awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Prokhorov (goal)
Goal:
E : Type u_1
inst : MeasurableSpace E
inst_1 : TopologicalSpace E
inst_2 : T2Space E
inst_3 : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K : (a : ℕ) → Set E) n)
A : ∀ (n : ℕ), IsCompact.{u_1} (X := E) ((partialSups K : (a : ℕ) → Set E) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ : FiniteMeasure E | μ.mass ≤ C ∧ ∀ (n : ℕ), (μ : (a : Set E) → ℝ≥0) (K n)ᶜ ≤ u n} ∈ f
ν : (n : ℕ) → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n) : Measure E).InnerRegular
νK : ∀ (n : ℕ), Eq (α := ℝ≥0) ((ν n : (a : Set E) → ℝ≥0) ((partialSups K : (a : ℕ) → Set E) n)ᶜ : ℝ≥0) (0 : ℝ≥0)
hν :
  ∀ (n : ℕ), Tendsto (fun (ρ : FiniteMeasure E) ↦ ρ.restrict (disjointed K n)) (↑f : Filter (FiniteMeasure E)) (𝓝 (ν n))
B : ((Measure.sum fun (n : ℕ) ↦ (↑(ν n) : Measure E) : Measure E) : (a : Set E) → ℝ≥0∞) univ ≤ (↑C : ℝ≥0∞)
μ : FiniteMeasure E :=
  Subtype.mk (p := fun (μ : Measure E) ↦ IsFiniteMeasure μ) (Measure.sum fun (n : ℕ) ↦ (↑(ν n) : Measure E)) ⋯
g : E →ᵇ ℝ
ε : ℝ
εpos : ε > 0
I1 :
  ∀ᶠ (n : ℕ) in atTop,
    dist (α := ℝ)
        (@integral _ _ _ _ inst (∑ i ∈ Finset.range (n + 1), (↑(ν i) : Measure E)) fun (x : E) ↦ (g : (a : E) → ℝ) x)
        (∫ (x : E), (g : (a : E) → ℝ) x ∂(↑μ : Measure E)) <
      ε / 3
I2 : ∀ᶠ (n : ℕ) in atTop, ‖g‖ * (↑(u n) : ℝ) < ε / 3
n : ℕ
hn :
  dist (α := ℝ)
      (@integral _ _ _ _ inst (∑ i ∈ Finset.range (n + 1), (↑(ν i) : Measure E)) fun (x : E) ↦ (g : (a : E) → ℝ) x)
      (∫ (x : E), (g : (a : E) → ℝ) x ∂(↑μ : Measure E)) <
    ε / 3
h'n : ‖g‖ * (↑(u n) : ℝ) < ε / 3
this_1 :
  Tendsto (β := ℝ)
    (fun (ρ : FiniteMeasure E) ↦
      @integral _ _ _ _ inst (↑(∑ i ∈ Finset.range (n + 1), ρ.restrict (disjointed K i)) : Measure E) fun (x : E) ↦
        (g : (a : E) → ℝ) x)
    (↑f : Filter (FiniteMeasure E))
    (𝓝 (@integral _ _ _ _ inst (↑(∑ i ∈ Finset.range (n + 1), ν i) : Measure E) fun (x : E) ↦ (g : (a : E) → ℝ) x))
ρ : FiniteMeasure E
hρ :
  dist (α := ℝ)
      (@integral _ _ _ _ inst (↑(∑ i ∈ Finset.range (n + 1), ρ.restrict (disjointed K i)) : Measure E) fun (x : E) ↦
        (g : (a : E) → ℝ) x)
      (@integral _ _ _ _ inst (↑(∑ i ∈ Finset.range (n + 1), ν i) : Measure E) fun (x : E) ↦ (g : (a : E) → ℝ) x) <
    ε / 3
h'ρ : ρ.mass ≤ C ∧ ∀ (n : ℕ), (ρ : (a : Set E) → ℝ≥0) (K n)ᶜ ≤ u n
this : ρ = ρ.restrict ((partialSups K : (a : ℕ) → Set E) n)ᶜ + ∑ i ∈ Finset.range (n + 1), ρ.restrict (disjointed K i)
⊢ (ae (↑(ρ.restrict ((partialSups K : (a : ℕ) → Set E) n)ᶜ) : Measure E)).EventuallyLE (β := ℝ)
    (fun (a : E) ↦ norm (E := ℝ) ((g : (a : E) → ℝ) a : ℝ)) fun (a : E) ↦ ‖g‖
have h₁ : ∀ (a : E),
  Membership.mem (γ := Set E)
    {x : E |
      (fun (x : E) ↦ LE.le (α := ℝ) ((fun (a : E) ↦ norm (E := ℝ) ((g : (a : E) → ℝ) a : ℝ)) x) ((fun (a : E) ↦ ‖g‖) x))
        x}
    a
Tactic: filter_upwards with x using norm_coe_le_norm g x
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Prokhorov (goal)
Goal:
E : Type u_1
inst : MeasurableSpace E
inst_1 : TopologicalSpace E
inst_2 : T2Space E
inst_3 : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K : (a : ℕ) → Set E) n)
A : ∀ (n : ℕ), IsCompact.{u_1} (X := E) ((partialSups K : (a : ℕ) → Set E) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ : FiniteMeasure E | μ.mass ≤ C ∧ ∀ (n : ℕ), (μ : (a : Set E) → ℝ≥0) (K n)ᶜ ≤ u n} ∈ f
ν : (n : ℕ) → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n) : Measure E).InnerRegular
νK : ∀ (n : ℕ), Eq (α := ℝ≥0) ((ν n : (a : Set E) → ℝ≥0) ((partialSups K : (a : ℕ) → Set E) n)ᶜ : ℝ≥0) (0 : ℝ≥0)
hν :
  ∀ (n : ℕ), Tendsto (fun (ρ : FiniteMeasure E) ↦ ρ.restrict (disjointed K n)) (↑f : Filter (FiniteMeasure E)) (𝓝 (ν n))
B : ((Measure.sum fun (n : ℕ) ↦ (↑(ν n) : Measure E) : Measure E) : (a : Set E) → ℝ≥0∞) univ ≤ (↑C : ℝ≥0∞)
μ : FiniteMeasure E :=
  Subtype.mk (p := fun (μ : Measure E) ↦ IsFiniteMeasure μ) (Measure.sum fun (n : ℕ) ↦ (↑(ν n) : Measure E)) ⋯
g : E →ᵇ ℝ
ε : ℝ
εpos : ε > 0
I1 :
  ∀ᶠ (n : ℕ) in atTop,
    dist (α := ℝ)
        (@integral _ _ _ _ inst (∑ i ∈ Finset.range (n + 1), (↑(ν i) : Measure E)) fun (x : E) ↦ (g : (a : E) → ℝ) x)
        (∫ (x : E), (g : (a : E) → ℝ) x ∂(↑μ : Measure E)) <
      ε / 3
I2 : ∀ᶠ (n : ℕ) in atTop, ‖g‖ * (↑(u n) : ℝ) < ε / 3
n : ℕ
hn :
  dist (α := ℝ)
      (@integral _ _ _ _ inst (∑ i ∈ Finset.range (n + 1), (↑(ν i) : Measure E)) fun (x : E) ↦ (g : (a : E) → ℝ) x)
      (∫ (x : E), (g : (a : E) → ℝ) x ∂(↑μ : Measure E)) <
    ε / 3
h'n : ‖g‖ * (↑(u n) : ℝ) < ε / 3
this_1 :
  Tendsto (β := ℝ)
    (fun (ρ : FiniteMeasure E) ↦
      @integral _ _ _ _ inst (↑(∑ i ∈ Finset.range (n + 1), ρ.restrict (disjointed K i)) : Measure E) fun (x : E) ↦
        (g : (a : E) → ℝ) x)
    (↑f : Filter (FiniteMeasure E))
    (𝓝 (@integral _ _ _ _ inst (↑(∑ i ∈ Finset.range (n + 1), ν i) : Measure E) fun (x : E) ↦ (g : (a : E) → ℝ) x))
ρ : FiniteMeasure E
hρ :
  dist (α := ℝ)
      (@integral _ _ _ _ inst (↑(∑ i ∈ Finset.range (n + 1), ρ.restrict (disjointed K i)) : Measure E) fun (x : E) ↦
        (g : (a : E) → ℝ) x)
      (@integral _ _ _ _ inst (↑(∑ i ∈ Finset.range (n + 1), ν i) : Measure E) fun (x : E) ↦ (g : (a : E) → ℝ) x) <
    ε / 3
h'ρ : ρ.mass ≤ C ∧ ∀ (n : ℕ), (ρ : (a : Set E) → ℝ≥0) (K n)ᶜ ≤ u n
this : ρ = ρ.restrict ((partialSups K : (a : ℕ) → Set E) n)ᶜ + ∑ i ∈ Finset.range (n + 1), ρ.restrict (disjointed K i)
h₁ :
  ∀ (a : E),
    Membership.mem (γ := Set E)
      {x : E |
        (fun (x : E) ↦
            LE.le (α := ℝ) ((fun (a : E) ↦ norm (E := ℝ) ((g : (a : E) → ℝ) a : ℝ)) x) ((fun (a : E) ↦ ‖g‖) x))
          x}
      a
⊢ (ae (↑(ρ.restrict ((partialSups K : (a : ℕ) → Set E) n)ᶜ) : Measure E)).EventuallyLE (β := ℝ)
    (fun (a : E) ↦ norm (E := ℝ) ((g : (a : E) → ℝ) a : ℝ)) fun (a : E) ↦ ‖g‖
have h₂ : ∀ (a : E), norm (E := ℝ) ((g : (a : E) → ℝ) a : ℝ) ≤ ‖g‖
Tactic: filter_upwards with x using norm_coe_le_norm g x
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Prokhorov (goal)
Goal:
E : Type u_1
inst : MeasurableSpace E
inst_1 : TopologicalSpace E
inst_2 : T2Space E
inst_3 : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K : (a : ℕ) → Set E) n)
A : ∀ (n : ℕ), IsCompact.{u_1} (X := E) ((partialSups K : (a : ℕ) → Set E) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ : FiniteMeasure E | μ.mass ≤ C ∧ ∀ (n : ℕ), (μ : (a : Set E) → ℝ≥0) (K n)ᶜ ≤ u n} ∈ f
ν : (n : ℕ) → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n) : Measure E).InnerRegular
νK : ∀ (n : ℕ), Eq (α := ℝ≥0) ((ν n : (a : Set E) → ℝ≥0) ((partialSups K : (a : ℕ) → Set E) n)ᶜ : ℝ≥0) (0 : ℝ≥0)
hν :
  ∀ (n : ℕ), Tendsto (fun (ρ : FiniteMeasure E) ↦ ρ.restrict (disjointed K n)) (↑f : Filter (FiniteMeasure E)) (𝓝 (ν n))
B : ((Measure.sum fun (n : ℕ) ↦ (↑(ν n) : Measure E) : Measure E) : (a : Set E) → ℝ≥0∞) univ ≤ (↑C : ℝ≥0∞)
μ : FiniteMeasure E :=
  Subtype.mk (p := fun (μ : Measure E) ↦ IsFiniteMeasure μ) (Measure.sum fun (n : ℕ) ↦ (↑(ν n) : Measure E)) ⋯
g : E →ᵇ ℝ
ε : ℝ
εpos : ε > 0
I1 :
  ∀ᶠ (n : ℕ) in atTop,
    dist (α := ℝ)
        (@integral _ _ _ _ inst (∑ i ∈ Finset.range (n + 1), (↑(ν i) : Measure E)) fun (x : E) ↦ (g : (a : E) → ℝ) x)
        (∫ (x : E), (g : (a : E) → ℝ) x ∂(↑μ : Measure E)) <
      ε / 3
I2 : ∀ᶠ (n : ℕ) in atTop, ‖g‖ * (↑(u n) : ℝ) < ε / 3
n : ℕ
hn :
  dist (α := ℝ)
      (@integral _ _ _ _ inst (∑ i ∈ Finset.range (n + 1), (↑(ν i) : Measure E)) fun (x : E) ↦ (g : (a : E) → ℝ) x)
      (∫ (x : E), (g : (a : E) → ℝ) x ∂(↑μ : Measure E)) <
    ε / 3
h'n : ‖g‖ * (↑(u n) : ℝ) < ε / 3
this_1 :
  Tendsto (β := ℝ)
    (fun (ρ : FiniteMeasure E) ↦
      @integral _ _ _ _ inst (↑(∑ i ∈ Finset.range (n + 1), ρ.restrict (disjointed K i)) : Measure E) fun (x : E) ↦
        (g : (a : E) → ℝ) x)
    (↑f : Filter (FiniteMeasure E))
    (𝓝 (@integral _ _ _ _ inst (↑(∑ i ∈ Finset.range (n + 1), ν i) : Measure E) fun (x : E) ↦ (g : (a : E) → ℝ) x))
ρ : FiniteMeasure E
hρ :
  dist (α := ℝ)
      (@integral _ _ _ _ inst (↑(∑ i ∈ Finset.range (n + 1), ρ.restrict (disjointed K i)) : Measure E) fun (x : E) ↦
        (g : (a : E) → ℝ) x)
      (@integral _ _ _ _ inst (↑(∑ i ∈ Finset.range (n + 1), ν i) : Measure E) fun (x : E) ↦ (g : (a : E) → ℝ) x) <
    ε / 3
h'ρ : ρ.mass ≤ C ∧ ∀ (n : ℕ), (ρ : (a : Set E) → ℝ≥0) (K n)ᶜ ≤ u n
this : ρ = ρ.restrict ((partialSups K : (a : ℕ) → Set E) n)ᶜ + ∑ i ∈ Finset.range (n + 1), ρ.restrict (disjointed K i)
⊢ ‖g‖ * (↑((ρ : (a : Set E) → ℝ≥0) ((partialSups K : (a : ℕ) → Set E) n)ᶜ) : ℝ) ≤
    ‖g‖ * (↑((ρ : (a : Set E) → ℝ≥0) (K n)ᶜ) : ℝ)
have hbc : K n ⊆ (partialSups K : (a : ℕ) → Set E) n
Tactic: gcongr
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Prokhorov (goal)
Goal:
E : Type u_1
inst : MeasurableSpace E
inst_1 : TopologicalSpace E
inst_2 : T2Space E
inst_3 : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K : (a : ℕ) → Set E) n)
A : ∀ (n : ℕ), IsCompact.{u_1} (X := E) ((partialSups K : (a : ℕ) → Set E) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ : FiniteMeasure E | μ.mass ≤ C ∧ ∀ (n : ℕ), (μ : (a : Set E) → ℝ≥0) (K n)ᶜ ≤ u n} ∈ f
ν : (n : ℕ) → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n) : Measure E).InnerRegular
νK : ∀ (n : ℕ), Eq (α := ℝ≥0) ((ν n : (a : Set E) → ℝ≥0) ((partialSups K : (a : ℕ) → Set E) n)ᶜ : ℝ≥0) (0 : ℝ≥0)
hν :
  ∀ (n : ℕ), Tendsto (fun (ρ : FiniteMeasure E) ↦ ρ.restrict (disjointed K n)) (↑f : Filter (FiniteMeasure E)) (𝓝 (ν n))
B : ((Measure.sum fun (n : ℕ) ↦ (↑(ν n) : Measure E) : Measure E) : (a : Set E) → ℝ≥0∞) univ ≤ (↑C : ℝ≥0∞)
μ : FiniteMeasure E :=
  Subtype.mk (p := fun (μ : Measure E) ↦ IsFiniteMeasure μ) (Measure.sum fun (n : ℕ) ↦ (↑(ν n) : Measure E)) ⋯
g : E →ᵇ ℝ
ε : ℝ
εpos : ε > 0
I1 :
  ∀ᶠ (n : ℕ) in atTop,
    dist (α := ℝ)
        (@integral _ _ _ _ inst (∑ i ∈ Finset.range (n + 1), (↑(ν i) : Measure E)) fun (x : E) ↦ (g : (a : E) → ℝ) x)
        (∫ (x : E), (g : (a : E) → ℝ) x ∂(↑μ : Measure E)) <
      ε / 3
I2 : ∀ᶠ (n : ℕ) in atTop, ‖g‖ * (↑(u n) : ℝ) < ε / 3
n : ℕ
hn :
  dist (α := ℝ)
      (@integral _ _ _ _ inst (∑ i ∈ Finset.range (n + 1), (↑(ν i) : Measure E)) fun (x : E) ↦ (g : (a : E) → ℝ) x)
      (∫ (x : E), (g : (a : E) → ℝ) x ∂(↑μ : Measure E)) <
    ε / 3
h'n : ‖g‖ * (↑(u n) : ℝ) < ε / 3
this_1 :
  Tendsto (β := ℝ)
    (fun (ρ : FiniteMeasure E) ↦
      @integral _ _ _ _ inst (↑(∑ i ∈ Finset.range (n + 1), ρ.restrict (disjointed K i)) : Measure E) fun (x : E) ↦
        (g : (a : E) → ℝ) x)
    (↑f : Filter (FiniteMeasure E))
    (𝓝 (@integral _ _ _ _ inst (↑(∑ i ∈ Finset.range (n + 1), ν i) : Measure E) fun (x : E) ↦ (g : (a : E) → ℝ) x))
ρ : FiniteMeasure E
hρ :
  dist (α := ℝ)
      (@integral _ _ _ _ inst (↑(∑ i ∈ Finset.range (n + 1), ρ.restrict (disjointed K i)) : Measure E) fun (x : E) ↦
        (g : (a : E) → ℝ) x)
      (@integral _ _ _ _ inst (↑(∑ i ∈ Finset.range (n + 1), ν i) : Measure E) fun (x : E) ↦ (g : (a : E) → ℝ) x) <
    ε / 3
h'ρ : ρ.mass ≤ C ∧ ∀ (n : ℕ), (ρ : (a : Set E) → ℝ≥0) (K n)ᶜ ≤ u n
this : ρ = ρ.restrict ((partialSups K : (a : ℕ) → Set E) n)ᶜ + ∑ i ∈ Finset.range (n + 1), ρ.restrict (disjointed K i)
⊢ ‖g‖ * (↑((ρ : (a : Set E) → ℝ≥0) (K n)ᶜ) : ℝ) ≤ ‖g‖ * (↑(u n) : ℝ)
have hbc : (ρ : (a : Set E) → ℝ≥0) (K n)ᶜ ≤ u n
Tactic: gcongr
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Prokhorov (goal)
Goal:
E : Type u_1
inst : MeasurableSpace E
inst_1 : TopologicalSpace E
inst_2 : T2Space E
inst_3 : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K : (a : ℕ) → Set E) n)
A : ∀ (n : ℕ), IsCompact.{u_1} (X := E) ((partialSups K : (a : ℕ) → Set E) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ : FiniteMeasure E | μ.mass ≤ C ∧ ∀ (n : ℕ), (μ : (a : Set E) → ℝ≥0) (K n)ᶜ ≤ u n} ∈ f
ν : (n : ℕ) → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n) : Measure E).InnerRegular
νK : ∀ (n : ℕ), Eq (α := ℝ≥0) ((ν n : (a : Set E) → ℝ≥0) ((partialSups K : (a : ℕ) → Set E) n)ᶜ : ℝ≥0) (0 : ℝ≥0)
hν :
  ∀ (n : ℕ), Tendsto (fun (ρ : FiniteMeasure E) ↦ ρ.restrict (disjointed K n)) (↑f : Filter (FiniteMeasure E)) (𝓝 (ν n))
B : ((Measure.sum fun (n : ℕ) ↦ (↑(ν n) : Measure E) : Measure E) : (a : Set E) → ℝ≥0∞) univ ≤ (↑C : ℝ≥0∞)
μ : FiniteMeasure E :=
  Subtype.mk (p := fun (μ : Measure E) ↦ IsFiniteMeasure μ) (Measure.sum fun (n : ℕ) ↦ (↑(ν n) : Measure E)) ⋯
L : Tendsto id (↑f : Filter (FiniteMeasure E)) (𝓝 μ)
⊢ ∃ (x : FiniteMeasure E),
    Membership.mem (γ := Set (FiniteMeasure E))
        {μ : FiniteMeasure E | μ.mass ≤ C ∧ ∀ (n : ℕ), (μ : (a : Set E) → ℝ≥0) (K n)ᶜ ≤ u n} x ∧
      (↑f : Filter (FiniteMeasure E)) ≤ 𝓝 x
have refine_1 : μ.mass ≤ C
Tactic: refine ⟨μ, ⟨?_, fun n ↦ ?_⟩, L⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Prokhorov (goal)
Goal:
E : Type u_1
inst : MeasurableSpace E
inst_1 : TopologicalSpace E
inst_2 : T2Space E
inst_3 : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K : (a : ℕ) → Set E) n)
A : ∀ (n : ℕ), IsCompact.{u_1} (X := E) ((partialSups K : (a : ℕ) → Set E) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ : FiniteMeasure E | μ.mass ≤ C ∧ ∀ (n : ℕ), (μ : (a : Set E) → ℝ≥0) (K n)ᶜ ≤ u n} ∈ f
ν : (n : ℕ) → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n) : Measure E).InnerRegular
νK : ∀ (n : ℕ), Eq (α := ℝ≥0) ((ν n : (a : Set E) → ℝ≥0) ((partialSups K : (a : ℕ) → Set E) n)ᶜ : ℝ≥0) (0 : ℝ≥0)
hν :
  ∀ (n : ℕ), Tendsto (fun (ρ : FiniteMeasure E) ↦ ρ.restrict (disjointed K n)) (↑f : Filter (FiniteMeasure E)) (𝓝 (ν n))
B : ((Measure.sum fun (n : ℕ) ↦ (↑(ν n) : Measure E) : Measure E) : (a : Set E) → ℝ≥0∞) univ ≤ (↑C : ℝ≥0∞)
μ : FiniteMeasure E :=
  Subtype.mk (p := fun (μ : Measure E) ↦ IsFiniteMeasure μ) (Measure.sum fun (n : ℕ) ↦ (↑(ν n) : Measure E)) ⋯
L : Tendsto id (↑f : Filter (FiniteMeasure E)) (𝓝 μ)
refine_1 : μ.mass ≤ C
⊢ ∃ (x : FiniteMeasure E),
    Membership.mem (γ := Set (FiniteMeasure E))
        {μ : FiniteMeasure E | μ.mass ≤ C ∧ ∀ (n : ℕ), (μ : (a : Set E) → ℝ≥0) (K n)ᶜ ≤ u n} x ∧
      (↑f : Filter (FiniteMeasure E)) ≤ 𝓝 x
have refine_2 : ∀ (n : ℕ), (μ : (a : Set E) → ℝ≥0) (K n)ᶜ ≤ u n
Tactic: refine ⟨μ, ⟨?_, fun n ↦ ?_⟩, L⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Prokhorov (goal)
Goal:
E : Type u_1
inst : MeasurableSpace E
inst_1 : TopologicalSpace E
inst_2 : T2Space E
inst_3 : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K : (a : ℕ) → Set E) n)
A : ∀ (n : ℕ), IsCompact.{u_1} (X := E) ((partialSups K : (a : ℕ) → Set E) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ : FiniteMeasure E | μ.mass ≤ C ∧ ∀ (n : ℕ), (μ : (a : Set E) → ℝ≥0) (K n)ᶜ ≤ u n} ∈ f
ν : (n : ℕ) → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n) : Measure E).InnerRegular
νK : ∀ (n : ℕ), Eq (α := ℝ≥0) ((ν n : (a : Set E) → ℝ≥0) ((partialSups K : (a : ℕ) → Set E) n)ᶜ : ℝ≥0) (0 : ℝ≥0)
hν :
  ∀ (n : ℕ), Tendsto (fun (ρ : FiniteMeasure E) ↦ ρ.restrict (disjointed K n)) (↑f : Filter (FiniteMeasure E)) (𝓝 (ν n))
B : ((Measure.sum fun (n : ℕ) ↦ (↑(ν n) : Measure E) : Measure E) : (a : Set E) → ℝ≥0∞) univ ≤ (↑C : ℝ≥0∞)
μ : FiniteMeasure E :=
  Subtype.mk (p := fun (μ : Measure E) ↦ IsFiniteMeasure μ) (Measure.sum fun (n : ℕ) ↦ (↑(ν n) : Measure E)) ⋯
L : Tendsto id (↑f : Filter (FiniteMeasure E)) (𝓝 μ)
⊢ μ.mass ≤ C
have refine_1 : ENNReal.toNNReal (((Measure.sum fun (n : ℕ) ↦ (↑(ν n) : Measure E) : Measure E) : (a : Set E) → ℝ≥0∞) univ) ≤ C
Tactic: simp only [mass, mk_apply, μ]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Prokhorov (goal)
Goal:
E : Type u_1
inst : MeasurableSpace E
inst_1 : TopologicalSpace E
inst_2 : T2Space E
inst_3 : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K : (a : ℕ) → Set E) n)
A : ∀ (n : ℕ), IsCompact.{u_1} (X := E) ((partialSups K : (a : ℕ) → Set E) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ : FiniteMeasure E | μ.mass ≤ C ∧ ∀ (n : ℕ), (μ : (a : Set E) → ℝ≥0) (K n)ᶜ ≤ u n} ∈ f
ν : (n : ℕ) → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n) : Measure E).InnerRegular
νK : ∀ (n : ℕ), Eq (α := ℝ≥0) ((ν n : (a : Set E) → ℝ≥0) ((partialSups K : (a : ℕ) → Set E) n)ᶜ : ℝ≥0) (0 : ℝ≥0)
hν :
  ∀ (n : ℕ), Tendsto (fun (ρ : FiniteMeasure E) ↦ ρ.restrict (disjointed K n)) (↑f : Filter (FiniteMeasure E)) (𝓝 (ν n))
B : ((Measure.sum fun (n : ℕ) ↦ (↑(ν n) : Measure E) : Measure E) : (a : Set E) → ℝ≥0∞) univ ≤ (↑C : ℝ≥0∞)
μ : FiniteMeasure E :=
  Subtype.mk (p := fun (μ : Measure E) ↦ IsFiniteMeasure μ) (Measure.sum fun (n : ℕ) ↦ (↑(ν n) : Measure E)) ⋯
L : Tendsto id (↑f : Filter (FiniteMeasure E)) (𝓝 μ)
⊢ ENNReal.toNNReal (((Measure.sum fun (n : ℕ) ↦ (↑(ν n) : Measure E) : Measure E) : (a : Set E) → ℝ≥0∞) univ) ≤ C
have refine_1 : ENNReal.toNNReal (((Measure.sum fun (n : ℕ) ↦ (↑(ν n) : Measure E) : Measure E) : (a : Set E) → ℝ≥0∞) univ) ≤
  (↑C : ℝ≥0∞).toNNReal
Tactic: rw [show C = (C : ℝ≥0∞).toNNReal by simp]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Prokhorov (goal)
Goal:
E : Type u_1
inst : MeasurableSpace E
inst_1 : TopologicalSpace E
inst_2 : T2Space E
inst_3 : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K : (a : ℕ) → Set E) n)
A : ∀ (n : ℕ), IsCompact.{u_1} (X := E) ((partialSups K : (a : ℕ) → Set E) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ : FiniteMeasure E | μ.mass ≤ C ∧ ∀ (n : ℕ), (μ : (a : Set E) → ℝ≥0) (K n)ᶜ ≤ u n} ∈ f
ν : (n : ℕ) → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n) : Measure E).InnerRegular
νK : ∀ (n : ℕ), Eq (α := ℝ≥0) ((ν n : (a : Set E) → ℝ≥0) ((partialSups K : (a : ℕ) → Set E) n)ᶜ : ℝ≥0) (0 : ℝ≥0)
hν :
  ∀ (n : ℕ), Tendsto (fun (ρ : FiniteMeasure E) ↦ ρ.restrict (disjointed K n)) (↑f : Filter (FiniteMeasure E)) (𝓝 (ν n))
B : ((Measure.sum fun (n : ℕ) ↦ (↑(ν n) : Measure E) : Measure E) : (a : Set E) → ℝ≥0∞) univ ≤ (↑C : ℝ≥0∞)
μ : FiniteMeasure E :=
  Subtype.mk (p := fun (μ : Measure E) ↦ IsFiniteMeasure μ) (Measure.sum fun (n : ℕ) ↦ (↑(ν n) : Measure E)) ⋯
L : Tendsto id (↑f : Filter (FiniteMeasure E)) (𝓝 μ)
n : ℕ
⊢ (μ : (a : Set E) → ℝ≥0) (K n)ᶜ ≤ u n
have refine_2 : ∀ (h : NormalSpace E), (μ : (a : Set E) → ℝ≥0) (K n)ᶜ ≤ u n
Tactic: rcases h with h | h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Prokhorov (goal)
Goal:
E : Type u_1
inst : MeasurableSpace E
inst_1 : TopologicalSpace E
inst_2 : T2Space E
inst_3 : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K : (a : ℕ) → Set E) n)
A : ∀ (n : ℕ), IsCompact.{u_1} (X := E) ((partialSups K : (a : ℕ) → Set E) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ : FiniteMeasure E | μ.mass ≤ C ∧ ∀ (n : ℕ), (μ : (a : Set E) → ℝ≥0) (K n)ᶜ ≤ u n} ∈ f
ν : (n : ℕ) → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n) : Measure E).InnerRegular
νK : ∀ (n : ℕ), Eq (α := ℝ≥0) ((ν n : (a : Set E) → ℝ≥0) ((partialSups K : (a : ℕ) → Set E) n)ᶜ : ℝ≥0) (0 : ℝ≥0)
hν :
  ∀ (n : ℕ), Tendsto (fun (ρ : FiniteMeasure E) ↦ ρ.restrict (disjointed K n)) (↑f : Filter (FiniteMeasure E)) (𝓝 (ν n))
B : ((Measure.sum fun (n : ℕ) ↦ (↑(ν n) : Measure E) : Measure E) : (a : Set E) → ℝ≥0∞) univ ≤ (↑C : ℝ≥0∞)
μ : FiniteMeasure E :=
  Subtype.mk (p := fun (μ : Measure E) ↦ IsFiniteMeasure μ) (Measure.sum fun (n : ℕ) ↦ (↑(ν n) : Measure E)) ⋯
L : Tendsto id (↑f : Filter (FiniteMeasure E)) (𝓝 μ)
n : ℕ
refine_2 : ∀ (h : NormalSpace E), (μ : (a : Set E) → ℝ≥0) (K n)ᶜ ≤ u n
⊢ (μ : (a : Set E) → ℝ≥0) (K n)ᶜ ≤ u n
have refine_2₁ : ∀ (h : Monotone K), (μ : (a : Set E) → ℝ≥0) (K n)ᶜ ≤ u n
Tactic: rcases h with h | h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Prokhorov (goal)
Goal:
E : Type u_1
inst : MeasurableSpace E
inst_1 : TopologicalSpace E
inst_2 : T2Space E
inst_3 : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K : (a : ℕ) → Set E) n)
A : ∀ (n : ℕ), IsCompact.{u_1} (X := E) ((partialSups K : (a : ℕ) → Set E) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ : FiniteMeasure E | μ.mass ≤ C ∧ ∀ (n : ℕ), (μ : (a : Set E) → ℝ≥0) (K n)ᶜ ≤ u n} ∈ f
ν : (n : ℕ) → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n) : Measure E).InnerRegular
νK : ∀ (n : ℕ), Eq (α := ℝ≥0) ((ν n : (a : Set E) → ℝ≥0) ((partialSups K : (a : ℕ) → Set E) n)ᶜ : ℝ≥0) (0 : ℝ≥0)
hν :
  ∀ (n : ℕ), Tendsto (fun (ρ : FiniteMeasure E) ↦ ρ.restrict (disjointed K n)) (↑f : Filter (FiniteMeasure E)) (𝓝 (ν n))
B : ((Measure.sum fun (n : ℕ) ↦ (↑(ν n) : Measure E) : Measure E) : (a : Set E) → ℝ≥0∞) univ ≤ (↑C : ℝ≥0∞)
μ : FiniteMeasure E :=
  Subtype.mk (p := fun (μ : Measure E) ↦ IsFiniteMeasure μ) (Measure.sum fun (n : ℕ) ↦ (↑(ν n) : Measure E)) ⋯
L : Tendsto id (↑f : Filter (FiniteMeasure E)) (𝓝 μ)
n : ℕ
h : NormalSpace E
⊢ (↑μ : Measure E).InnerRegular
have simp : Measure.InnerRegular.{u_1} (α := E) (Measure.sum fun (n : ℕ) ↦ (↑(ν n) : Measure E))
Tactic: simp only [toMeasure_mk, μ]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Prokhorov (hypothesis)
Goal:
E : Type u_1
inst : MeasurableSpace E
inst_1 : TopologicalSpace E
inst_2 : T2Space E
inst_3 : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K : (a : ℕ) → Set E) n)
A : ∀ (n : ℕ), IsCompact.{u_1} (X := E) ((partialSups K : (a : ℕ) → Set E) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ : FiniteMeasure E | μ.mass ≤ C ∧ ∀ (n : ℕ), (μ : (a : Set E) → ℝ≥0) (K n)ᶜ ≤ u n} ∈ f
ν : (n : ℕ) → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n) : Measure E).InnerRegular
νK : ∀ (n : ℕ), Eq (α := ℝ≥0) ((ν n : (a : Set E) → ℝ≥0) ((partialSups K : (a : ℕ) → Set E) n)ᶜ : ℝ≥0) (0 : ℝ≥0)
hν :
  ∀ (n : ℕ), Tendsto (fun (ρ : FiniteMeasure E) ↦ ρ.restrict (disjointed K n)) (↑f : Filter (FiniteMeasure E)) (𝓝 (ν n))
B : ((Measure.sum fun (n : ℕ) ↦ (↑(ν n) : Measure E) : Measure E) : (a : Set E) → ℝ≥0∞) univ ≤ (↑C : ℝ≥0∞)
μ : FiniteMeasure E :=
  Subtype.mk (p := fun (μ : Measure E) ↦ IsFiniteMeasure μ) (Measure.sum fun (n : ℕ) ↦ (↑(ν n) : Measure E)) ⋯
L : Tendsto id (↑f : Filter (FiniteMeasure E)) (𝓝 μ)
n : ℕ
h : NormalSpace E
⊢ (μ : (a : Set E) → ℝ≥0) (K n)ᶜ ≤ u n
have this : (↑μ : Measure E).InnerRegular
Tactic: have : Measure.InnerRegular (μ : Measure E) := by simp only [toMeasure_mk, μ]; infer_instance