topology.algebra.order.liminf_limsup | Mathlib porting status

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Changes in mathlib3

ce64cd31

(last sync)

feat(topology/algebra/order/liminf_limsup): Eventual boundedness of neighborhoods (#18629)

Generalise bounded_le_nhds/bounded_ge_nhds using two ad hoc typeclasses. The goal here is to circumvent the fact that the product of order topologies is not an order topology, and apply those lemmas to ℝⁿ.

Diff

@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
+Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov, Yaël Dillies
 -/
 import algebra.big_operators.intervals
 import algebra.big_operators.order
@@ -16,68 +16,152 @@ import topology.order.basic
 
 > THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
 > Any changes to this file require a corresponding PR to mathlib4.
+
+## Main declarations
+
+* `bounded_le_nhds_class`: Typeclass stating that neighborhoods are eventually bounded above.
+* `bounded_ge_nhds_class`: Typeclass stating that neighborhoods are eventually bounded below.
+
+## Implementation notes
+
+The same lemmas are true in `ℝ`, `ℝ × ℝ`, `ι → ℝ`, `euclidean_space ι ℝ`. To avoid code
+duplication, we provide an ad hoc axiomatisation of the properties we need.
 -/
 
 open filter topological_space
 open_locale topology classical
 
 universes u v
-variables {α : Type u} {β : Type v}
+variables {ι α β R S : Type*} {π : ι → Type*}
 
-section liminf_limsup
+/-- Ad hoc typeclass stating that neighborhoods are eventually bounded above. -/
+class bounded_le_nhds_class (α : Type*) [preorder α] [topological_space α] : Prop :=
+(is_bounded_le_nhds (a : α) : (𝓝 a).is_bounded (≤))
+
+/-- Ad hoc typeclass stating that neighborhoods are eventually bounded below. -/
+class bounded_ge_nhds_class (α : Type*) [preorder α] [topological_space α] : Prop :=
+(is_bounded_ge_nhds (a : α) : (𝓝 a).is_bounded (≥))
+
+section preorder
+variables [preorder α] [preorder β] [topological_space α] [topological_space β]
 
-section order_closed_topology
-variables [semilattice_sup α] [topological_space α] [order_topology α]
+section bounded_le_nhds_class
+variables [bounded_le_nhds_class α] [bounded_le_nhds_class β] {f : filter ι} {u : ι → α} {a : α}
 
 lemma is_bounded_le_nhds (a : α) : (𝓝 a).is_bounded (≤) :=
-(is_top_or_exists_gt a).elim (λ h, ⟨a, eventually_of_forall h⟩) (λ ⟨b, hb⟩, ⟨b, ge_mem_nhds hb⟩)
+bounded_le_nhds_class.is_bounded_le_nhds _
 
-lemma filter.tendsto.is_bounded_under_le {f : filter β} {u : β → α} {a : α}
-  (h : tendsto u f (𝓝 a)) : f.is_bounded_under (≤) u :=
+lemma filter.tendsto.is_bounded_under_le (h : tendsto u f (𝓝 a)) :
+  f.is_bounded_under (≤) u :=
 (is_bounded_le_nhds a).mono h
 
-lemma filter.tendsto.bdd_above_range_of_cofinite {u : β → α} {a : α}
+lemma filter.tendsto.bdd_above_range_of_cofinite [is_directed α (≤)]
   (h : tendsto u cofinite (𝓝 a)) : bdd_above (set.range u) :=
 h.is_bounded_under_le.bdd_above_range_of_cofinite
 
-lemma filter.tendsto.bdd_above_range {u : ℕ → α} {a : α}
-  (h : tendsto u at_top (𝓝 a)) : bdd_above (set.range u) :=
+lemma filter.tendsto.bdd_above_range [is_directed α (≤)] {u : ℕ → α} (h : tendsto u at_top (𝓝 a)) :
+  bdd_above (set.range u) :=
 h.is_bounded_under_le.bdd_above_range
 
 lemma is_cobounded_ge_nhds (a : α) : (𝓝 a).is_cobounded (≥) :=
 (is_bounded_le_nhds a).is_cobounded_flip
 
-lemma filter.tendsto.is_cobounded_under_ge {f : filter β} {u : β → α} {a : α}
-  [ne_bot f] (h : tendsto u f (𝓝 a)) : f.is_cobounded_under (≥) u :=
+lemma filter.tendsto.is_cobounded_under_ge [ne_bot f] (h : tendsto u f (𝓝 a)) :
+  f.is_cobounded_under (≥) u :=
 h.is_bounded_under_le.is_cobounded_flip
 
-end order_closed_topology
+instance : bounded_ge_nhds_class αᵒᵈ := ⟨@is_bounded_le_nhds α _ _ _⟩
+
+instance : bounded_le_nhds_class (α × β) :=
+begin
+  refine ⟨λ x, _⟩,
+  obtain ⟨a, ha⟩ := is_bounded_le_nhds x.1,
+  obtain ⟨b, hb⟩ := is_bounded_le_nhds x.2,
+  rw [←@prod.mk.eta _ _ x, nhds_prod_eq],
+  exact ⟨(a, b), ha.prod_mk hb⟩,
+end
+
+instance [finite ι] [Π i, preorder (π i)] [Π i, topological_space (π i)]
+  [Π i, bounded_le_nhds_class (π i)] : bounded_le_nhds_class (Π i, π i) :=
+begin
+  refine ⟨λ x, _⟩,
+  rw nhds_pi,
+  choose f hf using λ i, is_bounded_le_nhds (x i),
+  exact ⟨f, eventually_pi hf⟩,
+end
+
+end bounded_le_nhds_class
 
-section order_closed_topology
-variables [semilattice_inf α] [topological_space α] [order_topology α]
+section bounded_ge_nhds_class
+variables [bounded_ge_nhds_class α] [bounded_ge_nhds_class β] {f : filter ι} {u : ι → α} {a : α}
 
-lemma is_bounded_ge_nhds (a : α) : (𝓝 a).is_bounded (≥) := @is_bounded_le_nhds αᵒᵈ _ _ _ a
+lemma is_bounded_ge_nhds (a : α) : (𝓝 a).is_bounded (≥) :=
+bounded_ge_nhds_class.is_bounded_ge_nhds _
 
-lemma filter.tendsto.is_bounded_under_ge {f : filter β} {u : β → α} {a : α}
-  (h : tendsto u f (𝓝 a)) : f.is_bounded_under (≥) u :=
+lemma filter.tendsto.is_bounded_under_ge (h : tendsto u f (𝓝 a)) :
+  f.is_bounded_under (≥) u :=
 (is_bounded_ge_nhds a).mono h
 
-lemma filter.tendsto.bdd_below_range_of_cofinite {u : β → α} {a : α}
+lemma filter.tendsto.bdd_below_range_of_cofinite [is_directed α (≥)]
   (h : tendsto u cofinite (𝓝 a)) : bdd_below (set.range u) :=
 h.is_bounded_under_ge.bdd_below_range_of_cofinite
 
-lemma filter.tendsto.bdd_below_range {u : ℕ → α} {a : α}
-  (h : tendsto u at_top (𝓝 a)) : bdd_below (set.range u) :=
+lemma filter.tendsto.bdd_below_range [is_directed α (≥)] {u : ℕ → α} (h : tendsto u at_top (𝓝 a)) :
+  bdd_below (set.range u) :=
 h.is_bounded_under_ge.bdd_below_range
 
 lemma is_cobounded_le_nhds (a : α) : (𝓝 a).is_cobounded (≤) :=
 (is_bounded_ge_nhds a).is_cobounded_flip
 
-lemma filter.tendsto.is_cobounded_under_le {f : filter β} {u : β → α} {a : α}
-  [ne_bot f] (h : tendsto u f (𝓝 a)) : f.is_cobounded_under (≤) u :=
+lemma filter.tendsto.is_cobounded_under_le [ne_bot f] (h : tendsto u f (𝓝 a)) :
+  f.is_cobounded_under (≤) u :=
 h.is_bounded_under_ge.is_cobounded_flip
 
-end order_closed_topology
+instance : bounded_le_nhds_class αᵒᵈ := ⟨@is_bounded_ge_nhds α _ _ _⟩
+
+instance : bounded_ge_nhds_class (α × β) :=
+begin
+  refine ⟨λ x, _⟩,
+  obtain ⟨a, ha⟩ := is_bounded_ge_nhds x.1,
+  obtain ⟨b, hb⟩ := is_bounded_ge_nhds x.2,
+  rw [←@prod.mk.eta _ _ x, nhds_prod_eq],
+  exact ⟨(a, b), ha.prod_mk hb⟩,
+end
+
+instance [finite ι] [Π i, preorder (π i)] [Π i, topological_space (π i)]
+  [Π i, bounded_ge_nhds_class (π i)] : bounded_ge_nhds_class (Π i, π i) :=
+begin
+  refine ⟨λ x, _⟩,
+  rw nhds_pi,
+  choose f hf using λ i, is_bounded_ge_nhds (x i),
+  exact ⟨f, eventually_pi hf⟩,
+end
+
+end bounded_ge_nhds_class
+
+@[priority 100] -- See note [lower instance priority]
+instance order_top.to_bounded_le_nhds_class [order_top α] : bounded_le_nhds_class α :=
+⟨λ a, is_bounded_le_of_top⟩
+
+@[priority 100] -- See note [lower instance priority]
+instance order_bot.to_bounded_ge_nhds_class [order_bot α] : bounded_ge_nhds_class α :=
+⟨λ a, is_bounded_ge_of_bot⟩
+
+@[priority 100] -- See note [lower instance priority]
+instance order_topology.to_bounded_le_nhds_class [is_directed α (≤)] [order_topology α] :
+  bounded_le_nhds_class α :=
+⟨λ a, (is_top_or_exists_gt a).elim (λ h, ⟨a, eventually_of_forall h⟩) $ Exists.imp $ λ b,
+  ge_mem_nhds⟩
+
+@[priority 100] -- See note [lower instance priority]
+instance order_topology.to_bounded_ge_nhds_class [is_directed α (≥)] [order_topology α] :
+  bounded_ge_nhds_class α :=
+⟨λ a, (is_bot_or_exists_lt a).elim (λ h, ⟨a, eventually_of_forall h⟩) $ Exists.imp $ λ b,
+  le_mem_nhds⟩
+
+end preorder
+
+section liminf_limsup
 
 section conditionally_complete_linear_order
 variables [conditionally_complete_linear_order α]
@@ -224,7 +308,7 @@ end liminf_limsup
 
 section monotone
 
-variables {ι R S : Type*} {F : filter ι} [ne_bot F]
+variables {F : filter ι} [ne_bot F]
   [complete_linear_order R] [topological_space R] [order_topology R]
   [complete_linear_order S] [topological_space S] [order_topology S]
 
@@ -334,7 +418,7 @@ open_locale topology
 
 open filter set
 
-variables {ι : Type*} {R : Type*} [complete_linear_order R] [topological_space R] [order_topology R]
+variables [complete_linear_order R] [topological_space R] [order_topology R]
 
 lemma infi_eq_of_forall_le_of_tendsto {x : R} {as : ι → R}
   (x_le : ∀ i, x ≤ as i) {F : filter ι} [filter.ne_bot F] (as_lim : filter.tendsto as F (𝓝 x)) :
@@ -349,7 +433,7 @@ lemma supr_eq_of_forall_le_of_tendsto {x : R} {as : ι → R}
   (⨆ i, as i) = x :=
 @infi_eq_of_forall_le_of_tendsto ι (order_dual R) _ _ _ x as le_x F _ as_lim
 
-lemma Union_Ici_eq_Ioi_of_lt_of_tendsto {ι : Type*} (x : R) {as : ι → R} (x_lt : ∀ i, x < as i)
+lemma Union_Ici_eq_Ioi_of_lt_of_tendsto (x : R) {as : ι → R} (x_lt : ∀ i, x < as i)
   {F : filter ι} [filter.ne_bot F] (as_lim : filter.tendsto as F (𝓝 x)) :
   (⋃ (i : ι), Ici (as i)) = Ioi x :=
 begin
@@ -361,10 +445,10 @@ begin
   exact Union_Ici_eq_Ioi_infi obs,
 end
 
-lemma Union_Iic_eq_Iio_of_lt_of_tendsto {ι : Type*} (x : R) {as : ι → R} (lt_x : ∀ i, as i < x)
+lemma Union_Iic_eq_Iio_of_lt_of_tendsto (x : R) {as : ι → R} (lt_x : ∀ i, as i < x)
   {F : filter ι} [filter.ne_bot F] (as_lim : filter.tendsto as F (𝓝 x)) :
   (⋃ (i : ι), Iic (as i)) = Iio x :=
-@Union_Ici_eq_Ioi_of_lt_of_tendsto (order_dual R) _ _ _ ι x as lt_x F _ as_lim
+@Union_Ici_eq_Ioi_of_lt_of_tendsto ι Rᵒᵈ _ _ _ _ _ lt_x F _ as_lim
 
 end infi_and_supr

ffde2d8a

chore(order/liminf_limsup): Generalise and move lemmas (#18628)

Generalise lemmas from semilattices to codirected orders. Move topology-less lemmas from topology.algebra.order.liminf_limsup to order.liminf_limsup. Also turn arguments to bdd_above_insert and friends implicit.

Diff

@@ -51,19 +51,6 @@ lemma filter.tendsto.is_cobounded_under_ge {f : filter β} {u : β → α} {a :
   [ne_bot f] (h : tendsto u f (𝓝 a)) : f.is_cobounded_under (≥) u :=
 h.is_bounded_under_le.is_cobounded_flip
 
-lemma is_bounded_le_at_bot (α : Type*) [hα : nonempty α] [preorder α] :
-  (at_bot : filter α).is_bounded (≤) :=
-is_bounded_iff.2 ⟨set.Iic hα.some, mem_at_bot _, hα.some, λ x hx, hx⟩
-
-lemma filter.tendsto.is_bounded_under_le_at_bot {α : Type*} [nonempty α] [preorder α]
-  {f : filter β} {u : β → α} (h : tendsto u f at_bot) :
-  f.is_bounded_under (≤) u :=
-(is_bounded_le_at_bot α).mono h
-
-lemma bdd_above_range_of_tendsto_at_top_at_bot {α : Type*} [nonempty α] [semilattice_sup α]
-  {u : ℕ → α} (hx : tendsto u at_top at_bot) : bdd_above (set.range u) :=
-(filter.tendsto.is_bounded_under_le_at_bot hx).bdd_above_range
-
 end order_closed_topology
 
 section order_closed_topology
@@ -90,33 +77,11 @@ lemma filter.tendsto.is_cobounded_under_le {f : filter β} {u : β → α} {a :
   [ne_bot f] (h : tendsto u f (𝓝 a)) : f.is_cobounded_under (≤) u :=
 h.is_bounded_under_ge.is_cobounded_flip
 
-lemma is_bounded_ge_at_top (α : Type*) [hα : nonempty α] [preorder α] :
-  (at_top : filter α).is_bounded (≥) :=
-is_bounded_le_at_bot αᵒᵈ
-
-lemma filter.tendsto.is_bounded_under_ge_at_top {α : Type*} [nonempty α] [preorder α]
-  {f : filter β} {u : β → α} (h : tendsto u f at_top) :
-  f.is_bounded_under (≥) u :=
-(is_bounded_ge_at_top α).mono h
-
-lemma bdd_below_range_of_tendsto_at_top_at_top {α : Type*} [nonempty α] [semilattice_inf α]
-  {u : ℕ → α} (hx : tendsto u at_top at_top) : bdd_below (set.range u) :=
-(filter.tendsto.is_bounded_under_ge_at_top hx).bdd_below_range
-
 end order_closed_topology
 
 section conditionally_complete_linear_order
 variables [conditionally_complete_linear_order α]
 
-theorem lt_mem_sets_of_Limsup_lt {f : filter α} {b} (h : f.is_bounded (≤)) (l : f.Limsup < b) :
-  ∀ᶠ a in f, a < b :=
-let ⟨c, (h : ∀ᶠ a in f, a ≤ c), hcb⟩ := exists_lt_of_cInf_lt h l in
-mem_of_superset h $ assume a hac, lt_of_le_of_lt hac hcb
-
-theorem gt_mem_sets_of_Liminf_gt : ∀ {f : filter α} {b}, f.is_bounded (≥) → b < f.Liminf →
-  ∀ᶠ a in f, b < a :=
-@lt_mem_sets_of_Limsup_lt αᵒᵈ _
-
 variables [topological_space α] [order_topology α]
 
 /-- If the liminf and the limsup of a filter coincide, then this filter converges to

topology.algebra.order.liminf_limsup ⟷ Mathlib.Topology.Algebra.Order.LiminfLimsup

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

Renames

Dependencies 8 + 339

`topology.algebra.order.liminf_limsup` ⟷ `Mathlib.Topology.Algebra.Order.LiminfLimsup`