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[Merged by Bors] - feat(data/real/basic): add real.supr_nonneg etc #19096

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[Merged by Bors] - feat(data/real/basic): add real.supr_nonneg etc #19096

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3f46aeb
feat(data/real/basic): add `real.supr_nonneg` etc
urkud May 25, 2023
a94eced
Fix
urkud May 25, 2023
9271baa
Protect new lemmas
urkud May 26, 2023
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29 changes: 24 additions & 5 deletions src/data/real/basic.lean
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Original file line number Diff line number Diff line change
@@ -1,4 +1,4 @@
/-

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Changes to this file will need to be ported to mathlib 4!
Please consider retracting the changes to this file unless you are willing to immediately forward-port them.
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
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/--
As `0` is the default value for `real.Sup` of the empty set or sets which are not bounded above, it
suffices to show that `S` is bounded below by `0` to show that `0 ≤ Inf S`.
suffices to show that `S` is bounded below by `0` to show that `0 ≤ Sup S`.
-/
lemma Sup_nonneg (S : set ℝ) (hS : ∀ x ∈ S, (0:ℝ) ≤ x) : 0 ≤ Sup S :=
begin
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end

/--
As `0` is the default value for `real.Sup` of the empty set, it suffices to show that `S` is
bounded above by `0` to show that `Sup S ≤ 0`.
As `0` is the default value for `real.Sup` of the empty set or sets which are not bounded above, it
suffices to show that `f i` is nonnegative to show that `0 ≤ ⨆ i, f i`.
-/
lemma Sup_nonpos (S : set ℝ) (hS : ∀ x ∈ S, x ≤ (0:ℝ)) : Sup S ≤ 0 :=
protected lemma supr_nonneg {ι : Sort*} {f : ι → ℝ} (hf : ∀ i, 0 ≤ f i) : 0 ≤ ⨆ i, f i :=
Sup_nonneg _ $ set.forall_range_iff.2 hf

/--
As `0` is the default value for `real.Sup` of the empty set or sets which are not bounded above, it
suffices to show that all elements of `S` are bounded by a nonnagative number to show that `Sup S`
is bounded by this number.
-/
protected lemma Sup_le {S : set ℝ} {a : ℝ} (hS : ∀ x ∈ S, x ≤ a) (ha : 0 ≤ a) : Sup S ≤ a :=
begin
rcases S.eq_empty_or_nonempty with rfl | hS₂,
exacts [Sup_empty.le, cSup_le hS₂ hS],
exacts [Sup_empty.trans_le ha, cSup_le hS₂ hS],
end

protected lemma supr_le {ι : Sort*} {f : ι → ℝ} {a : ℝ} (hS : ∀ i, f i ≤ a) (ha : 0 ≤ a) :
(⨆ i, f i) ≤ a :=
real.Sup_le (set.forall_range_iff.2 hS) ha

/--
As `0` is the default value for `real.Sup` of the empty set, it suffices to show that `S` is
bounded above by `0` to show that `Sup S ≤ 0`.
-/
lemma Sup_nonpos (S : set ℝ) (hS : ∀ x ∈ S, x ≤ (0:ℝ)) : Sup S ≤ 0 :=
real.Sup_le hS le_rfl

/--
As `0` is the default value for `real.Inf` of the empty set, it suffices to show that `S` is
bounded below by `0` to show that `0 ≤ Inf S`.
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