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[Merged by Bors] - chore(Topology/Order): golf, deprecate #10554

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[Merged by Bors] - chore(Topology/Order): golf, deprecate #10554

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8 changes: 1 addition & 7 deletions Mathlib/Probability/Kernel/CondCdf.lean
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Expand Up @@ -351,13 +351,7 @@ theorem tendsto_preCDF_atBot_zero (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ
have h_exists : ∀ᵐ a ∂ρ.fst, ∃ l, Tendsto (fun r => preCDF ρ (-r) a) atTop (𝓝 l) := by
filter_upwards [monotone_preCDF ρ] with a ha
have h_anti : Antitone fun r => preCDF ρ (-r) a := fun p q hpq => ha (neg_le_neg hpq)
have h_tendsto :
Tendsto (fun r => preCDF ρ (-r) a) atTop atBot ∨
∃ l, Tendsto (fun r => preCDF ρ (-r) a) atTop (𝓝 l) :=
tendsto_of_antitone h_anti
cases' h_tendsto with h_bot h_tendsto
· exact ⟨0, Tendsto.mono_right h_bot atBot_le_nhds_bot⟩
· exact h_tendsto
exact ⟨_, tendsto_atTop_iInf h_anti⟩
classical
let F : α → ℝ≥0∞ := fun a =>
if h : ∃ l, Tendsto (fun r => preCDF ρ (-r) a) atTop (𝓝 l) then h.choose else 0
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16 changes: 5 additions & 11 deletions Mathlib/Topology/Order/Basic.lean
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Expand Up @@ -530,21 +530,15 @@ theorem Dense.topology_eq_generateFrom [DenselyOrdered α] {s : Set α} (hs : De
let _ := generateFrom (Ioi '' s ∪ Iio '' s)
exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <| .inr <| mem_image_of_mem _ h.1

@[deprecated OrderBot.atBot_eq] -- 2024-02-14
theorem atBot_le_nhds_bot [OrderBot α] : (atBot : Filter α) ≤ 𝓝 ⊥ := by
cases subsingleton_or_nontrivial α
· simp only [nhds_discrete, le_pure_iff, mem_atBot_sets, mem_singleton_iff,
eq_iff_true_of_subsingleton, imp_true_iff, exists_const]
have h : atBot.HasBasis (fun _ : α => True) Iic := @atBot_basis α _ _
have h_nhds : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) fun a => Iio a := @nhds_bot_basis α _ _ _ _ _
intro s
rw [h.mem_iff, h_nhds.mem_iff]
rintro ⟨a, ha_bot_lt, h_Iio_a_subset_s⟩
refine' ⟨⊥, trivial, _root_.trans _ h_Iio_a_subset_s⟩
simpa only [Iic_bot, singleton_subset_iff, mem_Iio]
rw [OrderBot.atBot_eq]
apply pure_le_nhds
#align at_bot_le_nhds_bot atBot_le_nhds_bot

@[deprecated OrderTop.atTop_eq] -- 2024-02-14
theorem atTop_le_nhds_top [OrderTop α] : (atTop : Filter α) ≤ 𝓝 ⊤ :=
@atBot_le_nhds_bot αᵒᵈ _ _ _ _
set_option linter.deprecated false in @atBot_le_nhds_bot αᵒᵈ _ _ _
#align at_top_le_nhds_top atTop_le_nhds_top

variable (α)
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