Zorn lemma for inclusion #978
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Co-authored-by: Takafumi Saikawa <[email protected]>
affeldt-aist
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CohenCyril
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classical/classical_sets.v
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| Section Zorn_subset. | ||
| Variables (T : Type) (P : set T -> Prop). | ||
| Let sigP := {x | P x}. | ||
| Let R (sA sB : sigP) := sval sA `<=` sval sB. | ||
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|
||
| Lemma Zorn_bigcup : | ||
| (forall F, total_on F R -> P (\bigcup_(x in F) sval x)) -> | ||
| exists A, P A /\ forall B, A `<` B -> ~ P B. | ||
| Proof. | ||
| move=> totR. | ||
| have {}totR F : total_on F R -> exists sB, forall sA, F sA -> R sA sB. | ||
| by move=> FR; exists (exist _ _ (totR _ FR)) => sA FsA; exact: bigcup_sup. | ||
| have [| | |sA sAmax] := Zorn _ _ _ totR. | ||
| - by move=> ?; exact: subset_refl. | ||
| - by move=> ? ? ?; exact: subset_trans. | ||
| - by move=> [A PA] [B PB]; rewrite /R /= => AB BA; exact/eq_exist/seteqP. | ||
| - exists (sval sA); case: sA => A PA in sAmax *; split => //= B AB PB. | ||
| have [BA] := sAmax (exist _ B PB) (properW AB). | ||
| by move: AB; rewrite BA; exact: properxx. | ||
| Qed. | ||
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| End Zorn_subset. | ||
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|
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It looks strange to me to introduce the artifact R in the statement.
I would rather phrase it like this:
Section Zorn_subset.
Variables (T : Type) (P : set (set T)).
Lemma Zorn_bigcup :
(forall F : set (set T), F `<=` P -> total_on F subset -> P (\bigcup_(X in F) X)) ->
exists A, P A /\ forall B, A `<` B -> ~ P B.
Proof.
move=> totP; pose R (sA sB : P) := sval sA `<=` sval sB.
have {}totR F (FR : total_on F R) : exists sB, forall sA, F sA -> R sA sB.
have FP : [set val x | x in F] `<=` P.
by move=> _ [X FX <-]; apply: set_mem; apply: valP.
have totF : total_on [set val x | x in F] subset.
by move=> _ _ [X FX <-] [Y FY <-]; apply: FR.
exists (SigSub (mem_set (totP _ FP totF))) => A FA; rewrite /R/=.
exact: (bigcup_sup (imageP val _)).
have [| | |sA sAmax] := Zorn _ _ _ totR.
- by move=> ?; exact: subset_refl.
- by move=> ? ? ?; exact: subset_trans.
- by move=> [A PA] [B PB]; rewrite /R /= => AB BA; exact/eq_exist/seteqP.
- exists (val sA); case: sA => A PA /= in sAmax *; split; first exact: set_mem.
move=> B AB PB; have [BA] := sAmax (SigSub (mem_set PB)) (properW AB).
by move: AB; rewrite BA; exact: properxx.
Qed.
End Zorn_subset.
Is it better or worse in the real usecases?
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The effect of the change can be observed in the diff of the commit
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I see... I still think it's worth it ...
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Then I guess we can merge?
Co-authored-by: Cyril Cohen <[email protected]>
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* Zorn lemma for inclusion Co-authored-by: Takafumi Saikawa <[email protected]> Co-authored-by: Cyril Cohen <[email protected]>
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* Zorn lemma for inclusion Co-authored-by: Takafumi Saikawa <[email protected]> Co-authored-by: Cyril Cohen <[email protected]>
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* Zorn lemma for inclusion Co-authored-by: Takafumi Saikawa <[email protected]> Co-authored-by: Cyril Cohen <[email protected]>
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Motivation for this change
Corollary of Zorn's lemma.
It has a new on its own (https://fr.wikipedia.org/wiki/Lemme_de_Zorn#Principes_de_maximalit%C3%A9_pour_l'inclusion) and happens to be exactly what we need for Vitali's lemma (PR #973 )
@t6s
Things done/to do
CHANGELOG_UNRELEASED.mdCompatibility with MathComp 2.0
TODO: HB portto make sure someone ports this PR tothe
hierarchy-builderbranch or I already opened an issue or PR (please cross reference).Automatic note to reviewers
Read this Checklist and put a milestone if possible.