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 |
|---|---|---|---|---|---|---|
𝕜 : Type u_1
inst✝¹ : IsROrC 𝕜
n : Type u_2
inst✝ : Fintype n
M : Matrix n n 𝕜
hM : PosDef M
x y : n → 𝕜
⊢ (starRingEnd 𝕜) (inner y x) = inner x y | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | dsimp only [Inner.inner] | /-- A positive definite matrix `M` induces a norm `‖x‖ = sqrt (re xᴴMx)`. -/
@[reducible]
noncomputable def NormedAddCommGroup.ofMatrix {M : Matrix n n 𝕜} (hM : M.PosDef) :
NormedAddCommGroup (n → 𝕜) :=
@InnerProductSpace.Core.toNormedAddCommGroup _ _ _ _ _
{ inner := fun x y => dotProduct (star x) (M.mulVe... | Mathlib.LinearAlgebra.Matrix.PosDef.393_0.RRbDg8T8pKv68Qo | /-- A positive definite matrix `M` induces a norm `‖x‖ = sqrt (re xᴴMx)`. -/
@[reducible]
noncomputable def NormedAddCommGroup.ofMatrix {M : Matrix n n 𝕜} (hM : M.PosDef) :
NormedAddCommGroup (n → 𝕜) | Mathlib_LinearAlgebra_Matrix_PosDef |
𝕜 : Type u_1
inst✝¹ : IsROrC 𝕜
n : Type u_2
inst✝ : Fintype n
M : Matrix n n 𝕜
hM : PosDef M
x y : n → 𝕜
⊢ (starRingEnd 𝕜) (star y ⬝ᵥ mulVec M x) = star x ⬝ᵥ mulVec M y | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [star_dotProduct, starRingEnd_apply, star_star, star_mulVec, dotProduct_mulVec,
hM.isHermitian.eq] | /-- A positive definite matrix `M` induces a norm `‖x‖ = sqrt (re xᴴMx)`. -/
@[reducible]
noncomputable def NormedAddCommGroup.ofMatrix {M : Matrix n n 𝕜} (hM : M.PosDef) :
NormedAddCommGroup (n → 𝕜) :=
@InnerProductSpace.Core.toNormedAddCommGroup _ _ _ _ _
{ inner := fun x y => dotProduct (star x) (M.mulVe... | Mathlib.LinearAlgebra.Matrix.PosDef.393_0.RRbDg8T8pKv68Qo | /-- A positive definite matrix `M` induces a norm `‖x‖ = sqrt (re xᴴMx)`. -/
@[reducible]
noncomputable def NormedAddCommGroup.ofMatrix {M : Matrix n n 𝕜} (hM : M.PosDef) :
NormedAddCommGroup (n → 𝕜) | Mathlib_LinearAlgebra_Matrix_PosDef |
𝕜 : Type u_1
inst✝¹ : IsROrC 𝕜
n : Type u_2
inst✝ : Fintype n
M : Matrix n n 𝕜
hM : PosDef M
x : n → 𝕜
⊢ 0 ≤ IsROrC.re (inner x x) | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | by_cases h : x = 0 | /-- A positive definite matrix `M` induces a norm `‖x‖ = sqrt (re xᴴMx)`. -/
@[reducible]
noncomputable def NormedAddCommGroup.ofMatrix {M : Matrix n n 𝕜} (hM : M.PosDef) :
NormedAddCommGroup (n → 𝕜) :=
@InnerProductSpace.Core.toNormedAddCommGroup _ _ _ _ _
{ inner := fun x y => dotProduct (star x) (M.mulVe... | Mathlib.LinearAlgebra.Matrix.PosDef.393_0.RRbDg8T8pKv68Qo | /-- A positive definite matrix `M` induces a norm `‖x‖ = sqrt (re xᴴMx)`. -/
@[reducible]
noncomputable def NormedAddCommGroup.ofMatrix {M : Matrix n n 𝕜} (hM : M.PosDef) :
NormedAddCommGroup (n → 𝕜) | Mathlib_LinearAlgebra_Matrix_PosDef |
case pos
𝕜 : Type u_1
inst✝¹ : IsROrC 𝕜
n : Type u_2
inst✝ : Fintype n
M : Matrix n n 𝕜
hM : PosDef M
x : n → 𝕜
h : x = 0
⊢ 0 ≤ IsROrC.re (inner x x) | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simp [h] | /-- A positive definite matrix `M` induces a norm `‖x‖ = sqrt (re xᴴMx)`. -/
@[reducible]
noncomputable def NormedAddCommGroup.ofMatrix {M : Matrix n n 𝕜} (hM : M.PosDef) :
NormedAddCommGroup (n → 𝕜) :=
@InnerProductSpace.Core.toNormedAddCommGroup _ _ _ _ _
{ inner := fun x y => dotProduct (star x) (M.mulVe... | Mathlib.LinearAlgebra.Matrix.PosDef.393_0.RRbDg8T8pKv68Qo | /-- A positive definite matrix `M` induces a norm `‖x‖ = sqrt (re xᴴMx)`. -/
@[reducible]
noncomputable def NormedAddCommGroup.ofMatrix {M : Matrix n n 𝕜} (hM : M.PosDef) :
NormedAddCommGroup (n → 𝕜) | Mathlib_LinearAlgebra_Matrix_PosDef |
case neg
𝕜 : Type u_1
inst✝¹ : IsROrC 𝕜
n : Type u_2
inst✝ : Fintype n
M : Matrix n n 𝕜
hM : PosDef M
x : n → 𝕜
h : ¬x = 0
⊢ 0 ≤ IsROrC.re (inner x x) | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | exact le_of_lt (hM.re_dotProduct_pos h) | /-- A positive definite matrix `M` induces a norm `‖x‖ = sqrt (re xᴴMx)`. -/
@[reducible]
noncomputable def NormedAddCommGroup.ofMatrix {M : Matrix n n 𝕜} (hM : M.PosDef) :
NormedAddCommGroup (n → 𝕜) :=
@InnerProductSpace.Core.toNormedAddCommGroup _ _ _ _ _
{ inner := fun x y => dotProduct (star x) (M.mulVe... | Mathlib.LinearAlgebra.Matrix.PosDef.393_0.RRbDg8T8pKv68Qo | /-- A positive definite matrix `M` induces a norm `‖x‖ = sqrt (re xᴴMx)`. -/
@[reducible]
noncomputable def NormedAddCommGroup.ofMatrix {M : Matrix n n 𝕜} (hM : M.PosDef) :
NormedAddCommGroup (n → 𝕜) | Mathlib_LinearAlgebra_Matrix_PosDef |
𝕜 : Type u_1
inst✝¹ : IsROrC 𝕜
n : Type u_2
inst✝ : Fintype n
M : Matrix n n 𝕜
hM : PosDef M
x : n → 𝕜
hx : star x ⬝ᵥ mulVec M x = 0
⊢ x = 0 | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | by_contra! h | /-- A positive definite matrix `M` induces a norm `‖x‖ = sqrt (re xᴴMx)`. -/
@[reducible]
noncomputable def NormedAddCommGroup.ofMatrix {M : Matrix n n 𝕜} (hM : M.PosDef) :
NormedAddCommGroup (n → 𝕜) :=
@InnerProductSpace.Core.toNormedAddCommGroup _ _ _ _ _
{ inner := fun x y => dotProduct (star x) (M.mulVe... | Mathlib.LinearAlgebra.Matrix.PosDef.393_0.RRbDg8T8pKv68Qo | /-- A positive definite matrix `M` induces a norm `‖x‖ = sqrt (re xᴴMx)`. -/
@[reducible]
noncomputable def NormedAddCommGroup.ofMatrix {M : Matrix n n 𝕜} (hM : M.PosDef) :
NormedAddCommGroup (n → 𝕜) | Mathlib_LinearAlgebra_Matrix_PosDef |
𝕜 : Type u_1
inst✝¹ : IsROrC 𝕜
n : Type u_2
inst✝ : Fintype n
M : Matrix n n 𝕜
hM : PosDef M
x : n → 𝕜
hx : star x ⬝ᵥ mulVec M x = 0
h : x ≠ 0
⊢ False | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simpa [hx, lt_irrefl] using hM.re_dotProduct_pos h | /-- A positive definite matrix `M` induces a norm `‖x‖ = sqrt (re xᴴMx)`. -/
@[reducible]
noncomputable def NormedAddCommGroup.ofMatrix {M : Matrix n n 𝕜} (hM : M.PosDef) :
NormedAddCommGroup (n → 𝕜) :=
@InnerProductSpace.Core.toNormedAddCommGroup _ _ _ _ _
{ inner := fun x y => dotProduct (star x) (M.mulVe... | Mathlib.LinearAlgebra.Matrix.PosDef.393_0.RRbDg8T8pKv68Qo | /-- A positive definite matrix `M` induces a norm `‖x‖ = sqrt (re xᴴMx)`. -/
@[reducible]
noncomputable def NormedAddCommGroup.ofMatrix {M : Matrix n n 𝕜} (hM : M.PosDef) :
NormedAddCommGroup (n → 𝕜) | Mathlib_LinearAlgebra_Matrix_PosDef |
𝕜 : Type u_1
inst✝¹ : IsROrC 𝕜
n : Type u_2
inst✝ : Fintype n
M : Matrix n n 𝕜
hM : PosDef M
⊢ ∀ (x y z : n → 𝕜), inner (x + y) z = inner x z + inner y z | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simp only [star_add, add_dotProduct, eq_self_iff_true, forall_const] | /-- A positive definite matrix `M` induces a norm `‖x‖ = sqrt (re xᴴMx)`. -/
@[reducible]
noncomputable def NormedAddCommGroup.ofMatrix {M : Matrix n n 𝕜} (hM : M.PosDef) :
NormedAddCommGroup (n → 𝕜) :=
@InnerProductSpace.Core.toNormedAddCommGroup _ _ _ _ _
{ inner := fun x y => dotProduct (star x) (M.mulVe... | Mathlib.LinearAlgebra.Matrix.PosDef.393_0.RRbDg8T8pKv68Qo | /-- A positive definite matrix `M` induces a norm `‖x‖ = sqrt (re xᴴMx)`. -/
@[reducible]
noncomputable def NormedAddCommGroup.ofMatrix {M : Matrix n n 𝕜} (hM : M.PosDef) :
NormedAddCommGroup (n → 𝕜) | Mathlib_LinearAlgebra_Matrix_PosDef |
𝕜 : Type u_1
inst✝¹ : IsROrC 𝕜
n : Type u_2
inst✝ : Fintype n
M : Matrix n n 𝕜
hM : PosDef M
x y : n → 𝕜
r : 𝕜
⊢ inner (r • x) y = (starRingEnd 𝕜) r * inner x y | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simp only | /-- A positive definite matrix `M` induces a norm `‖x‖ = sqrt (re xᴴMx)`. -/
@[reducible]
noncomputable def NormedAddCommGroup.ofMatrix {M : Matrix n n 𝕜} (hM : M.PosDef) :
NormedAddCommGroup (n → 𝕜) :=
@InnerProductSpace.Core.toNormedAddCommGroup _ _ _ _ _
{ inner := fun x y => dotProduct (star x) (M.mulVe... | Mathlib.LinearAlgebra.Matrix.PosDef.393_0.RRbDg8T8pKv68Qo | /-- A positive definite matrix `M` induces a norm `‖x‖ = sqrt (re xᴴMx)`. -/
@[reducible]
noncomputable def NormedAddCommGroup.ofMatrix {M : Matrix n n 𝕜} (hM : M.PosDef) :
NormedAddCommGroup (n → 𝕜) | Mathlib_LinearAlgebra_Matrix_PosDef |
𝕜 : Type u_1
inst✝¹ : IsROrC 𝕜
n : Type u_2
inst✝ : Fintype n
M : Matrix n n 𝕜
hM : PosDef M
x y : n → 𝕜
r : 𝕜
⊢ star (r • x) ⬝ᵥ mulVec M y = (starRingEnd 𝕜) r * star x ⬝ᵥ mulVec M y | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [← smul_eq_mul, ← smul_dotProduct, starRingEnd_apply, ← star_smul] | /-- A positive definite matrix `M` induces a norm `‖x‖ = sqrt (re xᴴMx)`. -/
@[reducible]
noncomputable def NormedAddCommGroup.ofMatrix {M : Matrix n n 𝕜} (hM : M.PosDef) :
NormedAddCommGroup (n → 𝕜) :=
@InnerProductSpace.Core.toNormedAddCommGroup _ _ _ _ _
{ inner := fun x y => dotProduct (star x) (M.mulVe... | Mathlib.LinearAlgebra.Matrix.PosDef.393_0.RRbDg8T8pKv68Qo | /-- A positive definite matrix `M` induces a norm `‖x‖ = sqrt (re xᴴMx)`. -/
@[reducible]
noncomputable def NormedAddCommGroup.ofMatrix {M : Matrix n n 𝕜} (hM : M.PosDef) :
NormedAddCommGroup (n → 𝕜) | Mathlib_LinearAlgebra_Matrix_PosDef |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
⊢ BaireSpace 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... | refine' ⟨fun f ho hd => _⟩ | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
⊢ Dense (⋂ n, f n) | /-
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 B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
⊢ Dense (⋂ n, f n) | /-
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 Bpos : ∀ n, 0 < B n := by
intro n
simp only [one_div, one_mul, ENNReal.inv_pos]
exact pow_ne_top two_ne_top | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
⊢ ∀ (n : ℕ), 0 < B n | /-
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... | intro n | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
n : ℕ
⊢ 0 < B n | /-
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 [one_div, one_mul, ENNReal.inv_pos] | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
n : ℕ
⊢ 2 ^ n ≠ ⊤ | /-
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 pow_ne_top two_ne_top | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
⊢ Dense (⋂ n, f n) | /-
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 : ∀ n x δ, δ ≠ 0 → ∃ y r, 0 < r ∧ r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := by
intro n x δ δpos
have : x ∈ closure (f n) := hd n x
rcases EMetric.mem_closure_iff.1 this (δ / 2) (ENNReal.half_pos δpos) with ⟨y, ys, xy⟩
rw [edist_comm] at xy
obtain ⟨r, rpos, hr⟩ : ∃ r > 0, closedBa... | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
⊢ ∀ (n : ℕ) (x : α) (δ : ℝ≥0∞), δ ≠ 0 → ∃ y r, 0 < r ∧ r ≤ B (n + 1) ∧ closedBall ... | /-
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... | intro n x δ δpos | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
n : ℕ
x : α
δ : ℝ≥0∞
δpos : δ ≠ 0
⊢ ∃ y r, 0 < r ∧ r ≤ B (n + 1) ∧ closedBall y r ... | /-
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 : x ∈ closure (f n) := hd n x | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
n : ℕ
x : α
δ : ℝ≥0∞
δpos : δ ≠ 0
this : x ∈ closure (f n)
⊢ ∃ y r, 0 < r ∧ r ≤ B ... | /-
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 EMetric.mem_closure_iff.1 this (δ / 2) (ENNReal.half_pos δpos) with ⟨y, ys, xy⟩ | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
n : ℕ
x : α
δ : ℝ≥0∞
δpos : δ ≠ 0
this : x ∈ closure (f n)
y : 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... | rw [edist_comm] at xy | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
n : ℕ
x : α
δ : ℝ≥0∞
δpos : δ ≠ 0
this : x ∈ closure (f n)
y : 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... | obtain ⟨r, rpos, hr⟩ : ∃ r > 0, closedBall y r ⊆ f n :=
nhds_basis_closed_eball.mem_iff.1 (isOpen_iff_mem_nhds.1 (ho n) y ys) | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case intro.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
n : ℕ
x : α
δ : ℝ≥0∞
δpos : δ ≠ 0
this : 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... | refine' ⟨y, min (min (δ / 2) r) (B (n + 1)), _, _, fun z hz => ⟨_, _⟩⟩ | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case intro.intro.intro.intro.refine'_1
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
n : ℕ
x : α
δ : ℝ≥0∞
δpos : δ ≠ 0
this : 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... | show 0 < min (min (δ / 2) r) (B (n + 1)) | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case intro.intro.intro.intro.refine'_1
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
n : ℕ
x : α
δ : ℝ≥0∞
δpos : δ ≠ 0
this : 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 lt_min (lt_min (ENNReal.half_pos δpos) rpos) (Bpos (n + 1)) | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case intro.intro.intro.intro.refine'_2
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
n : ℕ
x : α
δ : ℝ≥0∞
δpos : δ ≠ 0
this : 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... | show min (min (δ / 2) r) (B (n + 1)) ≤ B (n + 1) | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case intro.intro.intro.intro.refine'_2
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
n : ℕ
x : α
δ : ℝ≥0∞
δpos : δ ≠ 0
this : 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 min_le_right _ _ | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case intro.intro.intro.intro.refine'_3
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
n : ℕ
x : α
δ : ℝ≥0∞
δpos : δ ≠ 0
this : 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... | show z ∈ closedBall x δ | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case intro.intro.intro.intro.refine'_3
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
n : ℕ
x : α
δ : ℝ≥0∞
δpos : δ ≠ 0
this : 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
calc
edist z x ≤ edist z y + edist y x := edist_triangle _ _ _
_ ≤ min (min (δ / 2) r) (B (n + 1)) + δ / 2 := (add_le_add hz (le_of_lt xy))
_ ≤ δ / 2 + δ / 2 := (add_le_add (le_trans (min_le_left _ _) (min_le_left _ _)) le_rfl)
_ = δ := ENNReal.add_halves δ | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case intro.intro.intro.intro.refine'_4
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
n : ℕ
x : α
δ : ℝ≥0∞
δpos : δ ≠ 0
this : 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... | show z ∈ f n | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case intro.intro.intro.intro.refine'_4
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
n : ℕ
x : α
δ : ℝ≥0∞
δpos : δ ≠ 0
this : 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 hr (calc
edist z y ≤ min (min (δ / 2) r) (B (n + 1)) := hz
_ ≤ r := le_trans (min_le_left _ _) (min_le_right _ _)) | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
this : ∀ (n : ℕ) (x : α) (δ : ℝ≥0∞), δ ≠ 0 → ∃ y r, 0 < r ∧ r ≤ B (n + 1) ∧ closed... | /-
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... | choose! center radius Hpos HB Hball using this | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : α) (δ... | /-
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' fun x => (mem_closure_iff_nhds_basis nhds_basis_closed_eball).2 fun ε εpos => _ | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : α) (δ... | /-
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 F : ℕ → α × ℝ≥0∞ := fun n =>
Nat.recOn n (Prod.mk x (min ε (B 0))) fun n p => Prod.mk (center n p.1 p.2) (radius n p.1 p.2) | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : α) (δ... | /-
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 c : ℕ → α := fun n => (F n).1 | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : α) (δ... | /-
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 r : ℕ → ℝ≥0∞ := fun n => (F n).2 | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : α) (δ... | /-
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 rpos : ∀ n, 0 < r n := by
intro n
induction' n with n hn
exact lt_min εpos (Bpos 0)
exact Hpos n (c n) (r n) hn.ne' | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : α) (δ... | /-
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... | intro n | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : α) (δ... | /-
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... | induction' n with n hn | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case zero
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) ... | /-
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 lt_min εpos (Bpos 0) | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case succ
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) ... | /-
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 Hpos n (c n) (r n) hn.ne' | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : α) (δ... | /-
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 r0 : ∀ n, r n ≠ 0 := fun n => (rpos n).ne' | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : α) (δ... | /-
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 rB : ∀ n, r n ≤ B n := by
intro n
induction' n with n _
exact min_le_right _ _
exact HB n (c n) (r n) (r0 n) | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : α) (δ... | /-
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... | intro n | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : α) (δ... | /-
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... | induction' n with n _ | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case zero
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) ... | /-
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 min_le_right _ _ | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case succ
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) ... | /-
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 HB n (c n) (r n) (r0 n) | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : α) (δ... | /-
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 incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n :=
fun n => Hball n (c n) (r n) (r0 n) | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : α) (δ... | /-
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 cdist : ∀ n, edist (c n) (c (n + 1)) ≤ B n := by
intro n
rw [edist_comm]
have A : c (n + 1) ∈ closedBall (c (n + 1)) (r (n + 1)) := mem_closedBall_self
have I :=
calc
closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) :=
Subset.trans (incl n) (inter_subset_left _ _)... | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : α) (δ... | /-
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... | intro n | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : α) (δ... | /-
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 [edist_comm] | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : α) (δ... | /-
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 A : c (n + 1) ∈ closedBall (c (n + 1)) (r (n + 1)) := mem_closedBall_self | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : α) (δ... | /-
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 I :=
calc
closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) :=
Subset.trans (incl n) (inter_subset_left _ _)
_ ⊆ closedBall (c n) (B n) := closedBall_subset_closedBall (rB n) | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : α) (δ... | /-
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 I A | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : α) (δ... | /-
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 : CauchySeq c := cauchySeq_of_edist_le_geometric_two _ one_ne_top cdist | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : α) (δ... | /-
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 cauchySeq_tendsto_of_complete this with ⟨y, ylim⟩ | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ)... | /-
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... | use y | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (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 only [exists_prop, Set.mem_iInter] | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (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... | have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by
intro n
refine' Nat.le_induction _ fun m _ h => _
· exact Subset.refl _
· exact Subset.trans (incl m) (Subset.trans (inter_subset_left _ _) h) | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : α) (δ... | /-
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... | intro n | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : α) (δ... | /-
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' Nat.le_induction _ fun m _ h => _ | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case refine'_1
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n ... | /-
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 Subset.refl _ | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case refine'_2
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n ... | /-
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 Subset.trans (incl m) (Subset.trans (inter_subset_left _ _) h) | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (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... | have yball : ∀ n, y ∈ closedBall (c n) (r n) := by
intro n
refine' isClosed_ball.mem_of_tendsto ylim _
refine' (Filter.eventually_ge_atTop n).mono fun m hm => _
exact I n m hm mem_closedBall_self | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : α) (δ... | /-
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... | intro n | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : α) (δ... | /-
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' isClosed_ball.mem_of_tendsto ylim _ | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : α) (δ... | /-
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' (Filter.eventually_ge_atTop n).mono fun m hm => _ | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : α) (δ... | /-
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 I n m hm mem_closedBall_self | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (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... | constructor | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case h.left
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ... | /-
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... | show ∀ n, y ∈ f n | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case h.left
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ... | /-
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... | intro n | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case h.left
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ... | /-
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 : closedBall (c (n + 1)) (r (n + 1)) ⊆ f n :=
Subset.trans (incl n) (inter_subset_right _ _) | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case h.left
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ... | /-
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 this (yball (n + 1)) | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case h.right
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ... | /-
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... | show edist y x ≤ ε | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case h.right
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝¹ : PseudoEMetricSpace α
inst✝ : CompleteSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n => 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → α → ℝ≥0∞ → α
radius : ℕ → α → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ... | /-
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 le_trans (yball 0) (min_le_left _ _) | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority := 100) BaireSpace.of_pseudoEMetricSpace_completeSpace [PseudoEMetricSpace α]
[CompleteSpace α] : BaireS... | Mathlib.Topology.MetricSpace.Baire.49_0.GktojJRwRzEj9tj | /-- Baire theorems asserts that various topological spaces have the Baire property.
Two versions of these theorems are given.
The first states that complete `PseudoEMetricSpace`s are Baire. -/
instance (priority | 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 : T2Space α
inst✝ : LocallyCompactSpace α
⊢ BaireSpace 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... | constructor | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
[TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
| Mathlib.Topology.MetricSpace.Baire.148_0.GktojJRwRzEj9tj | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case baire_property
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : LocallyCompactSpace α
⊢ ∀ (f : ℕ → Set α), (∀ (n : ℕ), IsOpen (f n)) → (∀ (n : ℕ), Dense (f n)) → Dense (⋂ n, f n) | /-
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... | intro f ho hd | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
[TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
constructor
| Mathlib.Topology.MetricSpace.Baire.148_0.GktojJRwRzEj9tj | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case baire_property
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : LocallyCompactSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
⊢ Dense (⋂ n, f n) | /-
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_iff_inter_open.2 | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
[TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
constructor
intro f ho hd
/- To prove that an intersection of open dense subsets is dense, ... | Mathlib.Topology.MetricSpace.Baire.148_0.GktojJRwRzEj9tj | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case baire_property
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : LocallyCompactSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
⊢ ∀ (U : Set α), IsOpen U → Set.Nonempty U → Set.Nonempty (U ∩ ⋂ n, f n) | /-
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... | intro U U_open U_nonempty | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
[TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
constructor
intro f ho hd
/- To prove that an intersection of open dense subsets is dense, ... | Mathlib.Topology.MetricSpace.Baire.148_0.GktojJRwRzEj9tj | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case baire_property
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : LocallyCompactSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set α
U_open : IsOpen U
U_nonempty : Set.Nonempty U
⊢ Set.Nonempty (U ∩ ⋂ n, f n) | /-
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 exists_positiveCompacts_subset U_open U_nonempty with ⟨K₀, hK₀⟩ | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
[TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
constructor
intro f ho hd
/- To prove that an intersection of open dense subsets is dense, ... | Mathlib.Topology.MetricSpace.Baire.148_0.GktojJRwRzEj9tj | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case baire_property.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : LocallyCompactSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set α
U_open : IsOpen U
U_nonempty : Set.Nonempty U
K₀ : PositiveCompacts α
hK₀ : ↑K₀ ⊆ U
... | /-
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 : ∀ (n) (K : PositiveCompacts α), ∃ K' : PositiveCompacts α, ↑K' ⊆ f n ∩ interior K := by
refine' fun n K => exists_positiveCompacts_subset ((ho n).inter isOpen_interior) _
rw [inter_comm]
exact (hd n).inter_open_nonempty _ isOpen_interior K.interior_nonempty | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
[TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
constructor
intro f ho hd
/- To prove that an intersection of open dense subsets is dense, ... | Mathlib.Topology.MetricSpace.Baire.148_0.GktojJRwRzEj9tj | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority | 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 : T2Space α
inst✝ : LocallyCompactSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set α
U_open : IsOpen U
U_nonempty : Set.Nonempty U
K₀ : PositiveCompacts α
hK₀ : ↑K₀ ⊆ U
⊢ ∀ (n : ℕ) (K : PositiveC... | /-
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' fun n K => exists_positiveCompacts_subset ((ho n).inter isOpen_interior) _ | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
[TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
constructor
intro f ho hd
/- To prove that an intersection of open dense subsets is dense, ... | Mathlib.Topology.MetricSpace.Baire.148_0.GktojJRwRzEj9tj | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority | 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 : T2Space α
inst✝ : LocallyCompactSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set α
U_open : IsOpen U
U_nonempty : Set.Nonempty U
K₀ : PositiveCompacts α
hK₀ : ↑K₀ ⊆ U
n : ℕ
K : PositiveCompacts... | /-
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_comm] | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
[TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
constructor
intro f ho hd
/- To prove that an intersection of open dense subsets is dense, ... | Mathlib.Topology.MetricSpace.Baire.148_0.GktojJRwRzEj9tj | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority | 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 : T2Space α
inst✝ : LocallyCompactSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set α
U_open : IsOpen U
U_nonempty : Set.Nonempty U
K₀ : PositiveCompacts α
hK₀ : ↑K₀ ⊆ U
n : ℕ
K : PositiveCompacts... | /-
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 (hd n).inter_open_nonempty _ isOpen_interior K.interior_nonempty | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
[TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
constructor
intro f ho hd
/- To prove that an intersection of open dense subsets is dense, ... | Mathlib.Topology.MetricSpace.Baire.148_0.GktojJRwRzEj9tj | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case baire_property.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : LocallyCompactSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set α
U_open : IsOpen U
U_nonempty : Set.Nonempty U
K₀ : PositiveCompacts α
hK₀ : ↑K₀ ⊆ U
... | /-
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... | choose K_next hK_next using this | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
[TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
constructor
intro f ho hd
/- To prove that an intersection of open dense subsets is dense, ... | Mathlib.Topology.MetricSpace.Baire.148_0.GktojJRwRzEj9tj | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case baire_property.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : LocallyCompactSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set α
U_open : IsOpen U
U_nonempty : Set.Nonempty U
K₀ : PositiveCompacts α
hK₀ : ↑K₀ ⊆ U
... | /-
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 K : ℕ → PositiveCompacts α := fun n => Nat.recOn n K₀ K_next | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
[TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
constructor
intro f ho hd
/- To prove that an intersection of open dense subsets is dense, ... | Mathlib.Topology.MetricSpace.Baire.148_0.GktojJRwRzEj9tj | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case baire_property.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : LocallyCompactSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set α
U_open : IsOpen U
U_nonempty : Set.Nonempty U
K₀ : PositiveCompacts α
hK₀ : ↑K₀ ⊆ U
... | /-
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 hK_decreasing : ∀ n : ℕ, ((K (n + 1)).carrier) ⊆ (f n ∩ (K n).carrier) :=
fun n => (hK_next n (K n)).trans <| inter_subset_inter_right _ interior_subset | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
[TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
constructor
intro f ho hd
/- To prove that an intersection of open dense subsets is dense, ... | Mathlib.Topology.MetricSpace.Baire.148_0.GktojJRwRzEj9tj | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case baire_property.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : LocallyCompactSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set α
U_open : IsOpen U
U_nonempty : Set.Nonempty U
K₀ : PositiveCompacts α
hK₀ : ↑K₀ ⊆ U
... | /-
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 hK_subset : (⋂ n, (K n).carrier : Set α) ⊆ U ∩ ⋂ n, f n := by
intro x hx
simp only [mem_iInter] at hx
simp only [mem_inter_iff, mem_inter] at hx ⊢
refine' ⟨hK₀ <| hx 0, _⟩
simp only [mem_iInter]
exact fun n => (hK_decreasing n (hx (n + 1))).1 | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
[TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
constructor
intro f ho hd
/- To prove that an intersection of open dense subsets is dense, ... | Mathlib.Topology.MetricSpace.Baire.148_0.GktojJRwRzEj9tj | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority | 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 : T2Space α
inst✝ : LocallyCompactSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set α
U_open : IsOpen U
U_nonempty : Set.Nonempty U
K₀ : PositiveCompacts α
hK₀ : ↑K₀ ⊆ U
K_next : ℕ → PositiveCompa... | /-
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... | intro x hx | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
[TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
constructor
intro f ho hd
/- To prove that an intersection of open dense subsets is dense, ... | Mathlib.Topology.MetricSpace.Baire.148_0.GktojJRwRzEj9tj | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority | 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 : T2Space α
inst✝ : LocallyCompactSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set α
U_open : IsOpen U
U_nonempty : Set.Nonempty U
K₀ : PositiveCompacts α
hK₀ : ↑K₀ ⊆ U
K_next : ℕ → PositiveCompa... | /-
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 [mem_iInter] at hx | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
[TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
constructor
intro f ho hd
/- To prove that an intersection of open dense subsets is dense, ... | Mathlib.Topology.MetricSpace.Baire.148_0.GktojJRwRzEj9tj | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority | 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 : T2Space α
inst✝ : LocallyCompactSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set α
U_open : IsOpen U
U_nonempty : Set.Nonempty U
K₀ : PositiveCompacts α
hK₀ : ↑K₀ ⊆ U
K_next : ℕ → PositiveCompa... | /-
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 [mem_inter_iff, mem_inter] at hx ⊢ | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
[TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
constructor
intro f ho hd
/- To prove that an intersection of open dense subsets is dense, ... | Mathlib.Topology.MetricSpace.Baire.148_0.GktojJRwRzEj9tj | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority | 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 : T2Space α
inst✝ : LocallyCompactSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set α
U_open : IsOpen U
U_nonempty : Set.Nonempty U
K₀ : PositiveCompacts α
hK₀ : ↑K₀ ⊆ U
K_next : ℕ → PositiveCompa... | /-
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' ⟨hK₀ <| hx 0, _⟩ | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
[TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
constructor
intro f ho hd
/- To prove that an intersection of open dense subsets is dense, ... | Mathlib.Topology.MetricSpace.Baire.148_0.GktojJRwRzEj9tj | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority | 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 : T2Space α
inst✝ : LocallyCompactSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set α
U_open : IsOpen U
U_nonempty : Set.Nonempty U
K₀ : PositiveCompacts α
hK₀ : ↑K₀ ⊆ U
K_next : ℕ → PositiveCompa... | /-
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 [mem_iInter] | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
[TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
constructor
intro f ho hd
/- To prove that an intersection of open dense subsets is dense, ... | Mathlib.Topology.MetricSpace.Baire.148_0.GktojJRwRzEj9tj | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority | 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 : T2Space α
inst✝ : LocallyCompactSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set α
U_open : IsOpen U
U_nonempty : Set.Nonempty U
K₀ : PositiveCompacts α
hK₀ : ↑K₀ ⊆ U
K_next : ℕ → PositiveCompa... | /-
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 fun n => (hK_decreasing n (hx (n + 1))).1 | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
[TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
constructor
intro f ho hd
/- To prove that an intersection of open dense subsets is dense, ... | Mathlib.Topology.MetricSpace.Baire.148_0.GktojJRwRzEj9tj | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case baire_property.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : LocallyCompactSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set α
U_open : IsOpen U
U_nonempty : Set.Nonempty U
K₀ : PositiveCompacts α
hK₀ : ↑K₀ ⊆ U
... | /-
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 hK_nonempty : (⋂ n, (K n).carrier : Set α).Nonempty :=
IsCompact.nonempty_iInter_of_sequence_nonempty_compact_closed _
(fun n => (hK_decreasing n).trans (inter_subset_right _ _)) (fun n => (K n).nonempty)
(K 0).isCompact fun n => (K n).isCompact.isClosed | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
[TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
constructor
intro f ho hd
/- To prove that an intersection of open dense subsets is dense, ... | Mathlib.Topology.MetricSpace.Baire.148_0.GktojJRwRzEj9tj | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority | Mathlib_Topology_MetricSpace_Baire |
case baire_property.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : LocallyCompactSpace α
f : ℕ → Set α
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
U : Set α
U_open : IsOpen U
U_nonempty : Set.Nonempty U
K₀ : PositiveCompacts α
hK₀ : ↑K₀ ⊆ U
... | /-
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 hK_nonempty.mono hK_subset | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority := 100) BaireSpace.of_t2Space_locallyCompactSpace
[TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] : BaireSpace α := by
constructor
intro f ho hd
/- To prove that an intersection of open dense subsets is dense, ... | Mathlib.Topology.MetricSpace.Baire.148_0.GktojJRwRzEj9tj | /-- The second theorem states that locally compact spaces are Baire. -/
instance (priority | 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
⊢ 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... | rcases S.eq_empty_or_nonempty with h | h | /-- 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
| 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 inl
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 : 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... | simp [h] | /-- 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
· | 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
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
⊢ 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... | rcases hS.exists_eq_range h with ⟨f, 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]
· | 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
⊢ 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... | have F : ∀ n, f n ∈ S := fun n => by rw [hf]; 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 |
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