coveringP
to be compact A <-> near_covering A
. It seems a bit more idiomatic. I've made the updatee with that change.
Co-authored-by: affeldt-aist <[email protected]>
Co-authored-by: affeldt-aist <[email protected]>
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I just noticed that an intermediate hypothesis needn't be named but that's ok to remove this commit if you want.
Sounds good, that hypo doesn't need a name. |
* cluster1 proof * updating changelog * Update theories/topology.v Co-authored-by: affeldt-aist <[email protected]> * Update theories/topology.v Co-authored-by: affeldt-aist <[email protected]> * change to signature of compact_near_coveringP * no need to name intermediate hypo --------- Co-authored-by: affeldt-aist <[email protected]> Co-authored-by: Reynald Affeldt <[email protected]>
* cluster1 proof * updating changelog * Update theories/topology.v Co-authored-by: affeldt-aist <[email protected]> * Update theories/topology.v Co-authored-by: affeldt-aist <[email protected]> * change to signature of compact_near_coveringP * no need to name intermediate hypo --------- Co-authored-by: affeldt-aist <[email protected]> Co-authored-by: Reynald Affeldt <[email protected]>
Motivation for this change
This should be an easy review. We prove a classic topology result that in compact spaces, a filter
F
with one cluster pointx
converges tox
. This is another nice application of thenear_covering
definition. I claim this formalization is a good approximation of the classical proof:In coq, the
powerset_filter_from F
encodes "finite intersections" and the property(fun x W => ~ W x)
encodes the complements. Plugging these into the statement thatV \ U
is near_covering compact, we get a new goal to build the cover. Then given ay
with~U y
we knowy <> x
. So~ cluster F y
, and we get our neighborhood. The rest is just bookkeeping (pushing negations through quantifiers and handling the a vacuous~V t
case).Things done/to do
CHANGELOG_UNRELEASED.md
(do not edit former entries, only append new ones, be careful:
merge and rebase have a tendency to mess up
CHANGELOG_UNRELEASED.md
)Automatic note to reviewers
Read this Checklist and put a milestone if possible.