d901031e - mathematics_in

Authored by: Patrick Massot
Authored at: 2023-06-22 13:58:50 +0200
Committed by: Patrick Massot
Committed at: 2023-06-22 13:58:50 +0200
Object name: d901031e00f48220dfd6481eff6435f063282857
Parent: 0041fe37

Update jeu. 22 juin 2023 13:58:50 CEST

Changes

43 changed files (+626/-237)

MIL/C06_Abstract_Algebra/S01_Structures.lean
- @@ -1,7 +1,6 @@
 import Mathlib.Algebra.BigOperators.Ring import Mathlib.Data.Real.Basic - namespace C06S01 noncomputable section @[ext]
- @@ -45,11 +44,13 @@ def add' (a b : Point) : Point where
 z := a.z + b.z #check add myPoint1 myPoint2 + #check myPoint1.add myPoint2 end Point #check Point.add myPoint1 myPoint2 + #check myPoint1.add myPoint2 namespace Point
- @@ -97,14 +98,10 @@ example : ∀ a b : Point, add a b = add b a := fun ⟨xa, ya, za⟩ ⟨xb, yb,
 protected theorem add_assoc (a b c : Point) : (a.add b).add c = a.add (b.add c) := by sorry - def smul (r : ℝ) (a : Point) : Point := sorry - - theorem smul_distrib (r : ℝ) (a b : Point) : - (smul r a).add (smul r b) = smul r (a.add b) := by + theorem smul_distrib (r : ℝ) (a b : Point) : (smul r a).add (smul r b) = smul r (a.add b) := by sorry - end Point structure StandardTwoSimplex where
- @@ -141,9 +138,9 @@ def midpoint (a b : StandardTwoSimplex) : StandardTwoSimplex
 sum_eq := by field_simp; linarith [a.sum_eq, b.sum_eq] def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) - (a b : StandardTwoSimplex) : StandardTwoSimplex := + (a b : StandardTwoSimplex) : StandardTwoSimplex + where sorry - end end StandardTwoSimplex
- @@ -166,8 +163,7 @@ def midpoint (n : ℕ) (a b : StandardSimplex n) : StandardSimplex n
 · linarith [a.NonNeg i, b.NonNeg i] norm_num sum_eq_one := by - simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib, - a.sum_eq_one, b.sum_eq_one] + simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib, a.sum_eq_one, b.sum_eq_one] field_simp end StandardSimplex
- @@ -177,9 +173,11 @@ structure IsLinear (f : ℝ → ℝ) where
 preserves_mul : ∀ x c, f (c * x) = c * f x section + variable (f : ℝ → ℝ) (linf : IsLinear f) #check linf.is_additive + #check linf.preserves_mul end
- @@ -194,11 +192,15 @@ def PReal :=
 { y : ℝ // 0 < y } section + variable (x : PReal) #check x.val + #check x.property + #check x.1 + #check x.2 end
- @@ -212,6 +214,7 @@ def StandardSimplex' (n : ℕ) :=
 def StdSimplex := Σ n : ℕ, StandardSimplex n section + variable (s : StdSimplex) #check s.fst
MIL/C06_Abstract_Algebra/S02_Algebraic_Structures.lean
- @@ -1,6 +1,5 @@
  import Mathlib.Data.Real.Basic - namespace C06S02 structure Group₁ (α : Type _) where mul : α → α → α one : α
- @@ -15,6 +14,7 @@ structure Group₁Cat where
  str : Group₁ α section + variable (α β γ : Type _) variable (f : α ≃ β) (g : β ≃ γ)
- @@ -65,15 +65,15 @@ namespace Point
  def add (a b : Point) : Point := ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ - def neg (a : Point) : Point := sorry - - def zero : Point := sorry + def neg (a : point) : point := sorry - def add_group_point : AddGroup₁ point := sorry + def zero : point := sorry + def add_group_point : add_group₁ point := sorry end Point section + variable {α : Type _} (f g : Equiv.Perm α) (n : ℕ) #check f * g
- @@ -157,6 +157,7 @@ instance hasInvGroup₂ {α : Type _} [Group₂ α] : Inv α :=
  ⟨Group₂.inv⟩ section + variable {α : Type _} (f g : Equiv.Perm α) #check f * 1 * g⁻¹
MIL/C06_Abstract_Algebra/S03_Building_the_Gaussian_Integers.lean
- @@ -37,8 +37,7 @@ theorem add_def (x y : gaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ :=
 theorem neg_def (x : gaussInt) : -x = ⟨-x.re, -x.im⟩ := rfl - theorem mul_def (x y : gaussInt) : - x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := + theorem mul_def (x y : gaussInt) : x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := rfl @[simp]
- @@ -91,13 +90,13 @@ instance instCommRing : CommRing gaussInt where
 intros ext <;> simp <;> ring zero_add := by - intro + intros ext <;> simp add_zero := by - intro + intros ext <;> simp add_left_neg := by - intro + intros ext <;> simp add_comm := by intros
- @@ -106,10 +105,10 @@ instance instCommRing : CommRing gaussInt where
 intros ext <;> simp <;> ring one_mul := by - intro + intros ext <;> simp mul_one := by - intro + intros ext <;> simp left_distrib := by intros
- @@ -208,11 +207,9 @@ theorem div_def (x y : gaussInt) :
 theorem mod_def (x y : gaussInt) : x % y = x - y * (x / y) := rfl - theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : - (x % y).norm < y.norm := by + theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : (x % y).norm < y.norm := by have norm_y_pos : 0 < norm y := by rwa [norm_pos] - have : x % y * conj y = - ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ := by + have : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ := by rw [mod_def, sub_mul, Int.mod'_eq, Int.mod'_eq, sub_eq_add_neg, div_def, norm] ext <;> simp <;> ring have : norm (x % y) * norm y ≤ norm y / 2 * norm y := by
- @@ -241,8 +238,7 @@ theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) :
 theorem coe_natAbs_norm (x : gaussInt) : (x.norm.natAbs : ℤ) = x.norm := Int.natAbs_of_nonneg (norm_nonneg _) - theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : - (x % y).norm.natAbs < y.norm.natAbs := by + theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : (x % y).norm.natAbs < y.norm.natAbs := by apply Int.ofNat_lt.1 simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg] apply norm_mod_lt x hy
- @@ -260,8 +256,7 @@ instance : EuclideanDomain gaussInt :=
 { gaussInt.instCommRing with quotient := (· / ·) remainder := (· % ·) - quotient_mul_add_remainder_eq := - fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] + quotient_mul_add_remainder_eq := fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] quotient_zero := fun x => by simp [div_def, norm, Int.div'] rfl
MIL/C06_Abstract_Algebra/solutions/Solutions_S01_Structures.lean
- @@ -1,7 +1,6 @@
  import Mathlib.Algebra.BigOperators.Ring import Mathlib.Data.Real.Basic - namespace C06S01 noncomputable section @[ext]
- @@ -21,8 +20,7 @@ protected theorem add_assoc (a b c : Point) : (a.add b).add c = a.add (b.add c)
  def smul (r : ℝ) (a : Point) : Point := ⟨r * a.x, r * a.y, r * a.z⟩ - theorem smul_distrib (r : ℝ) (a b : Point) : - (smul r a).add (smul r b) = smul r (a.add b) := by + theorem smul_distrib (r : ℝ) (a b : Point) : (smul r a).add (smul r b) = smul r (a.add b) := by simp [add, smul, mul_add] end Point
- @@ -41,8 +39,8 @@ namespace StandardTwoSimplex
  noncomputable section def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) - (a b : StandardTwoSimplex) : StandardTwoSimplex - where + (a b : StandardTwoSimplex) : StandardTwoSimplex + where x := lambda * a.x + (1 - lambda) * b.x y := lambda * a.y + (1 - lambda) * b.y z := lambda * a.z + (1 - lambda) * b.z
- @@ -76,8 +74,7 @@ def midpoint (n : ℕ) (a b : StandardSimplex n) : StandardSimplex n
  · linarith [a.NonNeg i, b.NonNeg i] norm_num sum_eq_one := by - simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib, - a.sum_eq_one, b.sum_eq_one] + simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib, a.sum_eq_one, b.sum_eq_one] field_simp end StandardSimplex
MIL/C06_Abstract_Algebra/solutions/Solutions_S02_Algebraic_Structures.lean
- @@ -1,6 +1,5 @@
  import Mathlib.Data.Real.Basic - namespace C06S02 structure AddGroup₁ (α : Type _) where add : α → α → α zero : α
MIL/C06_Abstract_Algebra/solutions/Solutions_S03_Building_the_Gaussian_Integers.lean
- @@ -37,8 +37,7 @@ theorem add_def (x y : gaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ :=
 theorem neg_def (x : gaussInt) : -x = ⟨-x.re, -x.im⟩ := rfl - theorem mul_def (x y : gaussInt) : - x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := + theorem mul_def (x y : gaussInt) : x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := rfl @[simp]
- @@ -91,13 +90,13 @@ instance instCommRing : CommRing gaussInt where
 intros ext <;> simp <;> ring zero_add := by - intro + intros ext <;> simp add_zero := by - intro + intros ext <;> simp add_left_neg := by - intro + intros ext <;> simp add_comm := by intros
- @@ -106,10 +105,10 @@ instance instCommRing : CommRing gaussInt where
 intros ext <;> simp <;> ring one_mul := by - intro + intros ext <;> simp mul_one := by - intro + intros ext <;> simp left_distrib := by intros
- @@ -222,11 +221,9 @@ theorem div_def (x y : gaussInt) :
 theorem mod_def (x y : gaussInt) : x % y = x - y * (x / y) := rfl - theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : - (x % y).norm < y.norm := by + theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : (x % y).norm < y.norm := by have norm_y_pos : 0 < norm y := by rwa [norm_pos] - have : x % y * conj y = - ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ := by + have : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ := by rw [mod_def, sub_mul, Int.mod'_eq, Int.mod'_eq, sub_eq_add_neg, div_def, norm] ext <;> simp <;> ring have : norm (x % y) * norm y ≤ norm y / 2 * norm y := by
- @@ -255,8 +252,7 @@ theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) :
 theorem coe_natAbs_norm (x : gaussInt) : (x.norm.natAbs : ℤ) = x.norm := Int.natAbs_of_nonneg (norm_nonneg _) - theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : - (x % y).norm.natAbs < y.norm.natAbs := by + theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : (x % y).norm.natAbs < y.norm.natAbs := by apply Int.ofNat_lt.1 simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg] apply norm_mod_lt x hy
- @@ -274,8 +270,7 @@ instance : EuclideanDomain gaussInt :=
 { gaussInt.instCommRing with quotient := (· / ·) remainder := (· % ·) - quotient_mul_add_remainder_eq := - fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] + quotient_mul_add_remainder_eq := fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] quotient_zero := fun x => by simp [div_def, norm, Int.div'] rfl
MIL/C07_Hierarchies/S01_Basics.lean
- @@ -120,7 +120,7 @@ lemma dia_inv [Group₁ G] (a : G) : a ⋄ a⁻¹ = 𝟙 :=
  class AddSemigroup₃ (α : Type) extends Add α where - /-- Multiplication is associative -/ + /-- Addition is associative -/ add_assoc₃ : ∀ a b c : α, a + b + c = a + (b + c) @[to_additive AddSemigroup₃]
MIL/C07_Hierarchies/solutions/Solutions_S01_Basics.lean
- @@ -121,7 +121,7 @@ lemma dia_inv [Group₁ G] (a : G) : a ⋄ a⁻¹ = 𝟙 :=
  class AddSemigroup₃ (α : Type) extends Add α where - /-- Multiplication is associative -/ + /-- Addition is associative -/ add_assoc₃ : ∀ a b c : α, a + b + c = a + (b + c) @[to_additive AddSemigroup₃]
MIL/C07_Hierarchies/solutions/Solutions_S03_Subobjects.lean
- @@ -123,4 +123,3 @@ instance [CommMonoid M] (N : Submonoid M) : Monoid (M ⧸ N) where
  rintro ⟨a⟩ ; apply Quotient.sound ; dsimp only ; rw [one_mul] ; apply @Setoid.refl M N.Setoid mul_one := by rintro ⟨a⟩ ; apply Quotient.sound ; dsimp only ; rw [mul_one] ; apply @Setoid.refl M N.Setoid -
MIL/C07_Topology/S01_Filters.lean
- @@ -42,13 +42,14 @@ variable (f : ℝ → ℝ) (x₀ y₀ : ℝ)
 #check Tendsto (f ∘ (↑)) (comap ((↑) : ℚ → ℝ) (𝓝 x₀)) (𝓝 y₀) section + variable {α β γ : Type _} (F : Filter α) {m : γ → β} {n : β → α} #check (comap_comap : comap m (comap n F) = comap (n ∘ m) F) end - example : 𝓝 (x₀, y₀) = 𝓝 x₀ ×ˢ 𝓝 y₀ := + example : 𝓝 (x₀, y₀) = 𝓝 x₀ ×ᶠ 𝓝 y₀ := nhds_prod_eq #check le_inf_iff
- @@ -63,7 +64,7 @@ example (x₀ : ℝ) : HasBasis (𝓝 x₀) (fun ε : ℝ => 0 < ε) fun ε => I
 example (u : ℕ → ℝ) (x₀ : ℝ) : Tendsto u atTop (𝓝 x₀) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, u n ∈ Ioo (x₀ - ε) (x₀ + ε) := by - have : atTop.HasBasis (fun _ : ℕ => True) Ici := atTop_basis + have : atTop.HasBasis (fun n : ℕ => True) Ici := atTop_basis rw [this.tendsto_iff (nhds_basis_Ioo_pos x₀)] simp
- @@ -80,7 +81,9 @@ example (u v : ℕ → ℝ) (h : u =ᶠ[atTop] v) (x₀ : ℝ) :
 tendsto_congr' h #check @eventually_of_forall + #check @Eventually.mono + #check @Eventually.and example (P Q R : ℕ → Prop) (hP : ∀ᶠ n in atTop, P n) (hQ : ∀ᶠ n in atTop, Q n)
- @@ -96,7 +99,9 @@ example (P Q R : ℕ → Prop) (hP : ∀ᶠ n in atTop, P n) (hQ : ∀ᶠ n in a
 exact h'' ⟨h, h'⟩ #check mem_closure_iff_clusterPt + #check le_principal_iff + #check neBot_of_le example (u : ℕ → ℝ) (M : Set ℝ) (x : ℝ) (hux : Tendsto u atTop (𝓝 x))
MIL/C07_Topology/S02_Metric_Spaces.lean
- @@ -2,28 +2,33 @@ import Mathlib.Topology.Instances.Real
 import Mathlib.Analysis.NormedSpace.BanachSteinhaus open Set Filter + open Topology Filter variable {X : Type _} [MetricSpace X] (a b c : X) #check (dist a b : ℝ) + #check (dist_nonneg : 0 ≤ dist a b) + #check (dist_eq_zero : dist a b = 0 ↔ a = b) + #check (dist_comm a b : dist a b = dist b a) + #check (dist_triangle a b c : dist a c ≤ dist a b + dist b c) -- Note the next three lines are not quoted, their purpose is to make sure those things don't get renamed while we're looking elsewhere. #check EMetricSpace + #check PseudoMetricSpace + #check PseudoEMetricSpace - example {u : ℕ → X} {a : X} : - Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε := + example {u : ℕ → X} {a : X} : Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε := Metric.tendsto_atTop example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} : - Continuous f ↔ - ∀ x : X, ∀ ε > 0, ∃ δ > 0, ∀ x', dist x' x < δ → dist (f x') (f x) < ε := + Continuous f ↔ ∀ x : X, ∀ ε > 0, ∃ δ > 0, ∀ x', dist x' x < δ → dist (f x') (f x) < ε := Metric.continuous_iff example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) :
- @@ -81,8 +86,7 @@ example {s : Set X} (hs : IsClosed s) {u : ℕ → X} (hu : Tendsto u atTop (
 example {s : Set X} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, a ∈ Metric.ball b ε := Metric.mem_closure_iff - example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) : - a ∈ closure s := + example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) : a ∈ closure s := sorry example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.ball x ε ⊆ s :=
- @@ -98,18 +102,16 @@ example {s : Set X} (hs : IsCompact s) {u : ℕ → X} (hu : ∀ n, u n ∈ s) :
 ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := hs.tendsto_subseq hu - example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} - (hfs : ContinuousOn f s) : + example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} (hfs : ContinuousOn f s) : ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y := hs.exists_forall_le hs' hfs - example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} - (hfs : ContinuousOn f s) : + example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} (hfs : ContinuousOn f s) : ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x := hs.exists_forall_ge hs' hfs example {s : Set X} (hs : IsCompact s) : IsClosed s := - hs.isClosed + hs.IsClosed example {X : Type _} [MetricSpace X] [CompactSpace X] : IsCompact (univ : Set X) := isCompact_univ
- @@ -117,25 +119,20 @@ example {X : Type _} [MetricSpace X] [CompactSpace X] : IsCompact (univ : Set X)
 #check IsCompact.isClosed example {X : Type _} [MetricSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} : - UniformContinuous f ↔ - ∀ ε > 0, ∃ δ > 0, ∀ {a b : X}, dist a b < δ → dist (f a) (f b) < ε := + UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ {a b : X}, dist a b < δ → dist (f a) (f b) < ε := Metric.uniformContinuous_iff - example {X : Type _} [MetricSpace X] [CompactSpace X] - {Y : Type _} [MetricSpace Y] {f : X → Y} + example {X : Type _} [MetricSpace X] [CompactSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} (hf : Continuous f) : UniformContinuous f := sorry - example (u : ℕ → X) : - CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε := + example (u : ℕ → X) : CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε := Metric.cauchySeq_iff - example (u : ℕ → X) : - CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ n ≥ N, dist (u n) (u N) < ε := + example (u : ℕ → X) : CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ n ≥ N, dist (u n) (u N) < ε := Metric.cauchySeq_iff' - example [CompleteSpace X] (u : ℕ → X) (hu : CauchySeq u) : - ∃ x, Tendsto u atTop (𝓝 x) := + example [CompleteSpace X] (u : ℕ → X) (hu : CauchySeq u) : ∃ x, Tendsto u atTop (𝓝 x) := cauchySeq_tendsto_of_complete hu open BigOperators
- @@ -168,31 +165,28 @@ example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd :
 sorry /- Translate the density assumption into two functions `center` and `radius` associating to any n, x, δ, δpos a center and a positive radius such that - `closedBall center radius` is included both in `f n` and in `closedBall x δ`. + `closed_ball center radius` is included both in `f n` and in `closed_ball x δ`. We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ have : ∀ (n : ℕ) (x : X), - ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := + ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closed_ball y r ⊆ closed_ball x δ ∩ f n := by sorry choose! center radius Hpos HB Hball using this intro x - rw [mem_closure_iff_nhds_basis nhds_basis_closedBall] + rw [mem_closure_iff_nhds_basis nhds_basis_closed_ball] intro ε εpos - /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` - belonging to all `f n`. For this, we construct inductively a sequence - `F n = (c n, r n)` such that the closed ball `closedBall (c n) (r n)` is included - in the previous ball and in `f n`, and such that `r n` is small enough to ensure - that `c n` is a Cauchy sequence. Then `c n` converges to a limit which belongs - to all the `f n`. -/ + /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` belonging to all + `f n`. For this, we construct inductively a sequence `F n = (c n, r n)` such that the closed ball + `closed_ball (c n) (r n)` is included in the previous ball and in `f n`, and such that + `r n` is small enough to ensure that `c n` is a Cauchy sequence. Then `c n` converges to a + limit which belongs to all the `f n`. -/ let F : ℕ → X × ℝ := 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) + 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) let c : ℕ → X := fun n => (F n).1 let r : ℕ → ℝ := fun n => (F n).2 have rpos : ∀ n, 0 < r n := by sorry have rB : ∀ n, r n ≤ B n := by sorry - have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := by - sorry + have incl : ∀ n, closed_ball (c (n + 1)) (r (n + 1)) ⊆ closed_ball (c n) (r n) ∩ f n := by sorry have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by sorry have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`.
- @@ -200,7 +194,7 @@ example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd :
 -- this point `y` will be the desired point. We will check that it belongs to all -- `f n` and to `ball x ε`. use y - have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by sorry - have yball : ∀ n, y ∈ closedBall (c n) (r n) := by sorry + have I : ∀ n, ∀ m ≥ n, closed_ball (c m) (r m) ⊆ closed_ball (c n) (r n) := by sorry + have yball : ∀ n, y ∈ closed_ball (c n) (r n) := by sorry sorry
MIL/C07_Topology/S03_Topological_Spaces.lean
- @@ -1,9 +1,12 @@
 import Mathlib.Topology.Instances.Real import Mathlib.Analysis.NormedSpace.BanachSteinhaus - open Set Filter Topology + open Set Filter + + open Topology Filter section + variable {X : Type _} [TopologicalSpace X] example : IsOpen (univ : Set X) :=
- @@ -15,8 +18,7 @@ example : IsOpen (∅ : Set X) :=
 example {ι : Type _} {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋃ i, s i) := isOpen_iUnion hs - example {ι : Type _} [Fintype ι] {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : - IsOpen (⋂ i, s i) := + example {ι : Type _} [Fintype ι] {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋂ i, s i) := isOpen_iInter hs variable {Y : Type _} [TopologicalSpace Y]
- @@ -77,10 +79,8 @@ example (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) (f : X → Y) :
 Continuous f ↔ TopologicalSpace.coinduced f T_X ≤ T_Y := continuous_iff_coinduced_le - example {Z : Type _} (f : X → Y) (T_X : TopologicalSpace X) (T_Z : TopologicalSpace Z) - (g : Y → Z) : - @Continuous Y Z (TopologicalSpace.coinduced f T_X) T_Z g ↔ - @Continuous X Z T_X T_Z (g ∘ f) := by + example {Z : Type _} (f : X → Y) (T_X : TopologicalSpace X) (T_Z : TopologicalSpace Z) (g : Y → Z) : + @Continuous Y Z (TopologicalSpace.coinduced f T_X) T_Z g ↔ @Continuous X Z T_X T_Z (g ∘ f) := by rw [continuous_iff_coinduced_le, coinduced_compose, continuous_iff_coinduced_le] example (ι : Type _) (X : ι → Type _) (T_X : ∀ i, TopologicalSpace <| X i) :
- @@ -96,13 +96,11 @@ example [TopologicalSpace X] [RegularSpace X] (a : X) :
 (𝓝 a).HasBasis (fun s : Set X => s ∈ 𝓝 a ∧ IsClosed s) id := closed_nhds_basis a - example [TopologicalSpace X] {x : X} : - (𝓝 x).HasBasis (fun t : Set X => t ∈ 𝓝 x ∧ IsOpen t) id := + example [TopologicalSpace X] {x : X} : (𝓝 x).HasBasis (fun t : Set X => t ∈ 𝓝 x ∧ IsOpen t) id := nhds_basis_opens' x - theorem aux {X Y A : Type _} [TopologicalSpace X] {c : A → X} - {f : A → Y} {x : X} {F : Filter Y} - (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) : + theorem aux {X Y A : Type _} [TopologicalSpace X] {c : A → X} {f : A → Y} {x : X} {F : Filter Y} + (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) : ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := sorry
- @@ -114,8 +112,7 @@ example [TopologicalSpace X] [TopologicalSpace Y] [RegularSpace Y] {A : Set X}
 #check @HasBasis.tendsto_right_iff - example [TopologicalSpace X] [TopologicalSpace.FirstCountableTopology X] - {s : Set X} {a : X} : + example [TopologicalSpace X] [TopologicalSpace.FirstCountableTopology X] {s : Set X} {a : X} : a ∈ closure s ↔ ∃ u : ℕ → X, (∀ n, u n ∈ s) ∧ Tendsto u atTop (𝓝 a) := mem_closure_iff_seq_limit
- @@ -124,8 +121,7 @@ variable [TopologicalSpace X]
 example {F : Filter X} {x : X} : ClusterPt x F ↔ NeBot (𝓝 x ⊓ F) := Iff.rfl - example {s : Set X} : - IsCompact s ↔ ∀ (F : Filter X) [NeBot F], F ≤ 𝓟 s → ∃ a ∈ s, ClusterPt a F := + example {s : Set X} : IsCompact s ↔ ∀ (F : Filter X) [NeBot F], F ≤ 𝓟 s → ∃ a ∈ s, ClusterPt a F := Iff.rfl example [TopologicalSpace.FirstCountableTopology X] {s : Set X} {u : ℕ → X} (hs : IsCompact s)
MIL/C07_Topology/solutions/Solutions_S01_Filters.lean
- @@ -49,7 +49,7 @@ example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) :
 Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := calc Tendsto f atTop (𝓝 (x₀, y₀)) ↔ map f atTop ≤ 𝓝 (x₀, y₀) := Iff.rfl - _ ↔ map f atTop ≤ 𝓝 x₀ ×ˢ 𝓝 y₀ := by rw [nhds_prod_eq] + _ ↔ map f atTop ≤ 𝓝 x₀ ×ᶠ 𝓝 y₀ := by rw [nhds_prod_eq] _ ↔ map f atTop ≤ comap Prod.fst (𝓝 x₀) ⊓ comap Prod.snd (𝓝 y₀) := Iff.rfl _ ↔ map f atTop ≤ comap Prod.fst (𝓝 x₀) ∧ map f atTop ≤ comap Prod.snd (𝓝 y₀) := le_inf_iff _ ↔ map Prod.fst (map f atTop) ≤ 𝓝 x₀ ∧ map Prod.snd (map f atTop) ≤ 𝓝 y₀ := by
- @@ -63,9 +63,10 @@ example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) :
 Tendsto f atTop (𝓝 (x₀, y₀)) ↔ Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := by rw [nhds_prod_eq] - unfold Tendsto SProd.sprod Filter.instSProd Filter.prod - erw [le_inf_iff, ← map_le_iff_le_comap, map_map, ← map_le_iff_le_comap, map_map] + unfold Tendsto Filter.prod + rw [le_inf_iff, ← map_le_iff_le_comap, map_map, ← map_le_iff_le_comap, map_map] example (u : ℕ → ℝ) (M : Set ℝ) (x : ℝ) (hux : Tendsto u atTop (𝓝 x)) (huM : ∀ᶠ n in atTop, u n ∈ M) : x ∈ closure M := mem_closure_iff_clusterPt.mpr (neBot_of_le <| le_inf hux <| le_principal_iff.mpr huM) +
MIL/C07_Topology/solutions/Solutions_S02_Metric_Spaces.lean
- @@ -2,28 +2,33 @@ import Mathlib.Topology.Instances.Real
 import Mathlib.Analysis.NormedSpace.BanachSteinhaus open Set Filter + open Topology Filter variable {X : Type _} [MetricSpace X] (a b c : X) #check (dist a b : ℝ) + #check (dist_nonneg : 0 ≤ dist a b) + #check (dist_eq_zero : dist a b = 0 ↔ a = b) + #check (dist_comm a b : dist a b = dist b a) + #check (dist_triangle a b c : dist a c ≤ dist a b + dist b c) -- Note the next three lines are not quoted, their purpose is to make sure those things don't get renamed while we're looking elsewhere. #check EMetricSpace + #check PseudoMetricSpace + #check PseudoEMetricSpace - example {u : ℕ → X} {a : X} : - Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε := + example {u : ℕ → X} {a : X} : Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε := Metric.tendsto_atTop example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} : - Continuous f ↔ - ∀ x : X, ∀ ε > 0, ∃ δ > 0, ∀ x', dist x' x < δ → dist (f x') (f x) < ε := + Continuous f ↔ ∀ x : X, ∀ ε > 0, ∃ δ > 0, ∀ x', dist x' x < δ → dist (f x') (f x) < ε := Metric.continuous_iff example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) :
- @@ -84,8 +89,7 @@ example {s : Set X} (hs : IsClosed s) {u : ℕ → X} (hu : Tendsto u atTop (
 example {s : Set X} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, a ∈ Metric.ball b ε := Metric.mem_closure_iff - example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) : - a ∈ closure s := + example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) : a ∈ closure s := sorry example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) : a ∈ closure s := by
- @@ -110,18 +114,16 @@ example {s : Set X} (hs : IsCompact s) {u : ℕ → X} (hu : ∀ n, u n ∈ s) :
 ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := hs.tendsto_subseq hu - example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} - (hfs : ContinuousOn f s) : + example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} (hfs : ContinuousOn f s) : ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y := hs.exists_forall_le hs' hfs - example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} - (hfs : ContinuousOn f s) : + example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} (hfs : ContinuousOn f s) : ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x := hs.exists_forall_ge hs' hfs example {s : Set X} (hs : IsCompact s) : IsClosed s := - hs.isClosed + hs.IsClosed example {X : Type _} [MetricSpace X] [CompactSpace X] : IsCompact (univ : Set X) := isCompact_univ
- @@ -129,12 +131,10 @@ example {X : Type _} [MetricSpace X] [CompactSpace X] : IsCompact (univ : Set X)
 #check IsCompact.isClosed example {X : Type _} [MetricSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} : - UniformContinuous f ↔ - ∀ ε > 0, ∃ δ > 0, ∀ {a b : X}, dist a b < δ → dist (f a) (f b) < ε := + UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ {a b : X}, dist a b < δ → dist (f a) (f b) < ε := Metric.uniformContinuous_iff - example {X : Type _} [MetricSpace X] [CompactSpace X] - {Y : Type _} [MetricSpace Y] {f : X → Y} + example {X : Type _} [MetricSpace X] [CompactSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} (hf : Continuous f) : UniformContinuous f := sorry
- @@ -146,13 +146,13 @@ example {X : Type _} [MetricSpace X] [CompactSpace X] {Y : Type _} [MetricSpace
 have φ_cont : Continuous φ := hf.fst'.dist hf.snd' let K := { p : X × X | ε ≤ φ p } have K_closed : IsClosed K := isClosed_le continuous_const φ_cont - have K_cpct : IsCompact K := K_closed.isCompact + have K_cpct : IsCompact K := K_closed.is_compact cases' eq_empty_or_nonempty K with hK hK · use 1, by norm_num - intro x y _ + intro x y hxy have : (x, y) ∉ K := by simp [hK] - simpa using this - · rcases K_cpct.exists_forall_le hK continuous_dist.continuousOn with ⟨⟨x₀, x₁⟩, xx_in, H⟩ + simpa [K] + · rcases K_cpct.exists_forall_le hK continuous_dist.continuous_on with ⟨⟨x₀, x₁⟩, xx_in, H⟩ use dist x₀ x₁ constructor · change _ < _
- @@ -165,16 +165,13 @@ example {X : Type _} [MetricSpace X] [CompactSpace X] {Y : Type _} [MetricSpace
 intro hxx' exact H (x, x') hxx' - example (u : ℕ → X) : - CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε := + example (u : ℕ → X) : CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε := Metric.cauchySeq_iff - example (u : ℕ → X) : - CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ n ≥ N, dist (u n) (u N) < ε := + example (u : ℕ → X) : CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ n ≥ N, dist (u n) (u N) < ε := Metric.cauchySeq_iff' - example [CompleteSpace X] (u : ℕ → X) (hu : CauchySeq u) : - ∃ x, Tendsto u atTop (𝓝 x) := + example [CompleteSpace X] (u : ℕ → X) (hu : CauchySeq u) : ∃ x, Tendsto u atTop (𝓝 x) := cauchySeq_tendsto_of_complete hu open BigOperators
- @@ -208,8 +205,8 @@ example {u : ℕ → X} (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^
 simp_rw [← one_div_pow (2 : ℝ)] apply tendsto_pow_atTop_nhds_0_of_lt_1 <;> linarith exact tendsto_const_nhds - rcases(atTop_basis.tendsto_iff (nhds_basis_Ioo_pos (0 : ℝ))).mp this ε ε_pos with ⟨N, _, hN⟩ - exact ⟨N, by simpa using (hN N left_mem_Ici).2⟩ + rcases(atTop_basis.tendsto_iff (nhds_basis_Ioo_pos (0 : ℝ))).mp this ε ε_pos with ⟨N, H, hN⟩ + exact ⟨N, by simpa using (hN N le_rfl).2⟩ use N intro n hn obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn
- @@ -217,8 +214,8 @@ example {u : ℕ → X} (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^
 dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := by rw [dist_comm, add_zero] _ ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) := (dist_le_range_sum_dist (fun i => u (N + i)) k) - _ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := (sum_le_sum fun i _ => hu <| N + i) - _ = 1 / 2 ^ N * ∑ i in range k, (1 / 2 : ℝ) ^ i := by simp_rw [← one_div_pow, pow_add, ← mul_sum] + _ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := (sum_le_sum fun i hi => hu <| N + i) + _ = 1 / 2 ^ N * ∑ i in range k, (1 / 2) ^ i := by simp_rw [← one_div_pow, pow_add, ← mul_sum] _ ≤ 1 / 2 ^ N * 2 := (mul_le_mul_of_nonneg_left (sum_geometric_two_le _) (one_div_nonneg.mpr (pow_nonneg (zero_le_two : (0 : ℝ) ≤ 2) _)))
- @@ -234,31 +231,28 @@ example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd :
 sorry /- Translate the density assumption into two functions `center` and `radius` associating to any n, x, δ, δpos a center and a positive radius such that - `closedBall center radius` is included both in `f n` and in `closedBall x δ`. + `closed_ball center radius` is included both in `f n` and in `closed_ball x δ`. We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ have : ∀ (n : ℕ) (x : X), - ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := + ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closed_ball y r ⊆ closed_ball x δ ∩ f n := by sorry choose! center radius Hpos HB Hball using this intro x - rw [mem_closure_iff_nhds_basis nhds_basis_closedBall] + rw [mem_closure_iff_nhds_basis nhds_basis_closed_ball] intro ε εpos - /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` - belonging to all `f n`. For this, we construct inductively a sequence - `F n = (c n, r n)` such that the closed ball `closedBall (c n) (r n)` is included - in the previous ball and in `f n`, and such that `r n` is small enough to ensure - that `c n` is a Cauchy sequence. Then `c n` converges to a limit which belongs - to all the `f n`. -/ + /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` belonging to all + `f n`. For this, we construct inductively a sequence `F n = (c n, r n)` such that the closed ball + `closed_ball (c n) (r n)` is included in the previous ball and in `f n`, and such that + `r n` is small enough to ensure that `c n` is a Cauchy sequence. Then `c n` converges to a + limit which belongs to all the `f n`. -/ let F : ℕ → X × ℝ := 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) + 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) let c : ℕ → X := fun n => (F n).1 let r : ℕ → ℝ := fun n => (F n).2 have rpos : ∀ n, 0 < r n := by sorry have rB : ∀ n, r n ≤ B n := by sorry - have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := by - sorry + have incl : ∀ n, closed_ball (c (n + 1)) (r (n + 1)) ⊆ closed_ball (c n) (r n) ∩ f n := by sorry have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by sorry have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`.
- @@ -266,8 +260,8 @@ example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd :
 -- this point `y` will be the desired point. We will check that it belongs to all -- `f n` and to `ball x ε`. use y - have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by sorry - have yball : ∀ n, y ∈ closedBall (c n) (r n) := by sorry + have I : ∀ n, ∀ m ≥ n, closed_ball (c m) (r m) ⊆ closed_ball (c n) (r n) := by sorry + have yball : ∀ n, y ∈ closed_ball (c n) (r n) := by sorry sorry example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd : ∀ n, Dense (f n)) :
- @@ -276,23 +270,23 @@ example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd :
 have Bpos : ∀ n, 0 pow_pos sorry n /- Translate the density assumption into two functions `center` and `radius` associating to any n, x, δ, δpos a center and a positive radius such that - `closedBall center radius` is included both in `f n` and in `closedBall x δ`. + `closed_ball center radius` is included both in `f n` and in `closed_ball x δ`. We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ have : ∀ (n : ℕ) (x : X), - ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := by + ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closed_ball y r ⊆ closed_ball x δ ∩ f n := by intro n x δ δpos have : x ∈ closure (f n) := hd n x rcases Metric.mem_closure_iff.1 this (δ / 2) (half_pos δpos) with ⟨y, ys, xy⟩ rw [dist_comm] at xy - obtain ⟨r, rpos, hr⟩ : ∃ r > 0, closedBall y r ⊆ f n := - nhds_basis_closedBall.mem_iff.1 (isOpen_iff_mem_nhds.1 (ho n) y ys) + obtain ⟨r, rpos, hr⟩ : ∃ r > 0, closed_ball y r ⊆ f n := + nhds_basis_closed_ball.mem_iff.1 (isOpen_iff_mem_nhds.1 (ho n) y ys) refine' ⟨y, min (min (δ / 2) r) (B (n + 1)), _, _, fun z hz => ⟨_, _⟩⟩ show 0 < min (min (δ / 2) r) (B (n + 1)) exact lt_min (lt_min (half_pos δpos) rpos) (Bpos (n + 1)) show min (min (δ / 2) r) (B (n + 1)) ≤ B (n + 1) exact min_le_right _ _ - show z ∈ closedBall x δ + show z ∈ closed_ball x δ exact calc dist z x ≤ dist z y + dist y x := dist_triangle _ _ _
- @@ -308,10 +302,10 @@ example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd :
 _ ≤ r := (min_le_left _ _).trans (min_le_right _ _) ) choose! center radius Hpos HB Hball using this - refine' fun x => (mem_closure_iff_nhds_basis nhds_basis_closedBall).2 fun ε εpos => _ + refine' fun x => (mem_closure_iff_nhds_basis nhds_basis_closed_ball).2 fun ε εpos => _ /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` belonging to all `f n`. For this, we construct inductively a sequence `F n = (c n, r n)` such that the closed ball - `closedBall (c n) (r n)` is included in the previous ball and in `f n`, and such that + `closed_ball (c n) (r n)` is included in the previous ball and in `f n`, and such that `r n` is small enough to ensure that `c n` is a Cauchy sequence. Then `c n` converges to a limit which belongs to all the `f n`. -/ let F : ℕ → X × ℝ := fun n =>
- @@ -328,18 +322,18 @@ example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd :
 induction' n with n hn exact min_le_right _ _ exact HB n (c n) (r n) (rpos n) - have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := fun n => + have incl : ∀ n, closed_ball (c (n + 1)) (r (n + 1)) ⊆ closed_ball (c n) (r n) ∩ f n := fun n => Hball n (c n) (r n) (rpos n) have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by intro n rw [dist_comm] - have A : c (n + 1) ∈ closedBall (c (n + 1)) (r (n + 1)) := - mem_closedBall_self (rpos <| n + 1).le + have A : c (n + 1) ∈ closed_ball (c (n + 1)) (r (n + 1)) := + mem_closed_ball_self (rpos <| n + 1).le have I := calc - closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) := + closed_ball (c (n + 1)) (r (n + 1)) ⊆ closed_ball (c n) (r n) := (incl n).trans (inter_subset_left _ _) - _ ⊆ closedBall (c n) (B n) := closedBall_subset_closedBall (rB n) + _ ⊆ closed_ball (c n) (B n) := closed_ball_subset_closed_ball (rB n) exact I A have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist
- @@ -348,20 +342,20 @@ example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd :
 -- this point `y` will be the desired point. We will check that it belongs to all -- `f n` and to `ball x ε`. use y - have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by + have I : ∀ n, ∀ m ≥ n, closed_ball (c m) (r m) ⊆ closed_ball (c n) (r n) := by intro n refine' Nat.le_induction _ fun m hnm h => _ · exact Subset.rfl · exact (incl m).trans ((Set.inter_subset_left _ _).trans h) - have yball : ∀ n, y ∈ closedBall (c n) (r n) := by + have yball : ∀ n, y ∈ closed_ball (c n) (r n) := by intro n - refine' isClosed_ball.mem_of_tendsto ylim _ + refine' is_closed_ball.mem_of_tendsto ylim _ refine' (Filter.eventually_ge_atTop n).mono fun m hm => _ - exact I n m hm (mem_closedBall_self (rpos _).le) + exact I n m hm (mem_closed_ball_self (rpos _).le) constructor · suffices ∀ n, y ∈ f n by rwa [Set.mem_iInter] intro n - have : closedBall (c (n + 1)) (r (n + 1)) ⊆ f n := + have : closed_ball (c (n + 1)) (r (n + 1)) ⊆ f n := Subset.trans (incl n) (inter_subset_right _ _) exact this (yball (n + 1)) calc
MIL/C07_Topology/solutions/Solutions_S03_Topological_Spaces.lean
- @@ -1,9 +1,12 @@
 import Mathlib.Topology.Instances.Real import Mathlib.Analysis.NormedSpace.BanachSteinhaus - open Set Filter Topology + open Set Filter + + open Topology Filter section + variable {X : Type _} [TopologicalSpace X] example : IsOpen (univ : Set X) :=
- @@ -15,8 +18,7 @@ example : IsOpen (∅ : Set X) :=
 example {ι : Type _} {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋃ i, s i) := isOpen_iUnion hs - example {ι : Type _} [Fintype ι] {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : - IsOpen (⋂ i, s i) := + example {ι : Type _} [Fintype ι] {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋂ i, s i) := isOpen_iInter hs variable {Y : Type _} [TopologicalSpace Y]
- @@ -85,10 +87,8 @@ example (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) (f : X → Y) :
 Continuous f ↔ TopologicalSpace.coinduced f T_X ≤ T_Y := continuous_iff_coinduced_le - example {Z : Type _} (f : X → Y) (T_X : TopologicalSpace X) (T_Z : TopologicalSpace Z) - (g : Y → Z) : - @Continuous Y Z (TopologicalSpace.coinduced f T_X) T_Z g ↔ - @Continuous X Z T_X T_Z (g ∘ f) := by + example {Z : Type _} (f : X → Y) (T_X : TopologicalSpace X) (T_Z : TopologicalSpace Z) (g : Y → Z) : + @Continuous Y Z (TopologicalSpace.coinduced f T_X) T_Z g ↔ @Continuous X Z T_X T_Z (g ∘ f) := by rw [continuous_iff_coinduced_le, coinduced_compose, continuous_iff_coinduced_le] example (ι : Type _) (X : ι → Type _) (T_X : ∀ i, TopologicalSpace <| X i) :
- @@ -104,19 +104,16 @@ example [TopologicalSpace X] [RegularSpace X] (a : X) :
 (𝓝 a).HasBasis (fun s : Set X => s ∈ 𝓝 a ∧ IsClosed s) id := closed_nhds_basis a - example [TopologicalSpace X] {x : X} : - (𝓝 x).HasBasis (fun t : Set X => t ∈ 𝓝 x ∧ IsOpen t) id := + example [TopologicalSpace X] {x : X} : (𝓝 x).HasBasis (fun t : Set X => t ∈ 𝓝 x ∧ IsOpen t) id := nhds_basis_opens' x - theorem aux {X Y A : Type _} [TopologicalSpace X] {c : A → X} - {f : A → Y} {x : X} {F : Filter Y} - (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) : + theorem aux {X Y A : Type _} [TopologicalSpace X] {c : A → X} {f : A → Y} {x : X} {F : Filter Y} + (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) : ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := sorry - example {X Y A : Type _} [TopologicalSpace X] {c : A → X} - {f : A → Y} {x : X} {F : Filter Y} - (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) : + example {X Y A : Type _} [TopologicalSpace X] {c : A → X} {f : A → Y} {x : X} {F : Filter Y} + (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) : ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := by simpa [and_assoc] using ((nhds_basis_opens' x).comap c).tendsto_left_iff.mp h V' V'_in
- @@ -138,7 +135,7 @@ example [TopologicalSpace X] [TopologicalSpace Y] [T3Space Y] {A : Set X} (hA :
 · rw [continuous_iff_continuousAt] intro x suffices ∀ V' ∈ 𝓝 (φ x), IsClosed V' → φ ⁻¹' V' ∈ 𝓝 x by - simpa [ContinuousAt, (closed_nhds_basis (φ x)).tendsto_right_iff] + simp [ContinuousAt, (closed_nhds_basis _).tendsto_right_iff] intro V' V'_in V'_closed obtain ⟨V, V_in, V_op, hV⟩ : ∃ V ∈ 𝓝 x, IsOpen V ∧ (↑) ⁻¹' V ⊆ f ⁻¹' V' := aux (hφ x) V'_in suffices : ∀ y ∈ V, φ y ∈ V'
- @@ -150,10 +147,9 @@ example [TopologicalSpace X] [TopologicalSpace Y] [T3Space Y] {A : Set X} (hA :
 exact mem_of_superset (preimage_mem_comap hVx) hV · intro a have lim : Tendsto f (𝓝 a) (𝓝 <| φ a) := by simpa [nhds_induced] using hφ a - exact tendsto_nhds_unique lim f_cont.continuousAt + exact tendsto_nhds_unique limUnder f_cont.continuousAt - example [TopologicalSpace X] [TopologicalSpace.FirstCountableTopology X] - {s : Set X} {a : X} : + example [TopologicalSpace X] [TopologicalSpace.FirstCountableTopology X] {s : Set X} {a : X} : a ∈ closure s ↔ ∃ u : ℕ → X, (∀ n, u n ∈ s) ∧ Tendsto u atTop (𝓝 a) := mem_closure_iff_seq_limit
- @@ -162,8 +158,7 @@ variable [TopologicalSpace X]
 example {F : Filter X} {x : X} : ClusterPt x F ↔ NeBot (𝓝 x ⊓ F) := Iff.rfl - example {s : Set X} : - IsCompact s ↔ ∀ (F : Filter X) [NeBot F], F ≤ 𝓟 s → ∃ a ∈ s, ClusterPt a F := + example {s : Set X} : IsCompact s ↔ ∀ (F : Filter X) [NeBot F], F ≤ 𝓟 s → ∃ a ∈ s, ClusterPt a F := Iff.rfl example [TopologicalSpace.FirstCountableTopology X] {s : Set X} {u : ℕ → X} (hs : IsCompact s)
MIL/C08_Topology/S03_Topological_Spaces.lean
- @@ -152,4 +152,3 @@ example {ι : Type _} {s : Set X} (hs : IsCompact s) (U : ι → Set X) (hUo :
  example [CompactSpace X] : IsCompact (univ : Set X) := isCompact_univ -
MIL/C08_Topology/solutions/Solutions_S03_Topological_Spaces.lean
- @@ -204,4 +204,3 @@ example {ι : Type _} {s : Set X} (hs : IsCompact s) (U : ι → Set X) (hUo :
  example [CompactSpace X] : IsCompact (univ : Set X) := isCompact_univ -
MIL/C10_Integration_and_Measure_Theory/S01_Elementary_Integration.lean (new)
- @@ -0,0 +1,35 @@
  + import Mathlib.MeasureTheory.Integral.IntervalIntegral + import Mathlib.Analysis.SpecialFunctions.Integrals + import Mathlib.Analysis.Convolution + + local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) + + open Set Filter + + open Topology Filter + + noncomputable section + + open MeasureTheory intervalIntegral + + open Interval + -- this introduces the notation `[[a, b]]` for the segment from `min a b` to `max a b` + + example (a b : ℝ) : (∫ x in a..b, x) = (b ^ 2 - a ^ 2) / 2 := + integral_id + + example {a b : ℝ} (h : (0 : ℝ) ∉ [[a, b]]) : (∫ x in a..b, 1 / x) = Real.log (b / a) := + integral_one_div h + + example (f : ℝ → ℝ) (hf : Continuous f) (a b : ℝ) : deriv (fun u => ∫ x : ℝ in a..u, f x) b = f b := + (integral_hasStrictDerivAt_right (hf.intervalIntegrable _ _) (hf.stronglyMeasurableAtFilter _ _) + hf.continuousAt).hasDerivAt.deriv + + example {f : ℝ → ℝ} {a b : ℝ} {f' : ℝ → ℝ} (h : ∀ x ∈ [[a, b]], HasDerivAt f (f' x) x) + (h' : IntervalIntegrable f' volume a b) : (∫ y in a..b, f' y) = f b - f a := + integral_eq_sub_of_hasDerivAt h h' + + open Convolution + + example (f : ℝ → ℝ) (g : ℝ → ℝ) : f ⋆ g = fun x => ∫ t, f t * g (x - t) := + rfl
MIL/C10_Integration_and_Measure_Theory/S02_Measure_Theory.lean (new)
- @@ -0,0 +1,48 @@
  + import Mathlib.Analysis.NormedSpace.FiniteDimension + import Mathlib.Analysis.Convolution + import Mathlib.MeasureTheory.Function.Jacobian + import Mathlib.MeasureTheory.Integral.Bochner + import Mathlib.MeasureTheory.Measure.Lebesgue.Basic + + open Set Filter + + noncomputable section + + variable {α : Type _} [MeasurableSpace α] + + example : MeasurableSet (∅ : Set α) := + MeasurableSet.empty + + example : MeasurableSet (univ : Set α) := + MeasurableSet.univ + + example {s : Set α} (hs : MeasurableSet s) : MeasurableSet (sᶜ) := + hs.compl + + example : Encodable ℕ := by infer_instance + + example (n : ℕ) : Encodable (Fin n) := by infer_instance + + variable {ι : Type _} [Encodable ι] + + example {f : ι → Set α} (h : ∀ b, MeasurableSet (f b)) : MeasurableSet (⋃ b, f b) := + MeasurableSet.iUnion h + + example {f : ι → Set α} (h : ∀ b, MeasurableSet (f b)) : MeasurableSet (⋂ b, f b) := + MeasurableSet.iInter h + + open MeasureTheory + variable {μ : Measure α} + + example (s : Set α) : μ s = ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), μ t := + measure_eq_iInf s + + example (s : ι → Set α) : μ (⋃ i, s i) ≤ ∑' i, μ (s i) := + measure_iUnion_le s + + example {f : ℕ → Set α} (hmeas : ∀ i, MeasurableSet (f i)) (hdis : Pairwise (Disjoint on f)) : + μ (⋃ i, f i) = ∑' i, μ (f i) := + μ.m_iUnion hmeas hdis + + example {P : α → Prop} : (∀ᵐ x ∂μ, P x) ↔ ∀ᶠ x in μ.ae, P x := + Iff.rfl
MIL/C10_Integration_and_Measure_Theory/S03_Integration.lean (new)
- @@ -0,0 +1,61 @@
  + import Mathlib.Analysis.NormedSpace.FiniteDimension + import Mathlib.Analysis.Convolution + import Mathlib.MeasureTheory.Function.Jacobian + import Mathlib.MeasureTheory.Integral.Bochner + import Mathlib.MeasureTheory.Measure.Lebesgue.Basic + + open Set Filter + + open Topology Filter ENNReal + + open MeasureTheory + + noncomputable section + variable {α : Type _} [MeasurableSpace α] + variable {μ : Measure α} + + section + variable {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : α → E} + + example {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) : + (∫ a, f a + g a ∂μ) = (∫ a, f a ∂μ) + ∫ a, g a ∂μ := + integral_add hf hg + + example {s : Set α} (c : E) : (∫ x in s, c ∂μ) = (μ s).toReal • c := + set_integral_const c + + example {F : ℕ → α → E} {f : α → E} (bound : α → ℝ) (hmeas : ∀ n, AEStronglyMeasurable (F n) μ) + (hint : Integrable bound μ) (hbound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a) + (hlim : ∀ᵐ a ∂μ, Tendsto (fun n : ℕ => F n a) atTop (𝓝 (f a))) : + Tendsto (fun n => ∫ a, F n a ∂μ) atTop (𝓝 (∫ a, f a ∂μ)) := + tendsto_integral_of_dominated_convergence bound hmeas hint hbound hlim + + example {α : Type _} [MeasurableSpace α] {μ : Measure α} [SigmaFinite μ] {β : Type _} + [MeasurableSpace β] {ν : Measure β} [SigmaFinite ν] (f : α × β → E) + (hf : Integrable f (μ.prod ν)) : (∫ z, f z ∂ μ.prod ν) = ∫ x, ∫ y, f (x, y) ∂ν ∂μ := + integral_prod f hf + + end + + section + + open Convolution + + variable {𝕜 : Type _} {G : Type _} {E : Type _} {E' : Type _} {F : Type _} [NormedAddCommGroup E] + [NormedAddCommGroup E'] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] + [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] [MeasurableSpace G] [NormedSpace ℝ F] [CompleteSpace F] + [Sub G] + + example (f : G → E) (g : G → E') (L : E →L[𝕜] E' →L[𝕜] F) (μ : Measure G) : + f ⋆[L, μ] g = fun x => ∫ t, L (f t) (g (x - t)) ∂μ := + rfl + + end + + example {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] + [MeasurableSpace E] [BorelSpace E] (μ : Measure E) [μ.IsAddHaarMeasure] {F : Type _} + [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] {s : Set E} {f : E → E} + {f' : E → E →L[ℝ] E} (hs : MeasurableSet s) + (hf : ∀ x : E, x ∈ s → HasFDerivWithinAt f (f' x) s x) (h_inj : InjOn f s) (g : E → F) : + (∫ x in f '' s, g x ∂μ) = ∫ x in s, |(f' x).det| • g (f x) ∂μ := + integral_image_eq_integral_abs_det_fderiv_smul μ hs hf h_inj g
MIL/C10_Integration_and_Measure_Theory/solutions/Solutions_S01_Elementary_Integration.lean (new)
- @@ -0,0 +1,12 @@
  + import Mathlib.MeasureTheory.Integral.IntervalIntegral + import Mathlib.Analysis.SpecialFunctions.Integrals + import Mathlib.Analysis.Convolution + + local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) + + open Set Filter + + open Topology Filter + + noncomputable section +
MIL/C10_Integration_and_Measure_Theory/solutions/Solutions_S02_Measure_Theory.lean (new)
- @@ -0,0 +1,17 @@
  + import Mathlib.Analysis.NormedSpace.FiniteDimension + import Mathlib.Analysis.Convolution + import Mathlib.MeasureTheory.Function.Jacobian + import Mathlib.MeasureTheory.Integral.Bochner + import Mathlib.MeasureTheory.Measure.Lebesgue.Basic + + open Set Filter + + noncomputable section + + variable {α : Type _} [MeasurableSpace α] + + variable {ι : Type _} [Encodable ι] + + open MeasureTheory + variable {μ : Measure α} +
MIL/C10_Integration_and_Measure_Theory/solutions/Solutions_S03_Integration.lean (new)
- @@ -0,0 +1,16 @@
  + import Mathlib.Analysis.NormedSpace.FiniteDimension + import Mathlib.Analysis.Convolution + import Mathlib.MeasureTheory.Function.Jacobian + import Mathlib.MeasureTheory.Integral.Bochner + import Mathlib.MeasureTheory.Measure.Lebesgue.Basic + + open Set Filter + + open Topology Filter ENNReal + + open MeasureTheory + + noncomputable section + variable {α : Type _} [MeasurableSpace α] + variable {μ : Measure α} +
html/C01_Introduction.html
- @@ -58,6 +58,7 @@
 <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a></li> + <li class="toctree-l1"><a class="reference internal" href="C10_Integration_and_Measure_Theory.html">10. Integration and Measure Theory</a></li> </ul> <ul> <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li>
html/C02_Basics.html
- @@ -62,6 +62,7 @@
 <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a></li> + <li class="toctree-l1"><a class="reference internal" href="C10_Integration_and_Measure_Theory.html">10. Integration and Measure Theory</a></li> </ul> <ul> <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li>
- @@ -136,7 +137,7 @@ in VS Code.
 The symbol doesn’t appear until you hit space or the tab key. If you hover over a symbol when reading a Lean file, VS Code will show you the syntax that can be used to enter it. - If you are curious to see all available abreviations, you can hit Ctrl-Shift-p + If you are curious to see all available abbreviations, you can hit Ctrl-Shift-p and then type abbreviations to get access to the <code class="docutils literal notranslate">Lean 4: Show all abbreviations</code> command. If your keyboard does not have an easily accessible backslash, you can change the leading character by changing the
html/C03_Logic.html
- @@ -63,6 +63,7 @@
 <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a></li> + <li class="toctree-l1"><a class="reference internal" href="C10_Integration_and_Measure_Theory.html">10. Integration and Measure Theory</a></li> </ul> <ul> <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li>
html/C04_Sets_and_Functions.html
- @@ -60,6 +60,7 @@
 <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a></li> + <li class="toctree-l1"><a class="reference internal" href="C10_Integration_and_Measure_Theory.html">10. Integration and Measure Theory</a></li> </ul> <ul> <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li>
html/C05_Number_Theory.html
- @@ -60,6 +60,7 @@
 <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a></li> + <li class="toctree-l1"><a class="reference internal" href="C10_Integration_and_Measure_Theory.html">10. Integration and Measure Theory</a></li> </ul> <ul> <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li>
- @@ -411,7 +412,7 @@ In Lean, the natural numbers are declared as follows.
 | succ (n : Nat) : Nat </pre></div> </div> - You can find this in the library by writting <code class="docutils literal notranslate">#check Nat</code> and + You can find this in the library by writing <code class="docutils literal notranslate">#check Nat</code> and then using <code class="docutils literal notranslate">ctrl-click</code> on the identifier <code class="docutils literal notranslate">Nat</code>. The command specifies that <code class="docutils literal notranslate">Nat</code> is the datatype generated freely and inductively by the two constructors <code class="docutils literal notranslate">zero : Nat</code> and
- @@ -845,7 +846,7 @@ and abandoning the computational interpretation.
 We use the command <code class="docutils literal notranslate">open Finset</code> to avail ourselves of shorter names for the relevant theorems. Unlike the case with sets, most equivalences involving finsets do not hold definitionally, - so they need to be expanded manually using equivalances like + so they need to be expanded manually using equivalences like <code class="docutils literal notranslate">Finset.subset_iff</code>, <code class="docutils literal notranslate">Finset.mem_union</code>, <code class="docutils literal notranslate">Finset.mem_inter</code>, and <code class="docutils literal notranslate">Finset.mem_sdiff</code>. The <code class="docutils literal notranslate">ext</code> tactic can still be used to reduce show that two finite sets are equal by showing
html/C06_Structures.html
- @@ -60,6 +60,7 @@
 <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a></li> + <li class="toctree-l1"><a class="reference internal" href="C10_Integration_and_Measure_Theory.html">10. Integration and Measure Theory</a></li> </ul> <ul> <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li>
html/C07_Hierarchies.html
- @@ -59,6 +59,7 @@
 </li> <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a></li> + <li class="toctree-l1"><a class="reference internal" href="C10_Integration_and_Measure_Theory.html">10. Integration and Measure Theory</a></li> </ul> <ul> <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li>
- @@ -262,7 +263,7 @@ too). An easy way to see what are the fields of our classes is to check their co
 </pre></div> </div> So we see that <code class="docutils literal notranslate">Monoid₁</code> takes <code class="docutils literal notranslate">Semigroup₁ α</code> argument as expected but then it won’t - take a would-be overlapping <code class="docutils literal notranslate">DiaOneClass₁ α</code> argument but instead tears it appart and includes + take a would-be overlapping <code class="docutils literal notranslate">DiaOneClass₁ α</code> argument but instead tears it apart and includes only the non-overlapping parts. And it also auto-generated an instance <code class="docutils literal notranslate">Monoid₁.toDiaOneClass₁</code> which is not a field but has the expected signature which, from the end-user point of view, restores the symmetry between the two extended classes <code class="docutils literal notranslate">Semigroup₁</code> and <code class="docutils literal notranslate">DiaOneClass₁</code>.
- @@ -320,7 +321,7 @@ assumes every type has only one instance of each type class. There are various
 ways to solve this issue. Surprisingly mathlib uses the naive idea to duplicate everything for additive and multiplicative theories with the help of some code-generating attribute. Structures and classes are defined in both additive and multiplicative notation - with an attibute <code class="docutils literal notranslate">to_additive</code> linking them. In case of multiple inheritance like for + with an attribute <code class="docutils literal notranslate">to_additive</code> linking them. In case of multiple inheritance like for semi-groups, the auto-generated “symmetry-restoring” instances need also to be marked. This is a bit technical you don’t need to understand details. The important point is that lemmas are then only stated in multiplicative notation and marked with the attribute <code class="docutils literal notranslate">to_additive</code>
- @@ -328,7 +329,7 @@ to generate the additive version as <code class="docutils literal notranslate"><
 version <code class="docutils literal notranslate">left_neg_eq_right_neg'</code>. In order to check the name of this additive version we used the <code class="docutils literal notranslate">whatsnew in</code> command on top of <code class="docutils literal notranslate">left_inv_eq_right_inv'</code>. <div class="highlight-lean notranslate"><div class="highlight"><pre>class AddSemigroup₃ (α : Type) extends Add α where - /-- Multiplication is associative -/ + /-- Addition is associative -/ add_assoc₃ : ∀ a b c : α, a + b + c = a + (b + c) @[to_additive AddSemigroup₃]
- @@ -376,7 +377,7 @@ groups, and then define rings.
 inv_mul : ∀ a : G, a⁻¹ * a = 1 </pre></div> </div> - We should remember to tagged lemmas with <code class="docutils literal notranslate">simp</code> when approriate. + We should remember to tagged lemmas with <code class="docutils literal notranslate">simp</code> when appropriate. <div class="highlight-lean notranslate"><div class="highlight"><pre>attribute [simp] Group₃.inv_mul AddGroup₃.neg_add </pre></div> </div>
- @@ -407,7 +408,7 @@ to the additive version as follows.
 class CommGroup₃ (G : Type) extends Group₃ G, CommMonoid₃ G </pre></div> </div> - We are now ready for rings. For demonstration puprposes we won’t assume that addition is + We are now ready for rings. For demonstration purposes we won’t assume that addition is commutative, and then immediately provide an instance of <code class="docutils literal notranslate">AddCommGroup₃</code>. Mathlib does not play this game, first because in practice this does not make any ring instance easier and also because Mathlib’s algebraic hierarchy goes through semi-rings which are like rings but without
- @@ -500,8 +501,8 @@ would have the signature appearing in the error message:
 <code class="docutils literal notranslate">(R : Type) → [inst : Ring₃ R] → {M : Type} → [self : Module₁ R M] → AddCommGroup₃ M</code>. With such an instance in the type class database, each time Lean would look for a <code class="docutils literal notranslate">AddCommGroup₃ M</code> instance for some <code class="docutils literal notranslate">M</code>, it would need to go hunting for a completely - unspecified type <code class="docutils literal notranslate">R``and a ``Ring₃ R</code> instance before embarking on the main quest of finding a - <code class="docutils literal notranslate">Module₁ R M</code> instance. Those two side-quests are represented by the meta-variables mentionned in + unspecified type <code class="docutils literal notranslate">R</code> and a <code class="docutils literal notranslate">Ring₃ R</code> instance before embarking on the main quest of finding a + <code class="docutils literal notranslate">Module₁ R M</code> instance. Those two side-quests are represented by the meta-variables mentioned in the error message and denoted by <code class="docutils literal notranslate">?R</code> and <code class="docutils literal notranslate">?inst✝</code> there. Such a <code class="docutils literal notranslate">Module₃.toAddCommGroup₃</code> instance would then be a huge trap for the instance resolution procedure and then <code class="docutils literal notranslate">class</code> command refuses to set it up.
- @@ -553,7 +554,7 @@ for <code class="docutils literal notranslate">ℤ<
 <code class="docutils literal notranslate">ℤ</code> is a ring. Those two module structure correspond to the same abelian group structure, but it is not obvious that they have the same scalar multiplication. They actually do, but this isn’t true by definition, it requires a proof. This is very bad news for the type class - instance resolution procedure and will lead to very frustating failures for users of this + instance resolution procedure and will lead to very frustrating failures for users of this hierarchy. When directly asked to find an instance, Lean will pick one, and we can see which one using: <div class="highlight-lean notranslate"><div class="highlight"><pre>#synth Module₁ ℤ ℤ -- abGrpModule ℤ
- @@ -567,10 +568,10 @@ going through either <code class="docutils literal notranslate">Monoid₁ α</code> to <code class="docutils literal notranslate">Dia₁ α</code> through either <code class="docutils literal notranslate">Semigroup₁ α</code> or <code class="docutils literal notranslate">DiaOneClass₁ α</code> and thanks to the work done by the <code class="docutils literal notranslate">class</code> command, the resulting two <code class="docutils literal notranslate">Dia₁ α</code> instances - are definitionnaly equal. In particular a diamond having a <code class="docutils literal notranslate">Prop</code>-valued class at the bottom - cannot be bad since any too proofs of the same statement are definitionnaly equal. + are definitionally equal. In particular a diamond having a <code class="docutils literal notranslate">Prop</code>-valued class at the bottom + cannot be bad since any too proofs of the same statement are definitionally equal. But the diamond we created with modules is definitely bad. The offending piece is the <code class="docutils literal notranslate">smul</code> - field which is data, not a proof, and we have two constructions that are not definitionnaly equal. + field which is data, not a proof, and we have two constructions that are not definitionally equal. The robust way of fixing this issue is to make sure that going from a rich structure to a poor structure is always done by forgetting data, not by defining data. This well-known pattern as been named “forgetful inheritance” and extensively discussed in
- @@ -627,10 +628,9 @@ and similar tricks. You are now ready to read the definition of monoids, groups,
 in mathlib. There are more complicated than what we have seen here, because they are part of a huge hierarchy, but all principles have been explained above. As an exercise, you can come back to the order relation hierarchy you built above and try - to incorportate a type class <code class="docutils literal notranslate">LT₁</code> carrying the Less-Than notation <code class="docutils literal notranslate"><₁</code> and make sure + to incorporate a type class <code class="docutils literal notranslate">LT₁</code> carrying the Less-Than notation <code class="docutils literal notranslate"><₁</code> and make sure that every preorder comes with a <code class="docutils literal notranslate"><₁</code> which has a default value built from <code class="docutils literal notranslate">≤₁</code> and a <code class="docutils literal notranslate">Prop</code>-valued field asserting the natural relation between those two comparison operators. - TEXT. -/ </section> <section id="morphisms">
- @@ -670,7 +670,7 @@ is really a function that happens to be continuous. This is one reason why Mathl
 </div> We still have bundles continuous functions, which are convenient for instance to put a topology on a space of continuous functions, but they are not the primary tool to work with continuity. - By constrast, morphisms between monoids (or other algebraic structures) are bundled as in: + By contrast, morphisms between monoids (or other algebraic structures) are bundled as in: <div class="highlight-lean notranslate"><div class="highlight"><pre>@[ext] structure MonoidHom₁ (G H : Type) [Monoid G] [Monoid H] where toFun : G → H
- @@ -771,7 +771,7 @@ Hence we can retry defining our class, paying attention to the <code class="docu
 </pre></div> </div> As promised every lemma we prove about <code class="docutils literal notranslate">f : F</code> assuming an instance of <code class="docutils literal notranslate">MonoidHomClass₁ F</code> will - apply both to monoid morphims and ring morphisms. + apply both to monoid morphisms and ring morphisms. Let us see an example lemma and check it applies to both situations. <div class="highlight-lean notranslate"><div class="highlight"><pre>lemma map_inv_of_inv [Monoid M] [Monoid N] [MonoidHomClass₂ F M N] (f : F) {m m' : M} (h : m*m' = 1) : f m * f m' = 1 := by
- @@ -788,7 +788,7 @@ Let us see an example lemma and check it applies to both situations.
 But we haven’t. Everything is shifted one level of abstraction up. The type class resolution procedure won’t be looking for functions, it will be looking for either <code class="docutils literal notranslate">MonoidHom₁</code> or <code class="docutils literal notranslate">RingHom₁</code>. - One remaining issue with our approach is the presence of repeatitive code around the <code class="docutils literal notranslate">toFun</code> + One remaining issue with our approach is the presence of repetitive code around the <code class="docutils literal notranslate">toFun</code> field and the corresponding <code class="docutils literal notranslate">CoeFun</code> instance and <code class="docutils literal notranslate">coe</code> attribute. It would also be better to record that this pattern is used only for function with extra properties, meaning that the coercion to functions should be injective. So Mathlib adds one more layer of abstraction with
- @@ -972,7 +972,6 @@ to you.
 sorry </pre></div> </div> - -/ </section> </section>
html/C08_Topology.html
- @@ -72,6 +72,7 @@
 </ul> </li> <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a></li> + <li class="toctree-l1"><a class="reference internal" href="C10_Integration_and_Measure_Theory.html">10. Integration and Measure Theory</a></li> </ul> <ul> <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li>
- @@ -1001,7 +1002,7 @@ So the order relation we do use on <code class="docutils literal notranslate"><
 Iff.rfl </pre></div> </div> - Now we can recover continuity by combining the push-foward (or pull-back) operation with the order relation. + Now we can recover continuity by combining the push-forward (or pull-back) operation with the order relation. <div class="highlight-lean notranslate"><div class="highlight"><pre>example (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) (f : X → Y) : Continuous f ↔ TopologicalSpace.coinduced f T_X ≤ T_Y := continuous_iff_coinduced_le
- @@ -1028,7 +1029,7 @@ proves the existence of the product topology by abstract non-sense.
 We considered the case of <code class="docutils literal notranslate">ℝ → ℝ</code> above, but let’s now consider the general case of <code class="docutils literal notranslate">Π i, X i</code> for some <code class="docutils literal notranslate">ι : Type*</code> and <code class="docutils literal notranslate">X : ι → Type*</code>. We want, for any topological space <code class="docutils literal notranslate">Z</code> and any function <code class="docutils literal notranslate">f : Z → Π i, X i</code>, that <code class="docutils literal notranslate">f</code> is continuous if and only if <code class="docutils literal notranslate">(fun x ↦ x i) ∘ f</code> is continuous. - Let us explore that constraint “on papar” using notation $p_i$ for the projection + Let us explore that constraint “on paper” using notation $p_i$ for the projection <code class="docutils literal notranslate">(fun (x : Π i, X i) ↦ x i)</code>: <div class="math notranslate nohighlight"> \[\begin{split}(∀ i, p_i ∘ f \text{ continuous}) &⇔ ∀ i, (p_i ∘ f)_* T_Z ≤ T_{X_i} \\
html/C09_Differential_Calculus.html
- @@ -22,7 +22,7 @@
 <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="genindex.html" /> <link rel="search" title="Search" href="search.html" /> - <link rel="next" title="Index" href="genindex.html" /> + <link rel="next" title="10. Integration and Measure Theory" href="C10_Integration_and_Measure_Theory.html" /> <link rel="prev" title="8. Topology" href="C08_Topology.html" /> </head>
- @@ -65,6 +65,7 @@
 </li> </ul> </li> + <li class="toctree-l1"><a class="reference internal" href="C10_Integration_and_Measure_Theory.html">10. Integration and Measure Theory</a></li> </ul> <ul> <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li>
- @@ -447,7 +448,7 @@ see <code class="docutils literal notranslate">HasFDerivWithin
 </div> <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> <a href="C08_Topology.html" class="btn btn-neutral float-left" title="8. Topology" accesskey="p" rel="prev"> Previous</a> - <a href="genindex.html" class="btn btn-neutral float-right" title="Index" accesskey="n" rel="next">Next </a> + <a href="C10_Integration_and_Measure_Theory.html" class="btn btn-neutral float-right" title="10. Integration and Measure Theory" accesskey="n" rel="next">Next </a> </div> <hr/>
html/C10_Integration_and_Measure_Theory.html
- @@ -4,7 +4,7 @@
 <meta charset="utf-8" /><meta name="generator" content="Docutils 0.18.1: http://docutils.sourceforge.net/" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> - <title>Integration and Measure Theory — Mathematics in Lean 0.1 documentation</title> + <title>10. Integration and Measure Theory — Mathematics in Lean 0.1 documentation</title> <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> <link rel="stylesheet" href="_static/css/custom.css" type="text/css" />
- @@ -18,9 +18,12 @@
 <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> <script src="_static/doctools.js"></script> <script src="_static/sphinx_highlight.js"></script> + <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="genindex.html" /> - <link rel="search" title="Search" href="search.html" /> + <link rel="search" title="Search" href="search.html" /> + <link rel="next" title="Index" href="genindex.html" /> + <link rel="prev" title="9. Differential Calculus" href="C09_Differential_Calculus.html" /> </head> <body class="wy-body-for-nav">
- @@ -42,7 +45,7 @@
 </form> </div> </div><div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="Navigation menu"> - <ul> + <ul class="current"> <li class="toctree-l1"><a class="reference internal" href="C01_Introduction.html">1. Introduction</a></li> <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a></li> <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li>
- @@ -52,6 +55,12 @@
 <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li> <li class="toctree-l1"><a class="reference internal" href="C09_Differential_Calculus.html">9. Differential Calculus</a></li> + <li class="toctree-l1 current"><a class="current reference internal" href="#">10. Integration and Measure Theory</a><ul> + <li class="toctree-l2"><a class="reference internal" href="#elementary-integration">10.1. Elementary Integration</a></li> + <li class="toctree-l2"><a class="reference internal" href="#measure-theory">10.2. Measure Theory</a></li> + <li class="toctree-l2"><a class="reference internal" href="#integration">10.3. Integration</a></li> + </ul> + </li> </ul> <ul> <li class="toctree-l1"><a class="reference internal" href="genindex.html">Index</a></li>
- @@ -71,7 +80,7 @@
 <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> <li><a href="index.html" class="icon icon-home" aria-label="Home"></a></li> - <li class="breadcrumb-item active">Integration and Measure Theory</li> + <li class="breadcrumb-item active">10. Integration and Measure Theory</li> <li class="wy-breadcrumbs-aside"> <a href="_sources/C10_Integration_and_Measure_Theory.rst.txt" rel="nofollow"> View page source</a> </li>
- @@ -82,13 +91,207 @@
 <div itemprop="articleBody"> <section id="index-0"> - <h1>Integration and Measure Theory<a class="headerlink" href="#index-0" title="Permalink to this heading"></a></h1> + <h1>10. Integration and Measure Theory<a class="headerlink" href="#index-0" title="Permalink to this heading"></a></h1> + <section id="elementary-integration"> + <h2>10.1. Elementary Integration<a class="headerlink" href="#elementary-integration" title="Permalink to this heading"></a></h2> + We first focus on integration of functions on finite intervals in <code class="docutils literal notranslate">ℝ</code>. We can integrate + elementary functions. + <div class="highlight-lean notranslate"><div class="highlight"><pre>open MeasureTheory intervalIntegral + + open Interval + -- this introduces the notation `[[a, b]]` for the segment from `min a b` to `max a b` + + example (a b : ℝ) : (∫ x in a..b, x) = (b ^ 2 - a ^ 2) / 2 := + integral_id + + example {a b : ℝ} (h : (0 : ℝ) ∉ [[a, b]]) : (∫ x in a..b, 1 / x) = Real.log (b / a) := + integral_one_div h + </pre></div> + </div> + The fundamental theorem of calculus relates integration and differentiation. + Below we give simplified statements of the two parts of this theorem. The first part + says that integration provides an inverse to differentiation and the second one + specifies how to compute integrals of derivatives. + (These two parts are very closely related, but their optimal versions, + which are not shown here, are not equivalent.) + <div class="highlight-lean notranslate"><div class="highlight"><pre>example (f : ℝ → ℝ) (hf : Continuous f) (a b : ℝ) : deriv (fun u => ∫ x : ℝ in a..u, f x) b = f b := + (integral_hasStrictDerivAt_right (hf.intervalIntegrable _ _) (hf.stronglyMeasurableAtFilter _ _) + hf.continuousAt).hasDerivAt.deriv + + example {f : ℝ → ℝ} {a b : ℝ} {f' : ℝ → ℝ} (h : ∀ x ∈ [[a, b]], HasDerivAt f (f' x) x) + (h' : IntervalIntegrable f' volume a b) : (∫ y in a..b, f' y) = f b - f a := + integral_eq_sub_of_hasDerivAt h h' + </pre></div> + </div> + Convolution is also defined in mathlib and its basic properties are proved. + <div class="highlight-lean notranslate"><div class="highlight"><pre>open Convolution + + example (f : ℝ → ℝ) (g : ℝ → ℝ) : f ⋆ g = fun x => ∫ t, f t * g (x - t) := + rfl + </pre></div> + </div> + </section> + <section id="measure-theory"> + <h2>10.2. Measure Theory<a class="headerlink" href="#measure-theory" title="Permalink to this heading"></a></h2> + The general context for integration in mathlib is measure theory. Even the elementary + integrals of the previous section are in fact Bochner integrals. Bochner integration is + a generalization of Lebesgue integration where the target space can be any Banach space, + not necessarily finite dimensional. + The first component in the development of measure theory + is the notion of a $\sigma$-algebra of sets, which are called the + measurable sets. + The type class <code class="docutils literal notranslate">measurable_space</code> serves to equip a type with such a structure. + The sets <code class="docutils literal notranslate">empty</code> and <code class="docutils literal notranslate">univ</code> are measurable, + the complement of a measurable set is measurable, + and a countable union or intersection of measurable sets is measurable. + Note that these axioms are redundant; if you <code class="docutils literal notranslate">#print measurable_space</code>, + you will see the ones that mathlib uses. + As the examples below show, countability assumptions can be expressed using the + <code class="docutils literal notranslate">encodable</code> type class. + <div class="highlight-lean notranslate"><div class="highlight"><pre>variable {α : Type _} [MeasurableSpace α] + + example : MeasurableSet (∅ : Set α) := + MeasurableSet.empty + + example : MeasurableSet (univ : Set α) := + MeasurableSet.univ + + example {s : Set α} (hs : MeasurableSet s) : MeasurableSet (sᶜ) := + hs.compl + + example : Encodable ℕ := by infer_instance + + example (n : ℕ) : Encodable (Fin n) := by infer_instance + + variable {ι : Type _} [Encodable ι] + + example {f : ι → Set α} (h : ∀ b, MeasurableSet (f b)) : MeasurableSet (⋃ b, f b) := + MeasurableSet.iUnion h + + example {f : ι → Set α} (h : ∀ b, MeasurableSet (f b)) : MeasurableSet (⋂ b, f b) := + MeasurableSet.iInter h + </pre></div> + </div> + Once a type is measurable, we can measure it. On paper, a measure on a set + (or type) equipped with a + $\sigma$-algebra is a function from the measurable sets to + the extended non-negative reals that is + additive on countable disjoint unions. + In mathlib, we don’t want to carry around measurability assumptions + every time we write an application of the measure to a set. + So we extend the measure to any set <code class="docutils literal notranslate">s</code> + as the infimum of measures of measurable sets containing <code class="docutils literal notranslate">s</code>. + Of course, many lemmas still require + measurability assumptions, but not all. + <div class="highlight-lean notranslate"><div class="highlight"><pre>open MeasureTheory + variable {μ : Measure α} + + example (s : Set α) : μ s = ⨅ (t : Set α) (_ : s ⊆ t) (_ : MeasurableSet t), μ t := + measure_eq_iInf s + + example (s : ι → Set α) : μ (⋃ i, s i) ≤ ∑' i, μ (s i) := + measure_iUnion_le s + + example {f : ℕ → Set α} (hmeas : ∀ i, MeasurableSet (f i)) (hdis : Pairwise (Disjoint on f)) : + μ (⋃ i, f i) = ∑' i, μ (f i) := + μ.m_iUnion hmeas hdis + </pre></div> + </div> + Once a type has a measure associated with it, we say that a property <code class="docutils literal notranslate">P</code> + holds almost everywhere if the set of elements where the property fails + has measure 0. + The collection of properties that hold almost everywhere form a filter, + but mathlib introduces special notation for saying that a property holds + almost everywhere. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {P : α → Prop} : (∀ᵐ x ∂μ, P x) ↔ ∀ᶠ x in μ.ae, P x := + Iff.rfl + </pre></div> + </div> + </section> + <section id="integration"> + <h2>10.3. Integration<a class="headerlink" href="#integration" title="Permalink to this heading"></a></h2> + Now that we have measurable spaces and measures we can consider integrals. + As explained above, mathlib uses a very general notion of + integration that allows any Banach space as the target. + As usual, we don’t want our notation to + carry around assumptions, so we define integration in such a way + that an integral is equal to zero if the function in question is + not integrable. + Most lemmas having to do with integrals have integrability assumptions. + <div class="highlight-lean notranslate"><div class="highlight"><pre>section + variable {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : α → E} + + example {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) : + (∫ a, f a + g a ∂μ) = (∫ a, f a ∂μ) + ∫ a, g a ∂μ := + integral_add hf hg + </pre></div> + </div> + As an example of the complex interactions between our various conventions, let us see how to integrate constant functions. + Recall that a measure <code class="docutils literal notranslate">μ</code> takes values in <code class="docutils literal notranslate">ℝ≥0∞</code>, the type of extended non-negative reals. + There is a function <code class="docutils literal notranslate">ennreal.to_real : ℝ≥0∞ → ℝ</code> which sends <code class="docutils literal notranslate">⊤</code>, + the point at infinity, to zero. + For any <code class="docutils literal notranslate">s : Set α</code>, if <code class="docutils literal notranslate">μ s = ⊤</code>, then nonzero constant functions are not integrable on <code class="docutils literal notranslate">s</code>. + In that case, their integrals are equal to zero by definition, as is <code class="docutils literal notranslate">(μ s).to_real</code>. + So in all cases we have the following lemma. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {s : Set α} (c : E) : (∫ x in s, c ∂μ) = (μ s).toReal • c := + set_integral_const c + </pre></div> + </div> + We now quickly explain how to access the most important theorems in integration theory, starting + with the dominated convergence theorem. There are several versions in mathlib, + and here we only show the most basic one. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {F : ℕ → α → E} {f : α → E} (bound : α → ℝ) (hmeas : ∀ n, AEStronglyMeasurable (F n) μ) + (hint : Integrable bound μ) (hbound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a) + (hlim : ∀ᵐ a ∂μ, Tendsto (fun n : ℕ => F n a) atTop (𝓝 (f a))) : + Tendsto (fun n => ∫ a, F n a ∂μ) atTop (𝓝 (∫ a, f a ∂μ)) := + tendsto_integral_of_dominated_convergence bound hmeas hint hbound hlim + </pre></div> + </div> + Then we have Fubini’s theorem for integrals on product type. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {α : Type _} [MeasurableSpace α] {μ : Measure α} [SigmaFinite μ] {β : Type _} + [MeasurableSpace β] {ν : Measure β} [SigmaFinite ν] (f : α × β → E) + (hf : Integrable f (μ.prod ν)) : (∫ z, f z ∂ μ.prod ν) = ∫ x, ∫ y, f (x, y) ∂ν ∂μ := + integral_prod f hf + </pre></div> + </div> + There is a very general version of convolution that applies to any + continuous bilinear form. + <div class="highlight-lean notranslate"><div class="highlight"><pre>open Convolution + + variable {𝕜 : Type _} {G : Type _} {E : Type _} {E' : Type _} {F : Type _} [NormedAddCommGroup E] + [NormedAddCommGroup E'] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] + [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] [MeasurableSpace G] [NormedSpace ℝ F] [CompleteSpace F] + [Sub G] + + example (f : G → E) (g : G → E') (L : E →L[𝕜] E' →L[𝕜] F) (μ : Measure G) : + f ⋆[L, μ] g = fun x => ∫ t, L (f t) (g (x - t)) ∂μ := + rfl + </pre></div> + </div> + Finally, mathlib has a very general version of the change-of-variables formula. + In the statement below, <code class="docutils literal notranslate">borel_space E</code> means the + $\sigma$-algebra on <code class="docutils literal notranslate">E</code> is generated by the open sets of <code class="docutils literal notranslate">E</code>, + and <code class="docutils literal notranslate">is_add_haar_measure μ</code> means that the measure <code class="docutils literal notranslate">μ</code> is left-invariant, + gives finite mass to compact sets, and give positive mass to open sets. + <div class="highlight-lean notranslate"><div class="highlight"><pre>example {E : Type _} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] + [MeasurableSpace E] [BorelSpace E] (μ : Measure E) [μ.IsAddHaarMeasure] {F : Type _} + [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] {s : Set E} {f : E → E} + {f' : E → E →L[ℝ] E} (hs : MeasurableSet s) + (hf : ∀ x : E, x ∈ s → HasFDerivWithinAt f (f' x) s x) (h_inj : InjOn f s) (g : E → F) : + (∫ x in f '' s, g x ∂μ) = ∫ x in s, |(f' x).det| • g (f x) ∂μ := + integral_image_eq_integral_abs_det_fderiv_smul μ hs hf h_inj g + </pre></div> + </div> + </section> </section> </div> </div> - <footer> + <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> + <a href="C09_Differential_Calculus.html" class="btn btn-neutral float-left" title="9. 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  Integration and Measure Theory ============================== - .. - .. include:: C09_Integration_and_Measure_Theory/S01_Elementary_Integration.inc - .. include:: C09_Integration_and_Measure_Theory/S02_Measure_Theory.inc - .. include:: C09_Integration_and_Measure_Theory/S03_Integration.inc + + .. include:: C10_Integration_and_Measure_Theory/S01_Elementary_Integration.inc + .. include:: C10_Integration_and_Measure_Theory/S02_Measure_Theory.inc + .. include:: C10_Integration_and_Measure_Theory/S03_Integration.inc
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"Irrational Roots": [[4, "irrational-roots"]], "Induction and Recursion": [[4, "induction-and-recursion"]], "Infinitely Many Primes": [[4, "infinitely-many-primes"]], "Structures": [[5, "structures"]], "Defining structures": [[5, "defining-structures"]], "Algebraic Structures": [[5, "algebraic-structures"]], "Building the Gaussian Integers": [[5, "building-the-gaussian-integers"]], "Hierarchies": [[6, "hierarchies"]], "Morphisms": [[6, "morphisms"]], "Sub-objects": [[6, "sub-objects"]], "Topology": [[7, "index-0"]], "Filters": [[7, "filters"]], "Metric spaces": [[7, "metric-spaces"]], "Convergence and continuity": [[7, "convergence-and-continuity"]], "Balls, open sets and closed sets": [[7, "balls-open-sets-and-closed-sets"]], "Compactness": [[7, "compactness"], [7, "id5"]], "Uniformly continuous functions": [[7, "uniformly-continuous-functions"]], "Completeness": [[7, "completeness"]], "Topological spaces": [[7, "topological-spaces"]], "Fundamentals": [[7, "fundamentals"]], "Separation and countability": [[7, "separation-and-countability"]], "Differential Calculus": [[8, "index-0"]], "Elementary Differential Calculus": [[8, "elementary-differential-calculus"]], "Differential Calculus in Normed Spaces": [[8, "differential-calculus-in-normed-spaces"]], "Normed spaces": [[8, "id3"]], "Continuous linear maps": [[8, "continuous-linear-maps"]], "Asymptotic comparisons": [[8, "asymptotic-comparisons"]], "Differentiability": [[8, "differentiability"]], "Integration and Measure Theory": [[9, "index-0"]], "Elementary Integration": [[9, "elementary-integration"]], "Measure Theory": [[9, "measure-theory"]], "Integration": [[9, "integration"]], "Index": [[10, "index"]], "Mathematics in Lean": [[11, "mathematics-in-lean"]]}, "indexentries": {"check": [[0, "index-0"]], "commands": [[0, "index-0"]], "abel": [[1, "index-14"]], "absolute value": [[1, "index-23"]], "apply": [[1, "index-16"]], "calc": [[1, "index-3"]], "command": [[1, "index-9"]], "commutative ring": [[1, 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lake-manifest.json
- @@ -2,9 +2,15 @@
  "packagesDir": "lake-packages", "packages": [{"git": + {"url": "https://github.com/EdAyers/ProofWidgets4", + "subDir?": null, + "rev": "c43db94a8f495dad37829e9d7ad65483d68c86b8", + "name": "proofwidgets", + "inputRev?": "v0.0.11"}}, + {"git": {"url": "https://github.com/leanprover-community/mathlib4", "subDir?": null, - "rev": "2db7650f98b1ea3543e48b6b02aa9112e33bb68b", + "rev": "8a76b3e5aa06eb8babdf58b8e16afa708260fcf2", "name": "mathlib", "inputRev?": "master"}}, {"git":
- @@ -22,6 +28,6 @@
  {"git": {"url": "https://github.com/leanprover/std4", "subDir?": null, - "rev": "6932c4ea52914dc6b0488944e367459ddc4d01a6", + "rev": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "name": "std", "inputRev?": "main"}}]}
lean-toolchain
- @@ -1,1 +1,1 @@
  - leanprover/lean4:nightly-2023-05-31 + leanprover/lean4:nightly-2023-06-20
mathematics_in_lean.pdf