state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
S : Set (Set α)
ho : ∀ s ∈ S, IsOpen s
hS : Set.Countable S
hd : ∀ s ∈ S, Dense s
h : Set.Nonempty S
f : ℕ → Set α
hf : S = range f
n : ℕ
⊢ f n ∈ S | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | rw [hf] | /-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with ⋂₀. -/
theorem dense_sInter_of_isOpen {S : Set (Set α)} (ho : ∀ s ∈ S, IsOpen s) (hS : S.Countable)
(hd : ∀ s ∈ S, Dense s) : Dense (⋂₀ S) := by
rcases S.eq_empty_or_nonempty with h | h
· simp [h]
· rcases hS.exists_... | Mathlib.Topology.MetricSpace.Baire.195_0.GktojJRwRzEj9tj | /-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with ⋂₀. -/
theorem dense_sInter_of_isOpen {S : Set (Set α)} (ho : ∀ s ∈ S, IsOpen s) (hS : S.Countable)
(hd : ∀ s ∈ S, Dense s) : Dense (⋂₀ S) | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
S : Set (Set α)
ho : ∀ s ∈ S, IsOpen s
hS : Set.Countable S
hd : ∀ s ∈ S, Dense s
h : Set.Nonempty S
f : ℕ → Set α
hf : S = range f
n : ℕ
⊢ f n ∈ range f | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | exact mem_range_self _ | /-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with ⋂₀. -/
theorem dense_sInter_of_isOpen {S : Set (Set α)} (ho : ∀ s ∈ S, IsOpen s) (hS : S.Countable)
(hd : ∀ s ∈ S, Dense s) : Dense (⋂₀ S) := by
rcases S.eq_empty_or_nonempty with h | h
· simp [h]
· rcases hS.exists_... | Mathlib.Topology.MetricSpace.Baire.195_0.GktojJRwRzEj9tj | /-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with ⋂₀. -/
theorem dense_sInter_of_isOpen {S : Set (Set α)} (ho : ∀ s ∈ S, IsOpen s) (hS : S.Countable)
(hd : ∀ s ∈ S, Dense s) : Dense (⋂₀ S) | Mathlib_Topology_MetricSpace_Baire |
case inr.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
S : Set (Set α)
ho : ∀ s ∈ S, IsOpen s
hS : Set.Countable S
hd : ∀ s ∈ S, Dense s
h : Set.Nonempty S
f : ℕ → Set α
hf : S = range f
F : ∀ (n : ℕ), f n ∈ S
⊢ Dense (⋂₀ S) | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | rw [hf, sInter_range] | /-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with ⋂₀. -/
theorem dense_sInter_of_isOpen {S : Set (Set α)} (ho : ∀ s ∈ S, IsOpen s) (hS : S.Countable)
(hd : ∀ s ∈ S, Dense s) : Dense (⋂₀ S) := by
rcases S.eq_empty_or_nonempty with h | h
· simp [h]
· rcases hS.exists_... | Mathlib.Topology.MetricSpace.Baire.195_0.GktojJRwRzEj9tj | /-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with ⋂₀. -/
theorem dense_sInter_of_isOpen {S : Set (Set α)} (ho : ∀ s ∈ S, IsOpen s) (hS : S.Countable)
(hd : ∀ s ∈ S, Dense s) : Dense (⋂₀ S) | Mathlib_Topology_MetricSpace_Baire |
case inr.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
S : Set (Set α)
ho : ∀ s ∈ S, IsOpen s
hS : Set.Countable S
hd : ∀ s ∈ S, Dense s
h : Set.Nonempty S
f : ℕ → Set α
hf : S = range f
F : ∀ (n : ℕ), f n ∈ S
⊢ Dense (⋂ x, f x) | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | exact dense_iInter_of_isOpen_nat (fun n => ho _ (F n)) fun n => hd _ (F n) | /-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with ⋂₀. -/
theorem dense_sInter_of_isOpen {S : Set (Set α)} (ho : ∀ s ∈ S, IsOpen s) (hS : S.Countable)
(hd : ∀ s ∈ S, Dense s) : Dense (⋂₀ S) := by
rcases S.eq_empty_or_nonempty with h | h
· simp [h]
· rcases hS.exists_... | Mathlib.Topology.MetricSpace.Baire.195_0.GktojJRwRzEj9tj | /-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with ⋂₀. -/
theorem dense_sInter_of_isOpen {S : Set (Set α)} (ho : ∀ s ∈ S, IsOpen s) (hS : S.Countable)
(hd : ∀ s ∈ S, Dense s) : Dense (⋂₀ S) | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
S : Set β
f : β → Set α
ho : ∀ s ∈ S, IsOpen (f s)
hS : Set.Countable S
hd : ∀ s ∈ S, Dense (f s)
⊢ Dense (⋂ s ∈ S, f s) | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | rw [← sInter_image] | /-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with
an index set which is a countable set in any type. -/
theorem dense_biInter_of_isOpen {S : Set β} {f : β → Set α} (ho : ∀ s ∈ S, IsOpen (f s))
(hS : S.Countable) (hd : ∀ s ∈ S, Dense (f s)) : Dense (⋂ s ∈ S, f s) := by
| Mathlib.Topology.MetricSpace.Baire.206_0.GktojJRwRzEj9tj | /-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with
an index set which is a countable set in any type. -/
theorem dense_biInter_of_isOpen {S : Set β} {f : β → Set α} (ho : ∀ s ∈ S, IsOpen (f s))
(hS : S.Countable) (hd : ∀ s ∈ S, Dense (f s)) : Dense (⋂ s ∈ S, f s) | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
S : Set β
f : β → Set α
ho : ∀ s ∈ S, IsOpen (f s)
hS : Set.Countable S
hd : ∀ s ∈ S, Dense (f s)
⊢ Dense (⋂₀ ((fun s => f s) '' S)) | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | apply dense_sInter_of_isOpen | /-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with
an index set which is a countable set in any type. -/
theorem dense_biInter_of_isOpen {S : Set β} {f : β → Set α} (ho : ∀ s ∈ S, IsOpen (f s))
(hS : S.Countable) (hd : ∀ s ∈ S, Dense (f s)) : Dense (⋂ s ∈ S, f s) := by
r... | Mathlib.Topology.MetricSpace.Baire.206_0.GktojJRwRzEj9tj | /-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with
an index set which is a countable set in any type. -/
theorem dense_biInter_of_isOpen {S : Set β} {f : β → Set α} (ho : ∀ s ∈ S, IsOpen (f s))
(hS : S.Countable) (hd : ∀ s ∈ S, Dense (f s)) : Dense (⋂ s ∈ S, f s) | Mathlib_Topology_MetricSpace_Baire |
case ho
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
S : Set β
f : β → Set α
ho : ∀ s ∈ S, IsOpen (f s)
hS : Set.Countable S
hd : ∀ s ∈ S, Dense (f s)
⊢ ∀ s ∈ (fun s => f s) '' S, IsOpen s | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | rwa [ball_image_iff] | /-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with
an index set which is a countable set in any type. -/
theorem dense_biInter_of_isOpen {S : Set β} {f : β → Set α} (ho : ∀ s ∈ S, IsOpen (f s))
(hS : S.Countable) (hd : ∀ s ∈ S, Dense (f s)) : Dense (⋂ s ∈ S, f s) := by
r... | Mathlib.Topology.MetricSpace.Baire.206_0.GktojJRwRzEj9tj | /-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with
an index set which is a countable set in any type. -/
theorem dense_biInter_of_isOpen {S : Set β} {f : β → Set α} (ho : ∀ s ∈ S, IsOpen (f s))
(hS : S.Countable) (hd : ∀ s ∈ S, Dense (f s)) : Dense (⋂ s ∈ S, f s) | Mathlib_Topology_MetricSpace_Baire |
case hS
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
S : Set β
f : β → Set α
ho : ∀ s ∈ S, IsOpen (f s)
hS : Set.Countable S
hd : ∀ s ∈ S, Dense (f s)
⊢ Set.Countable ((fun s => f s) '' S) | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | exact hS.image _ | /-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with
an index set which is a countable set in any type. -/
theorem dense_biInter_of_isOpen {S : Set β} {f : β → Set α} (ho : ∀ s ∈ S, IsOpen (f s))
(hS : S.Countable) (hd : ∀ s ∈ S, Dense (f s)) : Dense (⋂ s ∈ S, f s) := by
r... | Mathlib.Topology.MetricSpace.Baire.206_0.GktojJRwRzEj9tj | /-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with
an index set which is a countable set in any type. -/
theorem dense_biInter_of_isOpen {S : Set β} {f : β → Set α} (ho : ∀ s ∈ S, IsOpen (f s))
(hS : S.Countable) (hd : ∀ s ∈ S, Dense (f s)) : Dense (⋂ s ∈ S, f s) | Mathlib_Topology_MetricSpace_Baire |
case hd
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
S : Set β
f : β → Set α
ho : ∀ s ∈ S, IsOpen (f s)
hS : Set.Countable S
hd : ∀ s ∈ S, Dense (f s)
⊢ ∀ s ∈ (fun s => f s) '' S, Dense s | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | rwa [ball_image_iff] | /-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with
an index set which is a countable set in any type. -/
theorem dense_biInter_of_isOpen {S : Set β} {f : β → Set α} (ho : ∀ s ∈ S, IsOpen (f s))
(hS : S.Countable) (hd : ∀ s ∈ S, Dense (f s)) : Dense (⋂ s ∈ S, f s) := by
r... | Mathlib.Topology.MetricSpace.Baire.206_0.GktojJRwRzEj9tj | /-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with
an index set which is a countable set in any type. -/
theorem dense_biInter_of_isOpen {S : Set β} {f : β → Set α} (ho : ∀ s ∈ S, IsOpen (f s))
(hS : S.Countable) (hd : ∀ s ∈ S, Dense (f s)) : Dense (⋂ s ∈ S, f s) | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : BaireSpace α
inst✝ : Encodable β
f : β → Set α
ho : ∀ (s : β), IsOpen (f s)
hd : ∀ (s : β), Dense (f s)
⊢ Dense (⋂ s, f s) | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | rw [← sInter_range] | /-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with
an index set which is an encodable type. -/
theorem dense_iInter_of_isOpen [Encodable β] {f : β → Set α} (ho : ∀ s, IsOpen (f s))
(hd : ∀ s, Dense (f s)) : Dense (⋂ s, f s) := by
| Mathlib.Topology.MetricSpace.Baire.217_0.GktojJRwRzEj9tj | /-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with
an index set which is an encodable type. -/
theorem dense_iInter_of_isOpen [Encodable β] {f : β → Set α} (ho : ∀ s, IsOpen (f s))
(hd : ∀ s, Dense (f s)) : Dense (⋂ s, f s) | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : BaireSpace α
inst✝ : Encodable β
f : β → Set α
ho : ∀ (s : β), IsOpen (f s)
hd : ∀ (s : β), Dense (f s)
⊢ Dense (⋂₀ range fun s => f s) | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | apply dense_sInter_of_isOpen | /-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with
an index set which is an encodable type. -/
theorem dense_iInter_of_isOpen [Encodable β] {f : β → Set α} (ho : ∀ s, IsOpen (f s))
(hd : ∀ s, Dense (f s)) : Dense (⋂ s, f s) := by
rw [← sInter_range]
| Mathlib.Topology.MetricSpace.Baire.217_0.GktojJRwRzEj9tj | /-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with
an index set which is an encodable type. -/
theorem dense_iInter_of_isOpen [Encodable β] {f : β → Set α} (ho : ∀ s, IsOpen (f s))
(hd : ∀ s, Dense (f s)) : Dense (⋂ s, f s) | Mathlib_Topology_MetricSpace_Baire |
case ho
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : BaireSpace α
inst✝ : Encodable β
f : β → Set α
ho : ∀ (s : β), IsOpen (f s)
hd : ∀ (s : β), Dense (f s)
⊢ ∀ s ∈ range fun s => f s, IsOpen s | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | rwa [forall_range_iff] | /-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with
an index set which is an encodable type. -/
theorem dense_iInter_of_isOpen [Encodable β] {f : β → Set α} (ho : ∀ s, IsOpen (f s))
(hd : ∀ s, Dense (f s)) : Dense (⋂ s, f s) := by
rw [← sInter_range]
apply dense_sInter_... | Mathlib.Topology.MetricSpace.Baire.217_0.GktojJRwRzEj9tj | /-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with
an index set which is an encodable type. -/
theorem dense_iInter_of_isOpen [Encodable β] {f : β → Set α} (ho : ∀ s, IsOpen (f s))
(hd : ∀ s, Dense (f s)) : Dense (⋂ s, f s) | Mathlib_Topology_MetricSpace_Baire |
case hS
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : BaireSpace α
inst✝ : Encodable β
f : β → Set α
ho : ∀ (s : β), IsOpen (f s)
hd : ∀ (s : β), Dense (f s)
⊢ Set.Countable (range fun s => f s) | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | exact countable_range _ | /-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with
an index set which is an encodable type. -/
theorem dense_iInter_of_isOpen [Encodable β] {f : β → Set α} (ho : ∀ s, IsOpen (f s))
(hd : ∀ s, Dense (f s)) : Dense (⋂ s, f s) := by
rw [← sInter_range]
apply dense_sInter_... | Mathlib.Topology.MetricSpace.Baire.217_0.GktojJRwRzEj9tj | /-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with
an index set which is an encodable type. -/
theorem dense_iInter_of_isOpen [Encodable β] {f : β → Set α} (ho : ∀ s, IsOpen (f s))
(hd : ∀ s, Dense (f s)) : Dense (⋂ s, f s) | Mathlib_Topology_MetricSpace_Baire |
case hd
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : BaireSpace α
inst✝ : Encodable β
f : β → Set α
ho : ∀ (s : β), IsOpen (f s)
hd : ∀ (s : β), Dense (f s)
⊢ ∀ s ∈ range fun s => f s, Dense s | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | rwa [forall_range_iff] | /-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with
an index set which is an encodable type. -/
theorem dense_iInter_of_isOpen [Encodable β] {f : β → Set α} (ho : ∀ s, IsOpen (f s))
(hd : ∀ s, Dense (f s)) : Dense (⋂ s, f s) := by
rw [← sInter_range]
apply dense_sInter_... | Mathlib.Topology.MetricSpace.Baire.217_0.GktojJRwRzEj9tj | /-- Baire theorem: a countable intersection of dense open sets is dense. Formulated here with
an index set which is an encodable type. -/
theorem dense_iInter_of_isOpen [Encodable β] {f : β → Set α} (ho : ∀ s, IsOpen (f s))
(hd : ∀ s, Dense (f s)) : Dense (⋂ s, f s) | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
s : Set α
⊢ s ∈ residual α ↔ ∃ t, ∃ (_ : t ⊆ s), IsGδ t ∧ Dense t | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | constructor | /-- A set is residual (comeagre) if and only if it includes a dense `Gδ` set. -/
theorem mem_residual {s : Set α} : s ∈ residual α ↔ ∃ (t : _) (_ : t ⊆ s), IsGδ t ∧ Dense t := by
| Mathlib.Topology.MetricSpace.Baire.228_0.GktojJRwRzEj9tj | /-- A set is residual (comeagre) if and only if it includes a dense `Gδ` set. -/
theorem mem_residual {s : Set α} : s ∈ residual α ↔ ∃ (t : _) (_ : t ⊆ s), IsGδ t ∧ Dense t | Mathlib_Topology_MetricSpace_Baire |
case mp
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
s : Set α
⊢ s ∈ residual α → ∃ t, ∃ (_ : t ⊆ s), IsGδ t ∧ Dense t | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | rw [mem_residual_iff] | /-- A set is residual (comeagre) if and only if it includes a dense `Gδ` set. -/
theorem mem_residual {s : Set α} : s ∈ residual α ↔ ∃ (t : _) (_ : t ⊆ s), IsGδ t ∧ Dense t := by
constructor
· | Mathlib.Topology.MetricSpace.Baire.228_0.GktojJRwRzEj9tj | /-- A set is residual (comeagre) if and only if it includes a dense `Gδ` set. -/
theorem mem_residual {s : Set α} : s ∈ residual α ↔ ∃ (t : _) (_ : t ⊆ s), IsGδ t ∧ Dense t | Mathlib_Topology_MetricSpace_Baire |
case mp
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
s : Set α
⊢ (∃ S, (∀ t ∈ S, IsOpen t) ∧ (∀ t ∈ S, Dense t) ∧ Set.Countable S ∧ ⋂₀ S ⊆ s) → ∃ t, ∃ (_ : t ⊆ s), IsGδ t ∧ Dense t | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | rintro ⟨S, hSo, hSd, Sct, Ss⟩ | /-- A set is residual (comeagre) if and only if it includes a dense `Gδ` set. -/
theorem mem_residual {s : Set α} : s ∈ residual α ↔ ∃ (t : _) (_ : t ⊆ s), IsGδ t ∧ Dense t := by
constructor
· rw [mem_residual_iff]
| Mathlib.Topology.MetricSpace.Baire.228_0.GktojJRwRzEj9tj | /-- A set is residual (comeagre) if and only if it includes a dense `Gδ` set. -/
theorem mem_residual {s : Set α} : s ∈ residual α ↔ ∃ (t : _) (_ : t ⊆ s), IsGδ t ∧ Dense t | Mathlib_Topology_MetricSpace_Baire |
case mp.intro.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
s : Set α
S : Set (Set α)
hSo : ∀ t ∈ S, IsOpen t
hSd : ∀ t ∈ S, Dense t
Sct : Set.Countable S
Ss : ⋂₀ S ⊆ s
⊢ ∃ t, ∃ (_ : t ⊆ s), IsGδ t ∧ Dense t | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | refine' ⟨_, Ss, ⟨_, fun t ht => hSo _ ht, Sct, rfl⟩, _⟩ | /-- A set is residual (comeagre) if and only if it includes a dense `Gδ` set. -/
theorem mem_residual {s : Set α} : s ∈ residual α ↔ ∃ (t : _) (_ : t ⊆ s), IsGδ t ∧ Dense t := by
constructor
· rw [mem_residual_iff]
rintro ⟨S, hSo, hSd, Sct, Ss⟩
| Mathlib.Topology.MetricSpace.Baire.228_0.GktojJRwRzEj9tj | /-- A set is residual (comeagre) if and only if it includes a dense `Gδ` set. -/
theorem mem_residual {s : Set α} : s ∈ residual α ↔ ∃ (t : _) (_ : t ⊆ s), IsGδ t ∧ Dense t | Mathlib_Topology_MetricSpace_Baire |
case mp.intro.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
s : Set α
S : Set (Set α)
hSo : ∀ t ∈ S, IsOpen t
hSd : ∀ t ∈ S, Dense t
Sct : Set.Countable S
Ss : ⋂₀ S ⊆ s
⊢ Dense (⋂₀ S) | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | exact dense_sInter_of_isOpen hSo Sct hSd | /-- A set is residual (comeagre) if and only if it includes a dense `Gδ` set. -/
theorem mem_residual {s : Set α} : s ∈ residual α ↔ ∃ (t : _) (_ : t ⊆ s), IsGδ t ∧ Dense t := by
constructor
· rw [mem_residual_iff]
rintro ⟨S, hSo, hSd, Sct, Ss⟩
refine' ⟨_, Ss, ⟨_, fun t ht => hSo _ ht, Sct, rfl⟩, _⟩
| Mathlib.Topology.MetricSpace.Baire.228_0.GktojJRwRzEj9tj | /-- A set is residual (comeagre) if and only if it includes a dense `Gδ` set. -/
theorem mem_residual {s : Set α} : s ∈ residual α ↔ ∃ (t : _) (_ : t ⊆ s), IsGδ t ∧ Dense t | Mathlib_Topology_MetricSpace_Baire |
case mpr
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
s : Set α
⊢ (∃ t, ∃ (_ : t ⊆ s), IsGδ t ∧ Dense t) → s ∈ residual α | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | rintro ⟨t, ts, ho, hd⟩ | /-- A set is residual (comeagre) if and only if it includes a dense `Gδ` set. -/
theorem mem_residual {s : Set α} : s ∈ residual α ↔ ∃ (t : _) (_ : t ⊆ s), IsGδ t ∧ Dense t := by
constructor
· rw [mem_residual_iff]
rintro ⟨S, hSo, hSd, Sct, Ss⟩
refine' ⟨_, Ss, ⟨_, fun t ht => hSo _ ht, Sct, rfl⟩, _⟩
exa... | Mathlib.Topology.MetricSpace.Baire.228_0.GktojJRwRzEj9tj | /-- A set is residual (comeagre) if and only if it includes a dense `Gδ` set. -/
theorem mem_residual {s : Set α} : s ∈ residual α ↔ ∃ (t : _) (_ : t ⊆ s), IsGδ t ∧ Dense t | Mathlib_Topology_MetricSpace_Baire |
case mpr.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
s t : Set α
ts : t ⊆ s
ho : IsGδ t
hd : Dense t
⊢ s ∈ residual α | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | exact mem_of_superset (residual_of_dense_Gδ ho hd) ts | /-- A set is residual (comeagre) if and only if it includes a dense `Gδ` set. -/
theorem mem_residual {s : Set α} : s ∈ residual α ↔ ∃ (t : _) (_ : t ⊆ s), IsGδ t ∧ Dense t := by
constructor
· rw [mem_residual_iff]
rintro ⟨S, hSo, hSd, Sct, Ss⟩
refine' ⟨_, Ss, ⟨_, fun t ht => hSo _ ht, Sct, rfl⟩, _⟩
exa... | Mathlib.Topology.MetricSpace.Baire.228_0.GktojJRwRzEj9tj | /-- A set is residual (comeagre) if and only if it includes a dense `Gδ` set. -/
theorem mem_residual {s : Set α} : s ∈ residual α ↔ ∃ (t : _) (_ : t ⊆ s), IsGδ t ∧ Dense t | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
p : α → Prop
⊢ (∀ᶠ (x : α) in residual α, p x) ↔ ∃ t, IsGδ t ∧ Dense t ∧ ∀ x ∈ t, p x | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | convert@mem_residual _ _ _ p | /-- A property holds on a residual (comeagre) set if and only if it holds on some dense `Gδ` set. -/
theorem eventually_residual {p : α → Prop} :
(∀ᶠ x in residual α, p x) ↔ ∃ t : Set α, IsGδ t ∧ Dense t ∧ ∀ x : α, x ∈ t → p x := by
-- this can probably be improved...
| Mathlib.Topology.MetricSpace.Baire.239_0.GktojJRwRzEj9tj | /-- A property holds on a residual (comeagre) set if and only if it holds on some dense `Gδ` set. -/
theorem eventually_residual {p : α → Prop} :
(∀ᶠ x in residual α, p x) ↔ ∃ t : Set α, IsGδ t ∧ Dense t ∧ ∀ x : α, x ∈ t → p x | Mathlib_Topology_MetricSpace_Baire |
case h.e'_2.h.e'_2.h.a
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
p : α → Prop
x✝ : Set α
⊢ (IsGδ x✝ ∧ Dense x✝ ∧ ∀ x ∈ x✝, p x) ↔ ∃ (_ : x✝ ⊆ p), IsGδ x✝ ∧ Dense x✝ | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | simp_rw [exists_prop, @and_comm ((_ : Set α) ⊆ p), and_assoc] | /-- A property holds on a residual (comeagre) set if and only if it holds on some dense `Gδ` set. -/
theorem eventually_residual {p : α → Prop} :
(∀ᶠ x in residual α, p x) ↔ ∃ t : Set α, IsGδ t ∧ Dense t ∧ ∀ x : α, x ∈ t → p x := by
-- this can probably be improved...
convert@mem_residual _ _ _ p
| Mathlib.Topology.MetricSpace.Baire.239_0.GktojJRwRzEj9tj | /-- A property holds on a residual (comeagre) set if and only if it holds on some dense `Gδ` set. -/
theorem eventually_residual {p : α → Prop} :
(∀ᶠ x in residual α, p x) ↔ ∃ t : Set α, IsGδ t ∧ Dense t ∧ ∀ x : α, x ∈ t → p x | Mathlib_Topology_MetricSpace_Baire |
case h.e'_2.h.e'_2.h.a
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
p : α → Prop
x✝ : Set α
⊢ (IsGδ x✝ ∧ Dense x✝ ∧ ∀ x ∈ x✝, p x) ↔ IsGδ x✝ ∧ Dense x✝ ∧ x✝ ⊆ p | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | rfl | /-- A property holds on a residual (comeagre) set if and only if it holds on some dense `Gδ` set. -/
theorem eventually_residual {p : α → Prop} :
(∀ᶠ x in residual α, p x) ↔ ∃ t : Set α, IsGδ t ∧ Dense t ∧ ∀ x : α, x ∈ t → p x := by
-- this can probably be improved...
convert@mem_residual _ _ _ p
simp_rw [exi... | Mathlib.Topology.MetricSpace.Baire.239_0.GktojJRwRzEj9tj | /-- A property holds on a residual (comeagre) set if and only if it holds on some dense `Gδ` set. -/
theorem eventually_residual {p : α → Prop} :
(∀ᶠ x in residual α, p x) ↔ ∃ t : Set α, IsGδ t ∧ Dense t ∧ ∀ x : α, x ∈ t → p x | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : BaireSpace α
inst✝ : Encodable β
f : β → Set α
ho : ∀ (s : β), IsGδ (f s)
hd : ∀ (s : β), Dense (f s)
⊢ Dense (⋂ s, f s) | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | rw [← sInter_range] | /-- Baire theorem: a countable intersection of dense Gδ sets is dense. Formulated here with
an index set which is an encodable type. -/
theorem dense_iInter_of_Gδ [Encodable β] {f : β → Set α} (ho : ∀ s, IsGδ (f s))
(hd : ∀ s, Dense (f s)) : Dense (⋂ s, f s) := by
| Mathlib.Topology.MetricSpace.Baire.261_0.GktojJRwRzEj9tj | /-- Baire theorem: a countable intersection of dense Gδ sets is dense. Formulated here with
an index set which is an encodable type. -/
theorem dense_iInter_of_Gδ [Encodable β] {f : β → Set α} (ho : ∀ s, IsGδ (f s))
(hd : ∀ s, Dense (f s)) : Dense (⋂ s, f s) | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : BaireSpace α
inst✝ : Encodable β
f : β → Set α
ho : ∀ (s : β), IsGδ (f s)
hd : ∀ (s : β), Dense (f s)
⊢ Dense (⋂₀ range fun s => f s) | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | exact dense_sInter_of_Gδ (forall_range_iff.2 ‹_›) (countable_range _) (forall_range_iff.2 ‹_›) | /-- Baire theorem: a countable intersection of dense Gδ sets is dense. Formulated here with
an index set which is an encodable type. -/
theorem dense_iInter_of_Gδ [Encodable β] {f : β → Set α} (ho : ∀ s, IsGδ (f s))
(hd : ∀ s, Dense (f s)) : Dense (⋂ s, f s) := by
rw [← sInter_range]
| Mathlib.Topology.MetricSpace.Baire.261_0.GktojJRwRzEj9tj | /-- Baire theorem: a countable intersection of dense Gδ sets is dense. Formulated here with
an index set which is an encodable type. -/
theorem dense_iInter_of_Gδ [Encodable β] {f : β → Set α} (ho : ∀ s, IsGδ (f s))
(hd : ∀ s, Dense (f s)) : Dense (⋂ s, f s) | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
S : Set β
f : (x : β) → x ∈ S → Set α
ho : ∀ (s : β) (H : s ∈ S), IsGδ (f s H)
hS : Set.Countable S
hd : ∀ (s : β) (H : s ∈ S), Dense (f s H)
⊢ Dense (⋂ s, ⋂ (h : s ∈ S), f s h) | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | rw [biInter_eq_iInter] | /-- Baire theorem: a countable intersection of dense Gδ sets is dense. Formulated here with
an index set which is a countable set in any type. -/
theorem dense_biInter_of_Gδ {S : Set β} {f : ∀ x ∈ S, Set α} (ho : ∀ s (H : s ∈ S), IsGδ (f s H))
(hS : S.Countable) (hd : ∀ s (H : s ∈ S), Dense (f s H)) : Dense (⋂ s ∈ ... | Mathlib.Topology.MetricSpace.Baire.271_0.GktojJRwRzEj9tj | /-- Baire theorem: a countable intersection of dense Gδ sets is dense. Formulated here with
an index set which is a countable set in any type. -/
theorem dense_biInter_of_Gδ {S : Set β} {f : ∀ x ∈ S, Set α} (ho : ∀ s (H : s ∈ S), IsGδ (f s H))
(hS : S.Countable) (hd : ∀ s (H : s ∈ S), Dense (f s H)) : Dense (⋂ s ∈ ... | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
S : Set β
f : (x : β) → x ∈ S → Set α
ho : ∀ (s : β) (H : s ∈ S), IsGδ (f s H)
hS : Set.Countable S
hd : ∀ (s : β) (H : s ∈ S), Dense (f s H)
⊢ Dense (⋂ x, f ↑x (_ : ↑x ∈ S)) | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | haveI := hS.toEncodable | /-- Baire theorem: a countable intersection of dense Gδ sets is dense. Formulated here with
an index set which is a countable set in any type. -/
theorem dense_biInter_of_Gδ {S : Set β} {f : ∀ x ∈ S, Set α} (ho : ∀ s (H : s ∈ S), IsGδ (f s H))
(hS : S.Countable) (hd : ∀ s (H : s ∈ S), Dense (f s H)) : Dense (⋂ s ∈ ... | Mathlib.Topology.MetricSpace.Baire.271_0.GktojJRwRzEj9tj | /-- Baire theorem: a countable intersection of dense Gδ sets is dense. Formulated here with
an index set which is a countable set in any type. -/
theorem dense_biInter_of_Gδ {S : Set β} {f : ∀ x ∈ S, Set α} (ho : ∀ s (H : s ∈ S), IsGδ (f s H))
(hS : S.Countable) (hd : ∀ s (H : s ∈ S), Dense (f s H)) : Dense (⋂ s ∈ ... | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
S : Set β
f : (x : β) → x ∈ S → Set α
ho : ∀ (s : β) (H : s ∈ S), IsGδ (f s H)
hS : Set.Countable S
hd : ∀ (s : β) (H : s ∈ S), Dense (f s H)
this : Encodable ↑S
⊢ Dense (⋂ x, f ↑x (_ : ↑x ∈ S)) | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | exact dense_iInter_of_Gδ (fun s => ho s s.2) fun s => hd s s.2 | /-- Baire theorem: a countable intersection of dense Gδ sets is dense. Formulated here with
an index set which is a countable set in any type. -/
theorem dense_biInter_of_Gδ {S : Set β} {f : ∀ x ∈ S, Set α} (ho : ∀ s (H : s ∈ S), IsGδ (f s H))
(hS : S.Countable) (hd : ∀ s (H : s ∈ S), Dense (f s H)) : Dense (⋂ s ∈ ... | Mathlib.Topology.MetricSpace.Baire.271_0.GktojJRwRzEj9tj | /-- Baire theorem: a countable intersection of dense Gδ sets is dense. Formulated here with
an index set which is a countable set in any type. -/
theorem dense_biInter_of_Gδ {S : Set β} {f : ∀ x ∈ S, Set α} (ho : ∀ s (H : s ∈ S), IsGδ (f s H))
(hS : S.Countable) (hd : ∀ s (H : s ∈ S), Dense (f s H)) : Dense (⋂ s ∈ ... | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
s t : Set α
hs : IsGδ s
ht : IsGδ t
hsc : Dense s
htc : Dense t
⊢ Dense (s ∩ t) | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | rw [inter_eq_iInter] | /-- Baire theorem: the intersection of two dense Gδ sets is dense. -/
theorem Dense.inter_of_Gδ {s t : Set α} (hs : IsGδ s) (ht : IsGδ t) (hsc : Dense s)
(htc : Dense t) : Dense (s ∩ t) := by
| Mathlib.Topology.MetricSpace.Baire.281_0.GktojJRwRzEj9tj | /-- Baire theorem: the intersection of two dense Gδ sets is dense. -/
theorem Dense.inter_of_Gδ {s t : Set α} (hs : IsGδ s) (ht : IsGδ t) (hsc : Dense s)
(htc : Dense t) : Dense (s ∩ t) | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
s t : Set α
hs : IsGδ s
ht : IsGδ t
hsc : Dense s
htc : Dense t
⊢ Dense (⋂ b, bif b then s else t) | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | apply dense_iInter_of_Gδ | /-- Baire theorem: the intersection of two dense Gδ sets is dense. -/
theorem Dense.inter_of_Gδ {s t : Set α} (hs : IsGδ s) (ht : IsGδ t) (hsc : Dense s)
(htc : Dense t) : Dense (s ∩ t) := by
rw [inter_eq_iInter]
| Mathlib.Topology.MetricSpace.Baire.281_0.GktojJRwRzEj9tj | /-- Baire theorem: the intersection of two dense Gδ sets is dense. -/
theorem Dense.inter_of_Gδ {s t : Set α} (hs : IsGδ s) (ht : IsGδ t) (hsc : Dense s)
(htc : Dense t) : Dense (s ∩ t) | Mathlib_Topology_MetricSpace_Baire |
case ho
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
s t : Set α
hs : IsGδ s
ht : IsGδ t
hsc : Dense s
htc : Dense t
⊢ ∀ (s_1 : Bool), IsGδ (bif s_1 then s else t) | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | simp [Bool.forall_bool, *] | /-- Baire theorem: the intersection of two dense Gδ sets is dense. -/
theorem Dense.inter_of_Gδ {s t : Set α} (hs : IsGδ s) (ht : IsGδ t) (hsc : Dense s)
(htc : Dense t) : Dense (s ∩ t) := by
rw [inter_eq_iInter]
apply dense_iInter_of_Gδ <;> | Mathlib.Topology.MetricSpace.Baire.281_0.GktojJRwRzEj9tj | /-- Baire theorem: the intersection of two dense Gδ sets is dense. -/
theorem Dense.inter_of_Gδ {s t : Set α} (hs : IsGδ s) (ht : IsGδ t) (hsc : Dense s)
(htc : Dense t) : Dense (s ∩ t) | Mathlib_Topology_MetricSpace_Baire |
case hd
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
s t : Set α
hs : IsGδ s
ht : IsGδ t
hsc : Dense s
htc : Dense t
⊢ ∀ (s_1 : Bool), Dense (bif s_1 then s else t) | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | simp [Bool.forall_bool, *] | /-- Baire theorem: the intersection of two dense Gδ sets is dense. -/
theorem Dense.inter_of_Gδ {s t : Set α} (hs : IsGδ s) (ht : IsGδ t) (hsc : Dense s)
(htc : Dense t) : Dense (s ∩ t) := by
rw [inter_eq_iInter]
apply dense_iInter_of_Gδ <;> | Mathlib.Topology.MetricSpace.Baire.281_0.GktojJRwRzEj9tj | /-- Baire theorem: the intersection of two dense Gδ sets is dense. -/
theorem Dense.inter_of_Gδ {s t : Set α} (hs : IsGδ s) (ht : IsGδ t) (hsc : Dense s)
(htc : Dense t) : Dense (s ∩ t) | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : BaireSpace α
inst✝ : Encodable ι
s : Set α
hs : IsGδ s
hd : Dense s
f : ι → Set α
hc : ∀ (i : ι), IsClosed (f i)
hU : s ⊆ ⋃ i, f i
⊢ Dense (⋃ i, interior (f i)) | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | let g i := (frontier (f i))ᶜ | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with `⋃`. -/
theorem IsGδ.dense_iUnion_interior_of_closed [Encodable ι] {s : Set α} (hs : IsGδ s) (hd : Dense s)
{f : ι → Set α} (hc : ∀ i, IsClosed (f i)) (hU : s ⊆ ⋃ i, f i) :
Dense (⋃ ... | Mathlib.Topology.MetricSpace.Baire.289_0.GktojJRwRzEj9tj | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with `⋃`. -/
theorem IsGδ.dense_iUnion_interior_of_closed [Encodable ι] {s : Set α} (hs : IsGδ s) (hd : Dense s)
{f : ι → Set α} (hc : ∀ i, IsClosed (f i)) (hU : s ⊆ ⋃ i, f i) :
Dense (⋃ ... | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : BaireSpace α
inst✝ : Encodable ι
s : Set α
hs : IsGδ s
hd : Dense s
f : ι → Set α
hc : ∀ (i : ι), IsClosed (f i)
hU : s ⊆ ⋃ i, f i
g : ι → Set α := fun i => (frontier (f i))ᶜ
⊢ Dense (⋃ i, interior (f i)) | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | have hgo : ∀ i, IsOpen (g i) := fun i => isClosed_frontier.isOpen_compl | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with `⋃`. -/
theorem IsGδ.dense_iUnion_interior_of_closed [Encodable ι] {s : Set α} (hs : IsGδ s) (hd : Dense s)
{f : ι → Set α} (hc : ∀ i, IsClosed (f i)) (hU : s ⊆ ⋃ i, f i) :
Dense (⋃ ... | Mathlib.Topology.MetricSpace.Baire.289_0.GktojJRwRzEj9tj | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with `⋃`. -/
theorem IsGδ.dense_iUnion_interior_of_closed [Encodable ι] {s : Set α} (hs : IsGδ s) (hd : Dense s)
{f : ι → Set α} (hc : ∀ i, IsClosed (f i)) (hU : s ⊆ ⋃ i, f i) :
Dense (⋃ ... | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : BaireSpace α
inst✝ : Encodable ι
s : Set α
hs : IsGδ s
hd : Dense s
f : ι → Set α
hc : ∀ (i : ι), IsClosed (f i)
hU : s ⊆ ⋃ i, f i
g : ι → Set α := fun i => (frontier (f i))ᶜ
hgo : ∀ (i : ι), IsOpen (g i)
⊢ Dense (⋃ i, interior (f ... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | have hgd : Dense (⋂ i, g i) := by
refine' dense_iInter_of_isOpen hgo fun i x => _
rw [closure_compl, interior_frontier (hc _)]
exact id | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with `⋃`. -/
theorem IsGδ.dense_iUnion_interior_of_closed [Encodable ι] {s : Set α} (hs : IsGδ s) (hd : Dense s)
{f : ι → Set α} (hc : ∀ i, IsClosed (f i)) (hU : s ⊆ ⋃ i, f i) :
Dense (⋃ ... | Mathlib.Topology.MetricSpace.Baire.289_0.GktojJRwRzEj9tj | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with `⋃`. -/
theorem IsGδ.dense_iUnion_interior_of_closed [Encodable ι] {s : Set α} (hs : IsGδ s) (hd : Dense s)
{f : ι → Set α} (hc : ∀ i, IsClosed (f i)) (hU : s ⊆ ⋃ i, f i) :
Dense (⋃ ... | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : BaireSpace α
inst✝ : Encodable ι
s : Set α
hs : IsGδ s
hd : Dense s
f : ι → Set α
hc : ∀ (i : ι), IsClosed (f i)
hU : s ⊆ ⋃ i, f i
g : ι → Set α := fun i => (frontier (f i))ᶜ
hgo : ∀ (i : ι), IsOpen (g i)
⊢ Dense (⋂ i, g i) | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | refine' dense_iInter_of_isOpen hgo fun i x => _ | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with `⋃`. -/
theorem IsGδ.dense_iUnion_interior_of_closed [Encodable ι] {s : Set α} (hs : IsGδ s) (hd : Dense s)
{f : ι → Set α} (hc : ∀ i, IsClosed (f i)) (hU : s ⊆ ⋃ i, f i) :
Dense (⋃ ... | Mathlib.Topology.MetricSpace.Baire.289_0.GktojJRwRzEj9tj | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with `⋃`. -/
theorem IsGδ.dense_iUnion_interior_of_closed [Encodable ι] {s : Set α} (hs : IsGδ s) (hd : Dense s)
{f : ι → Set α} (hc : ∀ i, IsClosed (f i)) (hU : s ⊆ ⋃ i, f i) :
Dense (⋃ ... | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : BaireSpace α
inst✝ : Encodable ι
s : Set α
hs : IsGδ s
hd : Dense s
f : ι → Set α
hc : ∀ (i : ι), IsClosed (f i)
hU : s ⊆ ⋃ i, f i
g : ι → Set α := fun i => (frontier (f i))ᶜ
hgo : ∀ (i : ι), IsOpen (g i)
i : ι
x : α
⊢ x ∈ closure ... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | rw [closure_compl, interior_frontier (hc _)] | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with `⋃`. -/
theorem IsGδ.dense_iUnion_interior_of_closed [Encodable ι] {s : Set α} (hs : IsGδ s) (hd : Dense s)
{f : ι → Set α} (hc : ∀ i, IsClosed (f i)) (hU : s ⊆ ⋃ i, f i) :
Dense (⋃ ... | Mathlib.Topology.MetricSpace.Baire.289_0.GktojJRwRzEj9tj | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with `⋃`. -/
theorem IsGδ.dense_iUnion_interior_of_closed [Encodable ι] {s : Set α} (hs : IsGδ s) (hd : Dense s)
{f : ι → Set α} (hc : ∀ i, IsClosed (f i)) (hU : s ⊆ ⋃ i, f i) :
Dense (⋃ ... | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : BaireSpace α
inst✝ : Encodable ι
s : Set α
hs : IsGδ s
hd : Dense s
f : ι → Set α
hc : ∀ (i : ι), IsClosed (f i)
hU : s ⊆ ⋃ i, f i
g : ι → Set α := fun i => (frontier (f i))ᶜ
hgo : ∀ (i : ι), IsOpen (g i)
i : ι
x : α
⊢ x ∈ ∅ᶜ | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | exact id | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with `⋃`. -/
theorem IsGδ.dense_iUnion_interior_of_closed [Encodable ι] {s : Set α} (hs : IsGδ s) (hd : Dense s)
{f : ι → Set α} (hc : ∀ i, IsClosed (f i)) (hU : s ⊆ ⋃ i, f i) :
Dense (⋃ ... | Mathlib.Topology.MetricSpace.Baire.289_0.GktojJRwRzEj9tj | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with `⋃`. -/
theorem IsGδ.dense_iUnion_interior_of_closed [Encodable ι] {s : Set α} (hs : IsGδ s) (hd : Dense s)
{f : ι → Set α} (hc : ∀ i, IsClosed (f i)) (hU : s ⊆ ⋃ i, f i) :
Dense (⋃ ... | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : BaireSpace α
inst✝ : Encodable ι
s : Set α
hs : IsGδ s
hd : Dense s
f : ι → Set α
hc : ∀ (i : ι), IsClosed (f i)
hU : s ⊆ ⋃ i, f i
g : ι → Set α := fun i => (frontier (f i))ᶜ
hgo : ∀ (i : ι), IsOpen (g i)
hgd : Dense (⋂ i, g i)
⊢ D... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | refine' (hd.inter_of_Gδ hs (isGδ_iInter_of_isOpen fun i => (hgo i)) hgd).mono _ | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with `⋃`. -/
theorem IsGδ.dense_iUnion_interior_of_closed [Encodable ι] {s : Set α} (hs : IsGδ s) (hd : Dense s)
{f : ι → Set α} (hc : ∀ i, IsClosed (f i)) (hU : s ⊆ ⋃ i, f i) :
Dense (⋃ ... | Mathlib.Topology.MetricSpace.Baire.289_0.GktojJRwRzEj9tj | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with `⋃`. -/
theorem IsGδ.dense_iUnion_interior_of_closed [Encodable ι] {s : Set α} (hs : IsGδ s) (hd : Dense s)
{f : ι → Set α} (hc : ∀ i, IsClosed (f i)) (hU : s ⊆ ⋃ i, f i) :
Dense (⋃ ... | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : BaireSpace α
inst✝ : Encodable ι
s : Set α
hs : IsGδ s
hd : Dense s
f : ι → Set α
hc : ∀ (i : ι), IsClosed (f i)
hU : s ⊆ ⋃ i, f i
g : ι → Set α := fun i => (frontier (f i))ᶜ
hgo : ∀ (i : ι), IsOpen (g i)
hgd : Dense (⋂ i, g i)
⊢ s... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | rintro x ⟨hxs, hxg⟩ | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with `⋃`. -/
theorem IsGδ.dense_iUnion_interior_of_closed [Encodable ι] {s : Set α} (hs : IsGδ s) (hd : Dense s)
{f : ι → Set α} (hc : ∀ i, IsClosed (f i)) (hU : s ⊆ ⋃ i, f i) :
Dense (⋃ ... | Mathlib.Topology.MetricSpace.Baire.289_0.GktojJRwRzEj9tj | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with `⋃`. -/
theorem IsGδ.dense_iUnion_interior_of_closed [Encodable ι] {s : Set α} (hs : IsGδ s) (hd : Dense s)
{f : ι → Set α} (hc : ∀ i, IsClosed (f i)) (hU : s ⊆ ⋃ i, f i) :
Dense (⋃ ... | Mathlib_Topology_MetricSpace_Baire |
case intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : BaireSpace α
inst✝ : Encodable ι
s : Set α
hs : IsGδ s
hd : Dense s
f : ι → Set α
hc : ∀ (i : ι), IsClosed (f i)
hU : s ⊆ ⋃ i, f i
g : ι → Set α := fun i => (frontier (f i))ᶜ
hgo : ∀ (i : ι), IsOpen (g i)
hgd : Dense (⋂ ... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | rw [mem_iInter] at hxg | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with `⋃`. -/
theorem IsGδ.dense_iUnion_interior_of_closed [Encodable ι] {s : Set α} (hs : IsGδ s) (hd : Dense s)
{f : ι → Set α} (hc : ∀ i, IsClosed (f i)) (hU : s ⊆ ⋃ i, f i) :
Dense (⋃ ... | Mathlib.Topology.MetricSpace.Baire.289_0.GktojJRwRzEj9tj | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with `⋃`. -/
theorem IsGδ.dense_iUnion_interior_of_closed [Encodable ι] {s : Set α} (hs : IsGδ s) (hd : Dense s)
{f : ι → Set α} (hc : ∀ i, IsClosed (f i)) (hU : s ⊆ ⋃ i, f i) :
Dense (⋃ ... | Mathlib_Topology_MetricSpace_Baire |
case intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : BaireSpace α
inst✝ : Encodable ι
s : Set α
hs : IsGδ s
hd : Dense s
f : ι → Set α
hc : ∀ (i : ι), IsClosed (f i)
hU : s ⊆ ⋃ i, f i
g : ι → Set α := fun i => (frontier (f i))ᶜ
hgo : ∀ (i : ι), IsOpen (g i)
hgd : Dense (⋂ ... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | rcases mem_iUnion.1 (hU hxs) with ⟨i, hi⟩ | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with `⋃`. -/
theorem IsGδ.dense_iUnion_interior_of_closed [Encodable ι] {s : Set α} (hs : IsGδ s) (hd : Dense s)
{f : ι → Set α} (hc : ∀ i, IsClosed (f i)) (hU : s ⊆ ⋃ i, f i) :
Dense (⋃ ... | Mathlib.Topology.MetricSpace.Baire.289_0.GktojJRwRzEj9tj | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with `⋃`. -/
theorem IsGδ.dense_iUnion_interior_of_closed [Encodable ι] {s : Set α} (hs : IsGδ s) (hd : Dense s)
{f : ι → Set α} (hc : ∀ i, IsClosed (f i)) (hU : s ⊆ ⋃ i, f i) :
Dense (⋃ ... | Mathlib_Topology_MetricSpace_Baire |
case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : BaireSpace α
inst✝ : Encodable ι
s : Set α
hs : IsGδ s
hd : Dense s
f : ι → Set α
hc : ∀ (i : ι), IsClosed (f i)
hU : s ⊆ ⋃ i, f i
g : ι → Set α := fun i => (frontier (f i))ᶜ
hgo : ∀ (i : ι), IsOpen (g i)
hgd : Den... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | exact mem_iUnion.2 ⟨i, self_diff_frontier (f i) ▸ ⟨hi, hxg _⟩⟩ | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with `⋃`. -/
theorem IsGδ.dense_iUnion_interior_of_closed [Encodable ι] {s : Set α} (hs : IsGδ s) (hd : Dense s)
{f : ι → Set α} (hc : ∀ i, IsClosed (f i)) (hU : s ⊆ ⋃ i, f i) :
Dense (⋃ ... | Mathlib.Topology.MetricSpace.Baire.289_0.GktojJRwRzEj9tj | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with `⋃`. -/
theorem IsGδ.dense_iUnion_interior_of_closed [Encodable ι] {s : Set α} (hs : IsGδ s) (hd : Dense s)
{f : ι → Set α} (hc : ∀ i, IsClosed (f i)) (hU : s ⊆ ⋃ i, f i) :
Dense (⋃ ... | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
t : Set ι
s : Set α
hs : IsGδ s
hd : Dense s
ht : Set.Countable t
f : ι → Set α
hc : ∀ i ∈ t, IsClosed (f i)
hU : s ⊆ ⋃ i ∈ t, f i
⊢ Dense (⋃ i ∈ t, interior (f i)) | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | haveI := ht.toEncodable | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with a union over a countable set in any type. -/
theorem IsGδ.dense_biUnion_interior_of_closed {t : Set ι} {s : Set α} (hs : IsGδ s) (hd : Dense s)
(ht : t.Countable) {f : ι → Set α} (hc : ∀... | Mathlib.Topology.MetricSpace.Baire.308_0.GktojJRwRzEj9tj | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with a union over a countable set in any type. -/
theorem IsGδ.dense_biUnion_interior_of_closed {t : Set ι} {s : Set α} (hs : IsGδ s) (hd : Dense s)
(ht : t.Countable) {f : ι → Set α} (hc : ∀... | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
t : Set ι
s : Set α
hs : IsGδ s
hd : Dense s
ht : Set.Countable t
f : ι → Set α
hc : ∀ i ∈ t, IsClosed (f i)
hU : s ⊆ ⋃ i ∈ t, f i
this : Encodable ↑t
⊢ Dense (⋃ i ∈ t, interior (f i)) | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | simp only [biUnion_eq_iUnion, SetCoe.forall'] at * | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with a union over a countable set in any type. -/
theorem IsGδ.dense_biUnion_interior_of_closed {t : Set ι} {s : Set α} (hs : IsGδ s) (hd : Dense s)
(ht : t.Countable) {f : ι → Set α} (hc : ∀... | Mathlib.Topology.MetricSpace.Baire.308_0.GktojJRwRzEj9tj | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with a union over a countable set in any type. -/
theorem IsGδ.dense_biUnion_interior_of_closed {t : Set ι} {s : Set α} (hs : IsGδ s) (hd : Dense s)
(ht : t.Countable) {f : ι → Set α} (hc : ∀... | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
t : Set ι
s : Set α
hs : IsGδ s
hd : Dense s
ht : Set.Countable t
f : ι → Set α
this : Encodable ↑t
hc : ∀ (x : ↑t), IsClosed (f ↑x)
hU : s ⊆ ⋃ x, f ↑x
⊢ Dense (⋃ x, interior (f ↑x)) | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | exact hs.dense_iUnion_interior_of_closed hd hc hU | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with a union over a countable set in any type. -/
theorem IsGδ.dense_biUnion_interior_of_closed {t : Set ι} {s : Set α} (hs : IsGδ s) (hd : Dense s)
(ht : t.Countable) {f : ι → Set α} (hc : ∀... | Mathlib.Topology.MetricSpace.Baire.308_0.GktojJRwRzEj9tj | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with a union over a countable set in any type. -/
theorem IsGδ.dense_biUnion_interior_of_closed {t : Set ι} {s : Set α} (hs : IsGδ s) (hd : Dense s)
(ht : t.Countable) {f : ι → Set α} (hc : ∀... | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : TopologicalSpace α
inst✝ : BaireSpace α
T : Set (Set α)
s : Set α
hs : IsGδ s
hd : Dense s
hc : Set.Countable T
hc' : ∀ t ∈ T, IsClosed t
hU : s ⊆ ⋃₀ T
⊢ s ⊆ ⋃ i ∈ T, i | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | rwa [← sUnion_eq_biUnion] | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with `⋃₀`. -/
theorem IsGδ.dense_sUnion_interior_of_closed {T : Set (Set α)} {s : Set α} (hs : IsGδ s)
(hd : Dense s) (hc : T.Countable) (hc' : ∀ t ∈ T, IsClosed t) (hU : s ⊆ ⋃₀ T) :
Dens... | Mathlib.Topology.MetricSpace.Baire.319_0.GktojJRwRzEj9tj | /-- If a countable family of closed sets cover a dense `Gδ` set, then the union of their interiors
is dense. Formulated here with `⋃₀`. -/
theorem IsGδ.dense_sUnion_interior_of_closed {T : Set (Set α)} {s : Set α} (hs : IsGδ s)
(hd : Dense s) (hc : T.Countable) (hc' : ∀ t ∈ T, IsClosed t) (hU : s ⊆ ⋃₀ T) :
Dens... | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝³ : TopologicalSpace α
inst✝² : BaireSpace α
inst✝¹ : Nonempty α
inst✝ : Encodable β
f : β → Set α
hc : ∀ (s : β), IsClosed (f s)
hU : ⋃ s, f s = univ
⊢ ∃ s, Set.Nonempty (interior (f s)) | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.GDelta
import Mathlib.Topology.Sets.Compacts
imp... | simpa using (dense_iUnion_interior_of_closed hc hU).nonempty | /-- One of the most useful consequences of Baire theorem: if a countable union of closed sets
covers the space, then one of the sets has nonempty interior. -/
theorem nonempty_interior_of_iUnion_of_closed [Nonempty α] [Encodable β] {f : β → Set α}
(hc : ∀ s, IsClosed (f s)) (hU : ⋃ s, f s = univ) : ∃ s, (interior <... | Mathlib.Topology.MetricSpace.Baire.349_0.GktojJRwRzEj9tj | /-- One of the most useful consequences of Baire theorem: if a countable union of closed sets
covers the space, then one of the sets has nonempty interior. -/
theorem nonempty_interior_of_iUnion_of_closed [Nonempty α] [Encodable β] {f : β → Set α}
(hc : ∀ s, IsClosed (f s)) (hU : ⋃ s, f s = univ) : ∃ s, (interior <... | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
inst✝³ : Ring β
inst✝² : LinearOrderedField α
inst✝¹ : Archimedean α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, abs' (f n) ≤ a
hnm : ∀ n ≥ m, f (Nat.succ n) ≤ f n
ε : α
ε0 : ε > 0
⊢ ∃ i, ∀ j ≥ i, abs' (f j - f i) < ε | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | let ⟨k, hk⟩ := Archimedean.arch a ε0 | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f := fun ε ε0 => by
| Mathlib.Data.Complex.Exponential.38_0.1bixbwhfBeJKySp | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝³ : Ring β
inst✝² : LinearOrderedField α
inst✝¹ : Archimedean α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, abs' (f n) ≤ a
hnm : ∀ n ≥ m, f (Nat.succ n) ≤ f n
ε : α
ε0 : ε > 0
k : ℕ
hk : a ≤ k • ε
⊢ ∃ i, ∀ j ≥ i, abs' (f j - f i) < ε | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have h : ∃ l, ∀ n ≥ m, a - l • ε < f n :=
⟨k + k + 1, fun n hnm =>
lt_of_lt_of_le
(show a - (k + (k + 1)) • ε < -|f n| from
lt_neg.1 <|
lt_of_le_of_lt (ham n hnm)
(by
rw [neg_sub, lt_sub_iff_add_lt, add_nsmul, add_nsmul, one_nsmul]
ex... | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f := fun ε ε0 => by
let ⟨k, hk⟩ := Archimedean.arch a ε0
| Mathlib.Data.Complex.Exponential.38_0.1bixbwhfBeJKySp | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝³ : Ring β
inst✝² : LinearOrderedField α
inst✝¹ : Archimedean α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, abs' (f n) ≤ a
hnm✝ : ∀ n ≥ m, f (Nat.succ n) ≤ f n
ε : α
ε0 : ε > 0
k : ℕ
hk : a ≤ k • ε
n : ℕ
hnm : n ≥ m
⊢ a < -(a - (k + (k + 1)) • ε) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [neg_sub, lt_sub_iff_add_lt, add_nsmul, add_nsmul, one_nsmul] | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f := fun ε ε0 => by
let ⟨k, hk⟩ := Archimedean.arch a ε0
have h : ∃ l, ∀ n ≥ m, a - l • ε < f n :=
⟨k + k + 1, fun n hnm =>
lt_of_lt_of_le
(show a - (k +... | Mathlib.Data.Complex.Exponential.38_0.1bixbwhfBeJKySp | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝³ : Ring β
inst✝² : LinearOrderedField α
inst✝¹ : Archimedean α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, abs' (f n) ≤ a
hnm✝ : ∀ n ≥ m, f (Nat.succ n) ≤ f n
ε : α
ε0 : ε > 0
k : ℕ
hk : a ≤ k • ε
n : ℕ
hnm : n ≥ m
⊢ a + a < k • ε + (k • ε + ε) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | exact add_lt_add_of_le_of_lt hk (lt_of_le_of_lt hk (lt_add_of_pos_right _ ε0)) | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f := fun ε ε0 => by
let ⟨k, hk⟩ := Archimedean.arch a ε0
have h : ∃ l, ∀ n ≥ m, a - l • ε < f n :=
⟨k + k + 1, fun n hnm =>
lt_of_lt_of_le
(show a - (k +... | Mathlib.Data.Complex.Exponential.38_0.1bixbwhfBeJKySp | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝³ : Ring β
inst✝² : LinearOrderedField α
inst✝¹ : Archimedean α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, abs' (f n) ≤ a
hnm : ∀ n ≥ m, f (Nat.succ n) ≤ f n
ε : α
ε0 : ε > 0
k : ℕ
hk : a ≤ k • ε
h : ∃ l, ∀ n ≥ m, a - l • ε < f n
⊢ ∃ i, ∀ j ≥ i, abs' (f j... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | let l := Nat.find h | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f := fun ε ε0 => by
let ⟨k, hk⟩ := Archimedean.arch a ε0
have h : ∃ l, ∀ n ≥ m, a - l • ε < f n :=
⟨k + k + 1, fun n hnm =>
lt_of_lt_of_le
(show a - (k +... | Mathlib.Data.Complex.Exponential.38_0.1bixbwhfBeJKySp | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝³ : Ring β
inst✝² : LinearOrderedField α
inst✝¹ : Archimedean α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, abs' (f n) ≤ a
hnm : ∀ n ≥ m, f (Nat.succ n) ≤ f n
ε : α
ε0 : ε > 0
k : ℕ
hk : a ≤ k • ε
h : ∃ l, ∀ n ≥ m, a - l • ε < f n
l : ℕ := Nat.find h
⊢ ∃ i... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have hl : ∀ n : ℕ, n ≥ m → f n > a - l • ε := Nat.find_spec h | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f := fun ε ε0 => by
let ⟨k, hk⟩ := Archimedean.arch a ε0
have h : ∃ l, ∀ n ≥ m, a - l • ε < f n :=
⟨k + k + 1, fun n hnm =>
lt_of_lt_of_le
(show a - (k +... | Mathlib.Data.Complex.Exponential.38_0.1bixbwhfBeJKySp | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝³ : Ring β
inst✝² : LinearOrderedField α
inst✝¹ : Archimedean α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, abs' (f n) ≤ a
hnm : ∀ n ≥ m, f (Nat.succ n) ≤ f n
ε : α
ε0 : ε > 0
k : ℕ
hk : a ≤ k • ε
h : ∃ l, ∀ n ≥ m, a - l • ε < f n
l : ℕ := Nat.find h
hl : ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have hl0 : l ≠ 0 := fun hl0 =>
not_lt_of_ge (ham m le_rfl)
(lt_of_lt_of_le (by have := hl m (le_refl m); simpa [hl0] using this) (le_abs_self (f m))) | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f := fun ε ε0 => by
let ⟨k, hk⟩ := Archimedean.arch a ε0
have h : ∃ l, ∀ n ≥ m, a - l • ε < f n :=
⟨k + k + 1, fun n hnm =>
lt_of_lt_of_le
(show a - (k +... | Mathlib.Data.Complex.Exponential.38_0.1bixbwhfBeJKySp | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝³ : Ring β
inst✝² : LinearOrderedField α
inst✝¹ : Archimedean α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, abs' (f n) ≤ a
hnm : ∀ n ≥ m, f (Nat.succ n) ≤ f n
ε : α
ε0 : ε > 0
k : ℕ
hk : a ≤ k • ε
h : ∃ l, ∀ n ≥ m, a - l • ε < f n
l : ℕ := Nat.find h
hl : ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have := hl m (le_refl m) | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f := fun ε ε0 => by
let ⟨k, hk⟩ := Archimedean.arch a ε0
have h : ∃ l, ∀ n ≥ m, a - l • ε < f n :=
⟨k + k + 1, fun n hnm =>
lt_of_lt_of_le
(show a - (k +... | Mathlib.Data.Complex.Exponential.38_0.1bixbwhfBeJKySp | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝³ : Ring β
inst✝² : LinearOrderedField α
inst✝¹ : Archimedean α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, abs' (f n) ≤ a
hnm : ∀ n ≥ m, f (Nat.succ n) ≤ f n
ε : α
ε0 : ε > 0
k : ℕ
hk : a ≤ k • ε
h : ∃ l, ∀ n ≥ m, a - l • ε < f n
l : ℕ := Nat.find h
hl : ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simpa [hl0] using this | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f := fun ε ε0 => by
let ⟨k, hk⟩ := Archimedean.arch a ε0
have h : ∃ l, ∀ n ≥ m, a - l • ε < f n :=
⟨k + k + 1, fun n hnm =>
lt_of_lt_of_le
(show a - (k +... | Mathlib.Data.Complex.Exponential.38_0.1bixbwhfBeJKySp | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝³ : Ring β
inst✝² : LinearOrderedField α
inst✝¹ : Archimedean α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, abs' (f n) ≤ a
hnm : ∀ n ≥ m, f (Nat.succ n) ≤ f n
ε : α
ε0 : ε > 0
k : ℕ
hk : a ≤ k • ε
h : ∃ l, ∀ n ≥ m, a - l • ε < f n
l : ℕ := Nat.find h
hl : ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | cases' not_forall.1 (Nat.find_min h (Nat.pred_lt hl0)) with i hi | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f := fun ε ε0 => by
let ⟨k, hk⟩ := Archimedean.arch a ε0
have h : ∃ l, ∀ n ≥ m, a - l • ε < f n :=
⟨k + k + 1, fun n hnm =>
lt_of_lt_of_le
(show a - (k +... | Mathlib.Data.Complex.Exponential.38_0.1bixbwhfBeJKySp | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f | Mathlib_Data_Complex_Exponential |
case intro
α : Type u_1
β : Type u_2
inst✝³ : Ring β
inst✝² : LinearOrderedField α
inst✝¹ : Archimedean α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, abs' (f n) ≤ a
hnm : ∀ n ≥ m, f (Nat.succ n) ≤ f n
ε : α
ε0 : ε > 0
k : ℕ
hk : a ≤ k • ε
h : ∃ l, ∀ n ≥ m, a - l • ε < f n
l : ℕ := Nat.f... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [not_imp, not_lt] at hi | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f := fun ε ε0 => by
let ⟨k, hk⟩ := Archimedean.arch a ε0
have h : ∃ l, ∀ n ≥ m, a - l • ε < f n :=
⟨k + k + 1, fun n hnm =>
lt_of_lt_of_le
(show a - (k +... | Mathlib.Data.Complex.Exponential.38_0.1bixbwhfBeJKySp | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f | Mathlib_Data_Complex_Exponential |
case intro
α : Type u_1
β : Type u_2
inst✝³ : Ring β
inst✝² : LinearOrderedField α
inst✝¹ : Archimedean α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, abs' (f n) ≤ a
hnm : ∀ n ≥ m, f (Nat.succ n) ≤ f n
ε : α
ε0 : ε > 0
k : ℕ
hk : a ≤ k • ε
h : ∃ l, ∀ n ≥ m, a - l • ε < f n
l : ℕ := Nat.f... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | exists i | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f := fun ε ε0 => by
let ⟨k, hk⟩ := Archimedean.arch a ε0
have h : ∃ l, ∀ n ≥ m, a - l • ε < f n :=
⟨k + k + 1, fun n hnm =>
lt_of_lt_of_le
(show a - (k +... | Mathlib.Data.Complex.Exponential.38_0.1bixbwhfBeJKySp | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f | Mathlib_Data_Complex_Exponential |
case intro
α : Type u_1
β : Type u_2
inst✝³ : Ring β
inst✝² : LinearOrderedField α
inst✝¹ : Archimedean α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, abs' (f n) ≤ a
hnm : ∀ n ≥ m, f (Nat.succ n) ≤ f n
ε : α
ε0 : ε > 0
k : ℕ
hk : a ≤ k • ε
h : ∃ l, ∀ n ≥ m, a - l • ε < f n
l : ℕ := Nat.f... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | intro j hj | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f := fun ε ε0 => by
let ⟨k, hk⟩ := Archimedean.arch a ε0
have h : ∃ l, ∀ n ≥ m, a - l • ε < f n :=
⟨k + k + 1, fun n hnm =>
lt_of_lt_of_le
(show a - (k +... | Mathlib.Data.Complex.Exponential.38_0.1bixbwhfBeJKySp | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f | Mathlib_Data_Complex_Exponential |
case intro
α : Type u_1
β : Type u_2
inst✝³ : Ring β
inst✝² : LinearOrderedField α
inst✝¹ : Archimedean α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, abs' (f n) ≤ a
hnm : ∀ n ≥ m, f (Nat.succ n) ≤ f n
ε : α
ε0 : ε > 0
k : ℕ
hk : a ≤ k • ε
h : ∃ l, ∀ n ≥ m, a - l • ε < f n
l : ℕ := Nat.f... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have hfij : f j ≤ f i := (Nat.rel_of_forall_rel_succ_of_le_of_le (· ≥ ·) hnm hi.1 hj).le | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f := fun ε ε0 => by
let ⟨k, hk⟩ := Archimedean.arch a ε0
have h : ∃ l, ∀ n ≥ m, a - l • ε < f n :=
⟨k + k + 1, fun n hnm =>
lt_of_lt_of_le
(show a - (k +... | Mathlib.Data.Complex.Exponential.38_0.1bixbwhfBeJKySp | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f | Mathlib_Data_Complex_Exponential |
case intro
α : Type u_1
β : Type u_2
inst✝³ : Ring β
inst✝² : LinearOrderedField α
inst✝¹ : Archimedean α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, abs' (f n) ≤ a
hnm : ∀ n ≥ m, f (Nat.succ n) ≤ f n
ε : α
ε0 : ε > 0
k : ℕ
hk : a ≤ k • ε
h : ∃ l, ∀ n ≥ m, a - l • ε < f n
l : ℕ := Nat.f... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [abs_of_nonpos (sub_nonpos.2 hfij), neg_sub, sub_lt_iff_lt_add'] | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f := fun ε ε0 => by
let ⟨k, hk⟩ := Archimedean.arch a ε0
have h : ∃ l, ∀ n ≥ m, a - l • ε < f n :=
⟨k + k + 1, fun n hnm =>
lt_of_lt_of_le
(show a - (k +... | Mathlib.Data.Complex.Exponential.38_0.1bixbwhfBeJKySp | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f | Mathlib_Data_Complex_Exponential |
case intro
α : Type u_1
β : Type u_2
inst✝³ : Ring β
inst✝² : LinearOrderedField α
inst✝¹ : Archimedean α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, abs' (f n) ≤ a
hnm : ∀ n ≥ m, f (Nat.succ n) ≤ f n
ε : α
ε0 : ε > 0
k : ℕ
hk : a ≤ k • ε
h : ∃ l, ∀ n ≥ m, a - l • ε < f n
l : ℕ := Nat.f... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | calc
f i ≤ a - Nat.pred l • ε := hi.2
_ = a - l • ε + ε := by
conv =>
rhs
rw [← Nat.succ_pred_eq_of_pos (Nat.pos_of_ne_zero hl0), succ_nsmul', sub_add,
add_sub_cancel]
_ < f j + ε := add_lt_add_right (hl j (le_trans hi.1 hj)) _ | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f := fun ε ε0 => by
let ⟨k, hk⟩ := Archimedean.arch a ε0
have h : ∃ l, ∀ n ≥ m, a - l • ε < f n :=
⟨k + k + 1, fun n hnm =>
lt_of_lt_of_le
(show a - (k +... | Mathlib.Data.Complex.Exponential.38_0.1bixbwhfBeJKySp | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝³ : Ring β
inst✝² : LinearOrderedField α
inst✝¹ : Archimedean α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, abs' (f n) ≤ a
hnm : ∀ n ≥ m, f (Nat.succ n) ≤ f n
ε : α
ε0 : ε > 0
k : ℕ
hk : a ≤ k • ε
h : ∃ l, ∀ n ≥ m, a - l • ε < f n
l : ℕ := Nat.find h
hl : ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | conv =>
rhs
rw [← Nat.succ_pred_eq_of_pos (Nat.pos_of_ne_zero hl0), succ_nsmul', sub_add,
add_sub_cancel] | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f := fun ε ε0 => by
let ⟨k, hk⟩ := Archimedean.arch a ε0
have h : ∃ l, ∀ n ≥ m, a - l • ε < f n :=
⟨k + k + 1, fun n hnm =>
lt_of_lt_of_le
(show a - (k +... | Mathlib.Data.Complex.Exponential.38_0.1bixbwhfBeJKySp | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝³ : Ring β
inst✝² : LinearOrderedField α
inst✝¹ : Archimedean α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, abs' (f n) ≤ a
hnm : ∀ n ≥ m, f (Nat.succ n) ≤ f n
ε : α
ε0 : ε > 0
k : ℕ
hk : a ≤ k • ε
h : ∃ l, ∀ n ≥ m, a - l • ε < f n
l : ℕ := Nat.find h
hl : ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rhs
rw [← Nat.succ_pred_eq_of_pos (Nat.pos_of_ne_zero hl0), succ_nsmul', sub_add,
add_sub_cancel] | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f := fun ε ε0 => by
let ⟨k, hk⟩ := Archimedean.arch a ε0
have h : ∃ l, ∀ n ≥ m, a - l • ε < f n :=
⟨k + k + 1, fun n hnm =>
lt_of_lt_of_le
(show a - (k +... | Mathlib.Data.Complex.Exponential.38_0.1bixbwhfBeJKySp | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝³ : Ring β
inst✝² : LinearOrderedField α
inst✝¹ : Archimedean α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, abs' (f n) ≤ a
hnm : ∀ n ≥ m, f (Nat.succ n) ≤ f n
ε : α
ε0 : ε > 0
k : ℕ
hk : a ≤ k • ε
h : ∃ l, ∀ n ≥ m, a - l • ε < f n
l : ℕ := Nat.find h
hl : ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rhs
rw [← Nat.succ_pred_eq_of_pos (Nat.pos_of_ne_zero hl0), succ_nsmul', sub_add,
add_sub_cancel] | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f := fun ε ε0 => by
let ⟨k, hk⟩ := Archimedean.arch a ε0
have h : ∃ l, ∀ n ≥ m, a - l • ε < f n :=
⟨k + k + 1, fun n hnm =>
lt_of_lt_of_le
(show a - (k +... | Mathlib.Data.Complex.Exponential.38_0.1bixbwhfBeJKySp | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝³ : Ring β
inst✝² : LinearOrderedField α
inst✝¹ : Archimedean α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, abs' (f n) ≤ a
hnm : ∀ n ≥ m, f (Nat.succ n) ≤ f n
ε : α
ε0 : ε > 0
k : ℕ
hk : a ≤ k • ε
h : ∃ l, ∀ n ≥ m, a - l • ε < f n
l : ℕ := Nat.find h
hl : ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rhs | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f := fun ε ε0 => by
let ⟨k, hk⟩ := Archimedean.arch a ε0
have h : ∃ l, ∀ n ≥ m, a - l • ε < f n :=
⟨k + k + 1, fun n hnm =>
lt_of_lt_of_le
(show a - (k +... | Mathlib.Data.Complex.Exponential.38_0.1bixbwhfBeJKySp | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝³ : Ring β
inst✝² : LinearOrderedField α
inst✝¹ : Archimedean α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, abs' (f n) ≤ a
hnm : ∀ n ≥ m, f (Nat.succ n) ≤ f n
ε : α
ε0 : ε > 0
k : ℕ
hk : a ≤ k • ε
h : ∃ l, ∀ n ≥ m, a - l • ε < f n
l : ℕ := Nat.find h
hl : ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← Nat.succ_pred_eq_of_pos (Nat.pos_of_ne_zero hl0), succ_nsmul', sub_add,
add_sub_cancel] | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f := fun ε ε0 => by
let ⟨k, hk⟩ := Archimedean.arch a ε0
have h : ∃ l, ∀ n ≥ m, a - l • ε < f n :=
⟨k + k + 1, fun n hnm =>
lt_of_lt_of_le
(show a - (k +... | Mathlib.Data.Complex.Exponential.38_0.1bixbwhfBeJKySp | theorem isCauSeq_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n.succ ≤ f n) : IsCauSeq abs f | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝³ : Ring β
inst✝² : LinearOrderedField α
inst✝¹ : Archimedean α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, abs' (f n) ≤ a
hnm : ∀ n ≥ m, f n ≤ f (Nat.succ n)
⊢ IsCauSeq abs' f | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | refine'
@Eq.ndrecOn (ℕ → α) _ (IsCauSeq abs) _ _
(-⟨_, @isCauSeq_of_decreasing_bounded _ _ _ (fun n => -f n) a m (by simpa) (by simpa)⟩ :
CauSeq α abs).2 | theorem isCauSeq_of_mono_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n ≤ f n.succ) : IsCauSeq abs f := by
| Mathlib.Data.Complex.Exponential.72_0.1bixbwhfBeJKySp | theorem isCauSeq_of_mono_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n ≤ f n.succ) : IsCauSeq abs f | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝³ : Ring β
inst✝² : LinearOrderedField α
inst✝¹ : Archimedean α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, abs' (f n) ≤ a
hnm : ∀ n ≥ m, f n ≤ f (Nat.succ n)
⊢ ∀ n ≥ m, abs' ((fun n => -f n) n) ≤ a | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simpa | theorem isCauSeq_of_mono_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n ≤ f n.succ) : IsCauSeq abs f := by
refine'
@Eq.ndrecOn (ℕ → α) _ (IsCauSeq abs) _ _
(-⟨_, @isCauSeq_of_decreasing_bounded _ _ _ (fun n => -f n) a m (by | Mathlib.Data.Complex.Exponential.72_0.1bixbwhfBeJKySp | theorem isCauSeq_of_mono_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n ≤ f n.succ) : IsCauSeq abs f | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝³ : Ring β
inst✝² : LinearOrderedField α
inst✝¹ : Archimedean α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, abs' (f n) ≤ a
hnm : ∀ n ≥ m, f n ≤ f (Nat.succ n)
⊢ ∀ n ≥ m, (fun n => -f n) (Nat.succ n) ≤ (fun n => -f n) n | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simpa | theorem isCauSeq_of_mono_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n ≤ f n.succ) : IsCauSeq abs f := by
refine'
@Eq.ndrecOn (ℕ → α) _ (IsCauSeq abs) _ _
(-⟨_, @isCauSeq_of_decreasing_bounded _ _ _ (fun n => -f n) a m (by simpa) (by | Mathlib.Data.Complex.Exponential.72_0.1bixbwhfBeJKySp | theorem isCauSeq_of_mono_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n ≤ f n.succ) : IsCauSeq abs f | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝³ : Ring β
inst✝² : LinearOrderedField α
inst✝¹ : Archimedean α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, abs' (f n) ≤ a
hnm : ∀ n ≥ m, f n ≤ f (Nat.succ n)
⊢ ↑(-{ val := fun n => -f n, property := (_ : IsCauSeq abs' fun n => -f n) }) = f | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | ext | theorem isCauSeq_of_mono_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n ≤ f n.succ) : IsCauSeq abs f := by
refine'
@Eq.ndrecOn (ℕ → α) _ (IsCauSeq abs) _ _
(-⟨_, @isCauSeq_of_decreasing_bounded _ _ _ (fun n => -f n) a m (by simpa) (by simpa)⟩ :
CauSeq α abs)... | Mathlib.Data.Complex.Exponential.72_0.1bixbwhfBeJKySp | theorem isCauSeq_of_mono_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n ≤ f n.succ) : IsCauSeq abs f | Mathlib_Data_Complex_Exponential |
case h
α : Type u_1
β : Type u_2
inst✝³ : Ring β
inst✝² : LinearOrderedField α
inst✝¹ : Archimedean α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, abs' (f n) ≤ a
hnm : ∀ n ≥ m, f n ≤ f (Nat.succ n)
x✝ : ℕ
⊢ ↑(-{ val := fun n => -f n, property := (_ : IsCauSeq abs' fun n => -f n) }) x✝ = ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | exact neg_neg _ | theorem isCauSeq_of_mono_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n ≤ f n.succ) : IsCauSeq abs f := by
refine'
@Eq.ndrecOn (ℕ → α) _ (IsCauSeq abs) _ _
(-⟨_, @isCauSeq_of_decreasing_bounded _ _ _ (fun n => -f n) a m (by simpa) (by simpa)⟩ :
CauSeq α abs)... | Mathlib.Data.Complex.Exponential.72_0.1bixbwhfBeJKySp | theorem isCauSeq_of_mono_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)
(hnm : ∀ n ≥ m, f n ≤ f n.succ) : IsCauSeq abs f | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : Ring β
inst✝¹ : LinearOrderedField α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : ℕ → α
n : ℕ
⊢ (∀ (m : ℕ), n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs' fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | intro hm hg ε ε0 | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i := by
| Mathlib.Data.Complex.Exponential.89_0.1bixbwhfBeJKySp | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : Ring β
inst✝¹ : LinearOrderedField α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ g m
hg : IsCauSeq abs' fun n => ∑ i in range n, g i
ε : α
ε0 : ε > 0
⊢ ∃ i, ∀ j ≥ i, abv ((fun n => ∑ i in range n, f i) j - (fun n => ∑ i in rang... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | cases' hg (ε / 2) (div_pos ε0 (by norm_num)) with i hi | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i := by
intro hm hg ε ε0
| Mathlib.Data.Complex.Exponential.89_0.1bixbwhfBeJKySp | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : Ring β
inst✝¹ : LinearOrderedField α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ g m
hg : IsCauSeq abs' fun n => ∑ i in range n, g i
ε : α
ε0 : ε > 0
⊢ 0 < 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | norm_num | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i := by
intro hm hg ε ε0
cases' hg (ε / 2) (div_pos ε0 (by | Mathlib.Data.Complex.Exponential.89_0.1bixbwhfBeJKySp | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i | Mathlib_Data_Complex_Exponential |
case intro
α : Type u_1
β : Type u_2
inst✝² : Ring β
inst✝¹ : LinearOrderedField α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ g m
hg : IsCauSeq abs' fun n => ∑ i in range n, g i
ε : α
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, abs' ((fun n => ∑ i in range n, g i) j - (fun ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | exists max n i | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i := by
intro hm hg ε ε0
cases' hg (ε / 2) (div_pos ε0 (by norm_num)) with i hi
| Mathlib.Data.Complex.Exponential.89_0.1bixbwhfBeJKySp | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i | Mathlib_Data_Complex_Exponential |
case intro
α : Type u_1
β : Type u_2
inst✝² : Ring β
inst✝¹ : LinearOrderedField α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ g m
hg : IsCauSeq abs' fun n => ∑ i in range n, g i
ε : α
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, abs' ((fun n => ∑ i in range n, g i) j - (fun ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | intro j ji | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i := by
intro hm hg ε ε0
cases' hg (ε / 2) (div_pos ε0 (by norm_num)) with i hi
exists max n i
| Mathlib.Data.Complex.Exponential.89_0.1bixbwhfBeJKySp | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i | Mathlib_Data_Complex_Exponential |
case intro
α : Type u_1
β : Type u_2
inst✝² : Ring β
inst✝¹ : LinearOrderedField α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ g m
hg : IsCauSeq abs' fun n => ∑ i in range n, g i
ε : α
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, abs' ((fun n => ∑ i in range n, g i) j - (fun ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have hi₁ := hi j (le_trans (le_max_right n i) ji) | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i := by
intro hm hg ε ε0
cases' hg (ε / 2) (div_pos ε0 (by norm_num)) with i hi
exists max n i
intro j ... | Mathlib.Data.Complex.Exponential.89_0.1bixbwhfBeJKySp | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i | Mathlib_Data_Complex_Exponential |
case intro
α : Type u_1
β : Type u_2
inst✝² : Ring β
inst✝¹ : LinearOrderedField α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ g m
hg : IsCauSeq abs' fun n => ∑ i in range n, g i
ε : α
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, abs' ((fun n => ∑ i in range n, g i) j - (fun ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have hi₂ := hi (max n i) (le_max_right n i) | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i := by
intro hm hg ε ε0
cases' hg (ε / 2) (div_pos ε0 (by norm_num)) with i hi
exists max n i
intro j ... | Mathlib.Data.Complex.Exponential.89_0.1bixbwhfBeJKySp | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i | Mathlib_Data_Complex_Exponential |
case intro
α : Type u_1
β : Type u_2
inst✝² : Ring β
inst✝¹ : LinearOrderedField α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ g m
hg : IsCauSeq abs' fun n => ∑ i in range n, g i
ε : α
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, abs' ((fun n => ∑ i in range n, g i) j - (fun ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have sub_le :=
abs_sub_le (∑ k in range j, g k) (∑ k in range i, g k) (∑ k in range (max n i), g k) | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i := by
intro hm hg ε ε0
cases' hg (ε / 2) (div_pos ε0 (by norm_num)) with i hi
exists max n i
intro j ... | Mathlib.Data.Complex.Exponential.89_0.1bixbwhfBeJKySp | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i | Mathlib_Data_Complex_Exponential |
case intro
α : Type u_1
β : Type u_2
inst✝² : Ring β
inst✝¹ : LinearOrderedField α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ g m
hg : IsCauSeq abs' fun n => ∑ i in range n, g i
ε : α
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, abs' ((fun n => ∑ i in range n, g i) j - (fun ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have := add_lt_add hi₁ hi₂ | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i := by
intro hm hg ε ε0
cases' hg (ε / 2) (div_pos ε0 (by norm_num)) with i hi
exists max n i
intro j ... | Mathlib.Data.Complex.Exponential.89_0.1bixbwhfBeJKySp | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i | Mathlib_Data_Complex_Exponential |
case intro
α : Type u_1
β : Type u_2
inst✝² : Ring β
inst✝¹ : LinearOrderedField α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ g m
hg : IsCauSeq abs' fun n => ∑ i in range n, g i
ε : α
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, abs' ((fun n => ∑ i in range n, g i) j - (fun ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [abs_sub_comm (∑ k in range (max n i), g k), add_halves ε] at this | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i := by
intro hm hg ε ε0
cases' hg (ε / 2) (div_pos ε0 (by norm_num)) with i hi
exists max n i
intro j ... | Mathlib.Data.Complex.Exponential.89_0.1bixbwhfBeJKySp | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i | Mathlib_Data_Complex_Exponential |
case intro
α : Type u_1
β : Type u_2
inst✝² : Ring β
inst✝¹ : LinearOrderedField α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ g m
hg : IsCauSeq abs' fun n => ∑ i in range n, g i
ε : α
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, abs' ((fun n => ∑ i in range n, g i) j - (fun ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | refine' lt_of_le_of_lt (le_trans (le_trans _ (le_abs_self _)) sub_le) this | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i := by
intro hm hg ε ε0
cases' hg (ε / 2) (div_pos ε0 (by norm_num)) with i hi
exists max n i
intro j ... | Mathlib.Data.Complex.Exponential.89_0.1bixbwhfBeJKySp | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i | Mathlib_Data_Complex_Exponential |
case intro
α : Type u_1
β : Type u_2
inst✝² : Ring β
inst✝¹ : LinearOrderedField α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ g m
hg : IsCauSeq abs' fun n => ∑ i in range n, g i
ε : α
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, abs' ((fun n => ∑ i in range n, g i) j - (fun ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | generalize hk : j - max n i = k | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i := by
intro hm hg ε ε0
cases' hg (ε / 2) (div_pos ε0 (by norm_num)) with i hi
exists max n i
intro j ... | Mathlib.Data.Complex.Exponential.89_0.1bixbwhfBeJKySp | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i | Mathlib_Data_Complex_Exponential |
case intro
α : Type u_1
β : Type u_2
inst✝² : Ring β
inst✝¹ : LinearOrderedField α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ g m
hg : IsCauSeq abs' fun n => ∑ i in range n, g i
ε : α
ε0 : ε > 0
i : ℕ
hi : ∀ j ≥ i, abs' ((fun n => ∑ i in range n, g i) j - (fun ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | clear this hi₂ hi₁ hi ε0 ε hg sub_le | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i := by
intro hm hg ε ε0
cases' hg (ε / 2) (div_pos ε0 (by norm_num)) with i hi
exists max n i
intro j ... | Mathlib.Data.Complex.Exponential.89_0.1bixbwhfBeJKySp | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i | Mathlib_Data_Complex_Exponential |
case intro
α : Type u_1
β : Type u_2
inst✝² : Ring β
inst✝¹ : LinearOrderedField α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ g m
i j : ℕ
ji : j ≥ max n i
k : ℕ
hk : j - max n i = k
⊢ abv ((fun n => ∑ i in range n, f i) j - (fun n => ∑ i in range n, f i) (max n... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [tsub_eq_iff_eq_add_of_le ji] at hk | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i := by
intro hm hg ε ε0
cases' hg (ε / 2) (div_pos ε0 (by norm_num)) with i hi
exists max n i
intro j ... | Mathlib.Data.Complex.Exponential.89_0.1bixbwhfBeJKySp | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i | Mathlib_Data_Complex_Exponential |
case intro
α : Type u_1
β : Type u_2
inst✝² : Ring β
inst✝¹ : LinearOrderedField α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ g m
i j : ℕ
ji : j ≥ max n i
k : ℕ
hk : j = k + max n i
⊢ abv ((fun n => ∑ i in range n, f i) j - (fun n => ∑ i in range n, f i) (max n... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [hk] | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i := by
intro hm hg ε ε0
cases' hg (ε / 2) (div_pos ε0 (by norm_num)) with i hi
exists max n i
intro j ... | Mathlib.Data.Complex.Exponential.89_0.1bixbwhfBeJKySp | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i | Mathlib_Data_Complex_Exponential |
case intro
α : Type u_1
β : Type u_2
inst✝² : Ring β
inst✝¹ : LinearOrderedField α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ g m
i j : ℕ
ji : j ≥ max n i
k : ℕ
hk : j = k + max n i
⊢ abv ((fun n => ∑ i in range n, f i) (k + max n i) - (fun n => ∑ i in range n,... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | dsimp only | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i := by
intro hm hg ε ε0
cases' hg (ε / 2) (div_pos ε0 (by norm_num)) with i hi
exists max n i
intro j ... | Mathlib.Data.Complex.Exponential.89_0.1bixbwhfBeJKySp | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i | Mathlib_Data_Complex_Exponential |
case intro
α : Type u_1
β : Type u_2
inst✝² : Ring β
inst✝¹ : LinearOrderedField α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ g m
i j : ℕ
ji : j ≥ max n i
k : ℕ
hk : j = k + max n i
⊢ abv (∑ i in range (k + max n i), f i - ∑ i in range (max n i), f i) ≤
∑ k... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | clear hk ji j | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i := by
intro hm hg ε ε0
cases' hg (ε / 2) (div_pos ε0 (by norm_num)) with i hi
exists max n i
intro j ... | Mathlib.Data.Complex.Exponential.89_0.1bixbwhfBeJKySp | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i | Mathlib_Data_Complex_Exponential |
case intro
α : Type u_1
β : Type u_2
inst✝² : Ring β
inst✝¹ : LinearOrderedField α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ g m
i k : ℕ
⊢ abv (∑ i in range (k + max n i), f i - ∑ i in range (max n i), f i) ≤
∑ k in range (k + max n i), g k - ∑ k in range ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | induction' k with k' hi | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i := by
intro hm hg ε ε0
cases' hg (ε / 2) (div_pos ε0 (by norm_num)) with i hi
exists max n i
intro j ... | Mathlib.Data.Complex.Exponential.89_0.1bixbwhfBeJKySp | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i | Mathlib_Data_Complex_Exponential |
case intro.zero
α : Type u_1
β : Type u_2
inst✝² : Ring β
inst✝¹ : LinearOrderedField α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ g m
i : ℕ
⊢ abv (∑ i in range (Nat.zero + max n i), f i - ∑ i in range (max n i), f i) ≤
∑ k in range (Nat.zero + max n i), g ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [abv_zero abv] | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i := by
intro hm hg ε ε0
cases' hg (ε / 2) (div_pos ε0 (by norm_num)) with i hi
exists max n i
intro j ... | Mathlib.Data.Complex.Exponential.89_0.1bixbwhfBeJKySp | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i | Mathlib_Data_Complex_Exponential |
case intro.succ
α : Type u_1
β : Type u_2
inst✝² : Ring β
inst✝¹ : LinearOrderedField α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ g m
i k' : ℕ
hi :
abv (∑ i in range (k' + max n i), f i - ∑ i in range (max n i), f i) ≤
∑ k in range (k' + max n i), g k - ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp only [Nat.succ_add, Nat.succ_eq_add_one, Finset.sum_range_succ_comm] | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i := by
intro hm hg ε ε0
cases' hg (ε / 2) (div_pos ε0 (by norm_num)) with i hi
exists max n i
intro j ... | Mathlib.Data.Complex.Exponential.89_0.1bixbwhfBeJKySp | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i | Mathlib_Data_Complex_Exponential |
case intro.succ
α : Type u_1
β : Type u_2
inst✝² : Ring β
inst✝¹ : LinearOrderedField α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ g m
i k' : ℕ
hi :
abv (∑ i in range (k' + max n i), f i - ∑ i in range (max n i), f i) ≤
∑ k in range (k' + max n i), g k - ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp only [add_assoc, sub_eq_add_neg] | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i := by
intro hm hg ε ε0
cases' hg (ε / 2) (div_pos ε0 (by norm_num)) with i hi
exists max n i
intro j ... | Mathlib.Data.Complex.Exponential.89_0.1bixbwhfBeJKySp | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i | Mathlib_Data_Complex_Exponential |
case intro.succ
α : Type u_1
β : Type u_2
inst✝² : Ring β
inst✝¹ : LinearOrderedField α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ g m
i k' : ℕ
hi :
abv (∑ i in range (k' + max n i), f i - ∑ i in range (max n i), f i) ≤
∑ k in range (k' + max n i), g k - ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | refine le_trans (abv_add _ _ _) ?_ | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i := by
intro hm hg ε ε0
cases' hg (ε / 2) (div_pos ε0 (by norm_num)) with i hi
exists max n i
intro j ... | Mathlib.Data.Complex.Exponential.89_0.1bixbwhfBeJKySp | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i | Mathlib_Data_Complex_Exponential |
case intro.succ
α : Type u_1
β : Type u_2
inst✝² : Ring β
inst✝¹ : LinearOrderedField α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ g m
i k' : ℕ
hi :
abv (∑ i in range (k' + max n i), f i - ∑ i in range (max n i), f i) ≤
∑ k in range (k' + max n i), g k - ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp only [sub_eq_add_neg] at hi | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i := by
intro hm hg ε ε0
cases' hg (ε / 2) (div_pos ε0 (by norm_num)) with i hi
exists max n i
intro j ... | Mathlib.Data.Complex.Exponential.89_0.1bixbwhfBeJKySp | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i | Mathlib_Data_Complex_Exponential |
case intro.succ
α : Type u_1
β : Type u_2
inst✝² : Ring β
inst✝¹ : LinearOrderedField α
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
g : ℕ → α
n : ℕ
hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ g m
i k' : ℕ
hi :
abv (∑ i in range (k' + max n i), f i + -∑ i in range (max n i), f i) ≤
∑ k in range (k' + max n i), g k +... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | refine add_le_add (hm _ (le_add_of_nonneg_of_le (Nat.zero_le _) (le_max_left _ _))) hi | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i := by
intro hm hg ε ε0
cases' hg (ε / 2) (div_pos ε0 (by norm_num)) with i hi
exists max n i
intro j ... | Mathlib.Data.Complex.Exponential.89_0.1bixbwhfBeJKySp | theorem isCauSeq_series_of_abv_le_of_isCauSeq {f : ℕ → β} {g : ℕ → α} (n : ℕ) :
(∀ m, n ≤ m → abv (f m) ≤ g m) →
(IsCauSeq abs fun n => ∑ i in range n, g i) → IsCauSeq abv fun n => ∑ i in range n, f i | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝⁴ : LinearOrderedField α
inst✝³ : Archimedean α
β : Type u_2
inst✝² : Ring β
inst✝¹ : Nontrivial β
abv : β → α
inst✝ : IsAbsoluteValue abv
x : β
hx1 : abv x < 1
h : abv x = 1
⊢ False | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [h, lt_irrefl] at hx1 | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m :=
have hx1' : abv x ≠ 1 := fun h => by | Mathlib.Data.Complex.Exponential.129_0.1bixbwhfBeJKySp | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m | Mathlib_Data_Complex_Exponential |
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