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fixes #1123 (unusable lemma) #1124

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Dec 26, 2023
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fixes #1123 (unusable lemma) #1124

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05aff01
fixes #1123
affeldt-aist Dec 21, 2023
a2ec35b
fix
affeldt-aist Dec 26, 2023
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18 changes: 11 additions & 7 deletions theories/ereal.v
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Original file line number Diff line number Diff line change
Expand Up @@ -533,15 +533,19 @@ Proof. by move=> A B AB; apply: ub_ereal_sup => x Ax; apply/ereal_sup_ub/AB. Qed
Lemma le_ereal_inf : {homo @ereal_inf R : A B / A `<=` B >-> B <= A}.
Proof. by move=> A B AB; apply: lb_ereal_inf => x Bx; exact/ereal_inf_lb/AB. Qed.

Lemma hasNub_ereal_sup (A : set (\bar R)) : ~ has_ubound A ->
A !=set0 -> ereal_sup A = +oo%E.
Lemma hasNub_ereal_sup (A : set R) : ~ has_ubound A ->
A !=set0 -> ereal_sup (EFin @` A) = +oo%E.
Proof.
move=> hasNubA A0.
apply/eqP; rewrite eq_le leey /= leNgt; apply: contra_notN hasNubA => Aoo.
by exists (ereal_sup A); exact: ereal_sup_ub.
move=> + A0; apply: contra_notP => /eqP; rewrite -ltey => Aoo.
exists (fine (ereal_sup (EFin @` A))) => x Ax.
rewrite -lee_fin -(@fineK _ x%:E)// lee_fin fine_le//; last first.
by apply: ereal_sup_ub => /=; exists x.
rewrite fin_numE// -ltey Aoo andbT.
apply/eqP => /ereal_sup_ninfty/(_ x%:E).
by have /[swap] /[apply]: (EFin @` A) x%:E by exists x.
Qed.

Lemma ereal_sup_EFin (A : set R) :
Lemma ereal_sup_EFin (A : set R) :
has_ubound A -> A !=set0 -> ereal_sup (EFin @` A) = (sup A)%:E.
Proof.
move=> has_ubA A0; apply/eqP; rewrite eq_le; apply/andP; split.
Expand All @@ -559,7 +563,7 @@ by rewrite -lee_fin fineK//; apply: ereal_sup_ub; exists r.
Qed.

Lemma ereal_inf_EFin (A : set R) : has_lbound A -> A !=set0 ->
ereal_inf (EFin @` A) = (inf A)%:E.
ereal_inf (EFin @` A) = (inf A)%:E.
Proof.
move=> has_lbA A0; rewrite /ereal_inf /inf EFinN; congr (- _)%E.
rewrite -ereal_sup_EFin; [|exact/has_lb_ubN|exact/nonemptyN].
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