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Zorn lemma for inclusion #978

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Jul 20, 2023
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Zorn lemma for inclusion #978

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83d886a
Zorn lemma for inclusion
affeldt-aist Jul 17, 2023
d6b34ae
Zorn inclusion by Cyril
affeldt-aist Jul 19, 2023
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3 changes: 3 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Original file line number Diff line number Diff line change
Expand Up @@ -54,6 +54,9 @@
- in `classical_sets.v`:
+ lemmas `properW`, `properxx`

- in `classical_sets.v`:
+ lemma `Zorn_bigcup`

### Changed

- moved from `lebesgue_measure.v` to `real_interval.v`:
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33 changes: 28 additions & 5 deletions classical/classical_sets.v
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Expand Up @@ -2677,12 +2677,12 @@ Lemma Zorn T (R : T -> T -> Prop) :
exists t, forall s, R t s -> s = t.
Proof.
move=> Rrefl Rtrans Rantisym Rtot_max.
set totR := ({A : set T | total_on A R}).
pose totR := {A : set T | total_on A R}.
set R' := fun A B : totR => sval A `<=` sval B.
have R'refl A : R' A A by [].
have R'trans A B C : R' A B -> R' B C -> R' A C by apply: subset_trans.
have R'antisym A B : R' A B -> R' B A -> A = B.
rewrite /R'; case: A; case: B => /= B totB A totA sAB sBA.
rewrite /R'; move: A B => [/= A totA] [/= B totB] sAB sBA.
by apply: eq_exist; rewrite predeqE=> ?; split=> [/sAB|/sBA].
have R'tot_lub A : total_on A R' -> exists t, (forall s, A s -> R' s t) /\
forall r, (forall s, A s -> R' s r) -> R' t r.
Expand All @@ -2693,7 +2693,7 @@ have R'tot_lub A : total_on A R' -> exists t, (forall s, A s -> R' s t) /\
by have /(_ _ _ Cs Ct) := svalP C.
by have /(_ _ _ Bs Bt) := svalP B.
exists (exist _ (\bigcup_(B in A) sval B) AUtot); split.
by move=> B ???; exists B.
by move=> B ? ? ?; exists B.
by move=> B Bub ? /= [? /Bub]; apply.
apply: contrapT => nomax.
have {}nomax t : exists s, R t s /\ s <> t.
Expand All @@ -2708,9 +2708,9 @@ have Astot : total_on (sval A `|` [set s]) R.
by move=> [/tub Rvt|->]; right=> //; apply: Rtrans Rts.
move=> [Av|->]; [apply: (svalP A)|left] => //.
by apply: Rtrans Rts; apply: tub.
exists (exist _ (sval A `|` [set s]) Astot); split; first by move=> ??; left.
exists (exist _ (sval A `|` [set s]) Astot); split; first by move=> ? ?; left.
split=> [AeAs|[B Btot] sAB sBAs].
have [/tub Rst|] := (pselect (sval A s)); first exact/snet/Rantisym.
have [/tub Rst|] := pselect (sval A s); first exact/snet/Rantisym.
by rewrite AeAs /=; apply; right.
have [Bs|nBs] := pselect (B s).
by right; apply: eq_exist; rewrite predeqE => r; split=> [/sBAs|[/sAB|->]].
Expand All @@ -2719,6 +2719,29 @@ apply: eq_exist; rewrite predeqE => r; split=> [Br|/sAB] //.
by have /sBAs [|ser] // := Br; rewrite ser in Br.
Qed.

Section Zorn_subset.
Variables (T : Type) (P : set T -> Prop).
Let sigP := {x | P x}.
Let R (sA sB : sigP) := sval sA `<=` sval sB.

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.

End Zorn_subset.

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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
f2a57df

file before: lines 2260-2270
file after: lines 2302-2311

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I see... I still think it's worth it ...

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Then I guess we can merge?

Definition premaximal T (R : T -> T -> Prop) (t : T) :=
forall s, R t s -> R s t.

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