ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Besicovitch.SatelliteConfig.inter
structure Besicovitch.SatelliteConfig (α : Type*) [MetricSpace α] (N : ℕ) (τ : ℝ) where
  c : Fin N.succ → α
  r : Fin N.succ → ℝ
  rpos : ∀ i, 0 < r i
  h : Pairwise fun i j =>
    r i ≤ dist (c i) (c j) ∧ r j ≤ τ * r i ∨ r j ≤ dist (c j) (c i) ∧ r i ≤ τ * r j
  hlast : ∀ i < last N, r i ≤ dist (c i) (c (last N)) ∧ r (last N) ≤ τ * r i
  inter : ∀ i < last N, dist (c i) (c (last N)) ≤ r i + r (last N)
Arguments: {α : Type u_1} [MetricSpace α] {N : ℕ} {τ : ℝ} (self : Besicovitch.SatelliteConfig α N τ) (i : Fin (N + 1))
Goal: i < Fin.last N → dist (self.c i) (self.c (Fin.last N)) ≤ self.r i + self.r (Fin.last N)
-- Besicovitch.SatelliteConfig.rpos
structure Besicovitch.SatelliteConfig (α : Type*) [MetricSpace α] (N : ℕ) (τ : ℝ) where
  c : Fin N.succ → α
  r : Fin N.succ → ℝ
  rpos : ∀ i, 0 < r i
  h : Pairwise fun i j =>
    r i ≤ dist (c i) (c j) ∧ r j ≤ τ * r i ∨ r j ≤ dist (c j) (c i) ∧ r i ≤ τ * r j
  hlast : ∀ i < last N, r i ≤ dist (c i) (c (last N)) ∧ r (last N) ≤ τ * r i
  inter : ∀ i < last N, dist (c i) (c (last N)) ≤ r i + r (last N)
Arguments: {α : Type u_1} [MetricSpace α] {N : ℕ} {τ : ℝ} (self : Besicovitch.SatelliteConfig α N τ) (i : Fin N.succ)
Goal: 0 < self.r i
-- HasBesicovitchCovering.no_satelliteConfig
class HasBesicovitchCovering (α : Type*) [MetricSpace α] : Prop where
  no_satelliteConfig : ∃ (N : ℕ) (τ : ℝ), 1 < τ ∧ IsEmpty (Besicovitch.SatelliteConfig α N τ)
Arguments: {α : Type u_1} [MetricSpace α] [self : HasBesicovitchCovering α]
Goal: ∃ N τ, 1 < τ ∧ IsEmpty (Besicovitch.SatelliteConfig α N τ)
-- Besicovitch.SatelliteConfig.inter'
theorem inter' (i : Fin N.succ) : dist (a.c i) (a.c (last N)) ≤ a.r i + a.r (last N) 
Arguments: {α : Type u_1} [MetricSpace α] {N : ℕ} {τ : ℝ} (a : Besicovitch.SatelliteConfig α N τ) (i : Fin N.succ)
Goal: dist (a.c i) (a.c (Fin.last N)) ≤ a.r i + a.r (Fin.last N)
-- Besicovitch.SatelliteConfig.hlast'
theorem hlast' (i : Fin N.succ) (h : 1 ≤ τ) : a.r (last N) ≤ τ * a.r i 
Arguments: {α : Type u_1} [MetricSpace α] {N : ℕ} {τ : ℝ} (a : Besicovitch.SatelliteConfig α N τ) (i : Fin N.succ) (h : 1 ≤ τ)
Goal: a.r (Fin.last N) ≤ τ * a.r i
-- Besicovitch.BallPackage.rpos
structure BallPackage (β : Type*) (α : Type*) where
  c : β → α
  r : β → ℝ
  rpos : ∀ b, 0 < r b
  r_bound : ℝ
  r_le : ∀ b, r b ≤ r_bound
Arguments: {β : Type u_1} {α : Type u_2} (self : Besicovitch.BallPackage β α) (b : β)
Goal: 0 < self.r b
-- Besicovitch.BallPackage.r_le
structure BallPackage (β : Type*) (α : Type*) where
  c : β → α
  r : β → ℝ
  rpos : ∀ b, 0 < r b
  r_bound : ℝ
  r_le : ∀ b, r b ≤ r_bound
Arguments: {β : Type u_1} {α : Type u_2} (self : Besicovitch.BallPackage β α) (b : β)
Goal: self.r b ≤ self.r_bound
-- Besicovitch.TauPackage.one_lt_tau
structure TauPackage (β : Type*) (α : Type*) extends BallPackage β α where
  τ : ℝ
  one_lt_tau : 1 < τ
Arguments: {β : Type u_1} {α : Type u_2} (self : Besicovitch.TauPackage β α)
Goal: 1 < self.τ
-- Besicovitch.TauPackage.monotone_iUnionUpTo
theorem monotone_iUnionUpTo : Monotone p.iUnionUpTo 
Arguments: {α : Type u_1} [MetricSpace α] {β : Type u} [Nonempty β] (p : Besicovitch.TauPackage β α)
Goal: Monotone p.iUnionUpTo
-- Besicovitch.TauPackage.lastStep_nonempty
theorem lastStep_nonempty :
    {i | ¬∃ b : β, p.c b ∉ p.iUnionUpTo i ∧ p.R i ≤ p.τ * p.r b}.Nonempty 
Arguments: {α : Type u_1} [MetricSpace α] {β : Type u} [Nonempty β] (p : Besicovitch.TauPackage β α)
Goal: {i | ¬∃ b, p.c b ∉ p.iUnionUpTo i ∧ p.R i ≤ p.τ * p.r b}.Nonempty
-- Besicovitch.TauPackage.mem_iUnionUpTo_lastStep
theorem mem_iUnionUpTo_lastStep (x : β) : p.c x ∈ p.iUnionUpTo p.lastStep 
Arguments: {α : Type u_1} [MetricSpace α] {β : Type u} [Nonempty β] (p : Besicovitch.TauPackage β α) (x : β)
Goal: p.c x ∈ p.iUnionUpTo p.lastStep
-- Besicovitch.TauPackage.color_lt
theorem color_lt {i : Ordinal.{u}} (hi : i < p.lastStep) {N : ℕ}
    (hN : IsEmpty (SatelliteConfig α N p.τ)) : p.color i < N 
Arguments: {α : Type u_1} [MetricSpace α] {β : Type u} [Nonempty β] (p : Besicovitch.TauPackage β α) {i : Ordinal.{u}} (hi : i < p.lastStep) {N : ℕ} (hN : IsEmpty (Besicovitch.SatelliteConfig α N p.τ))
Goal: p.color i < N
-- Besicovitch.exist_disjoint_covering_families
theorem exist_disjoint_covering_families {N : ℕ} {τ : ℝ} (hτ : 1 < τ)
    (hN : IsEmpty (SatelliteConfig α N τ)) (q : BallPackage β α) :
    ∃ s : Fin N → Set β,
      (∀ i : Fin N, (s i).PairwiseDisjoint fun j => closedBall (q.c j) (q.r j)) ∧
        range q.c ⊆ ⋃ i : Fin N, ⋃ j ∈ s i, ball (q.c j) (q.r j) 
Arguments: {α : Type u_1} [MetricSpace α] {β : Type u} {N : ℕ} {τ : ℝ} (hτ : 1 < τ) (hN : IsEmpty (Besicovitch.SatelliteConfig α N τ)) (q : Besicovitch.BallPackage β α)
Goal: ∃ s,
  (∀ (i : Fin N), (s i).PairwiseDisjoint fun j => Metric.closedBall (q.c j) (q.r j)) ∧
    Set.range q.c ⊆ ⋃ i, ⋃ j ∈ s i, Metric.ball (q.c j) (q.r j)
-- Besicovitch.exist_finset_disjoint_balls_large_measure
theorem exist_finset_disjoint_balls_large_measure (μ : Measure α) [IsFiniteMeasure μ] {N : ℕ}
    {τ : ℝ} (hτ : 1 < τ) (hN : IsEmpty (SatelliteConfig α N τ)) (s : Set α) (r : α → ℝ)
    (rpos : ∀ x ∈ s, 0 < r x) (rle : ∀ x ∈ s, r x ≤ 1) :
    ∃ t : Finset α, ↑t ⊆ s ∧ μ (s \ ⋃ x ∈ t, closedBall x (r x)) ≤ N / (N + 1) * μ s ∧
      (t : Set α).PairwiseDisjoint fun x => closedBall x (r x) 
Arguments: {α : Type u_1} [MetricSpace α] [SecondCountableTopology α] [MeasurableSpace α] [OpensMeasurableSpace α] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] {N : ℕ} {τ : ℝ} (hτ : 1 < τ) (hN : IsEmpty (Besicovitch.SatelliteConfig α N τ)) (s : Set α) (r : α → ℝ) (rpos : ∀ x ∈ s, 0 < r x) (rle : ∀ x ∈ s, r x ≤ 1)
Goal: ∃ t,
  ↑t ⊆ s ∧
    μ (s \ ⋃ x ∈ t, Metric.closedBall x (r x)) ≤ ↑N / (↑N + 1) * μ s ∧
      (↑t).PairwiseDisjoint fun x => Metric.closedBall x (r x)
-- Besicovitch.exists_disjoint_closedBall_covering_ae_of_finiteMeasure_aux
theorem exists_disjoint_closedBall_covering_ae_of_finiteMeasure_aux (μ : Measure α)
    [IsFiniteMeasure μ] (f : α → Set ℝ) (s : Set α)
    (hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty) :
    ∃ t : Set (α × ℝ), t.Countable ∧ (∀ p ∈ t, p.1 ∈ s) ∧ (∀ p ∈ t, p.2 ∈ f p.1) ∧
      μ (s \ ⋃ (p : α × ℝ) (_ : p ∈ t), closedBall p.1 p.2) = 0 ∧
        t.PairwiseDisjoint fun p => closedBall p.1 p.2 
Arguments: {α : Type u_1} [MetricSpace α] [SecondCountableTopology α] [MeasurableSpace α] [OpensMeasurableSpace α] [HasBesicovitchCovering α] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (f : α → Set ℝ) (s : Set α) (hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Set.Ioo 0 δ).Nonempty)
Goal: ∃ t,
  t.Countable ∧
    (∀ p ∈ t, p.1 ∈ s) ∧
      (∀ p ∈ t, p.2 ∈ f p.1) ∧
        μ (s \ ⋃ p ∈ t, Metric.closedBall p.1 p.2) = 0 ∧ t.PairwiseDisjoint fun p => Metric.closedBall p.1 p.2
-- Besicovitch.exists_disjoint_closedBall_covering_ae_aux
theorem exists_disjoint_closedBall_covering_ae_aux (μ : Measure α) [SFinite μ] (f : α → Set ℝ)
    (s : Set α) (hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty) :
    ∃ t : Set (α × ℝ), t.Countable ∧ (∀ p ∈ t, p.1 ∈ s) ∧ (∀ p ∈ t, p.2 ∈ f p.1) ∧
      μ (s \ ⋃ (p : α × ℝ) (_ : p ∈ t), closedBall p.1 p.2) = 0 ∧
        t.PairwiseDisjoint fun p => closedBall p.1 p.2 
Arguments: {α : Type u_1} [MetricSpace α] [SecondCountableTopology α] [MeasurableSpace α] [OpensMeasurableSpace α] [HasBesicovitchCovering α] (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ] (f : α → Set ℝ) (s : Set α) (hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Set.Ioo 0 δ).Nonempty)
Goal: ∃ t,
  t.Countable ∧
    (∀ p ∈ t, p.1 ∈ s) ∧
      (∀ p ∈ t, p.2 ∈ f p.1) ∧
        μ (s \ ⋃ p ∈ t, Metric.closedBall p.1 p.2) = 0 ∧ t.PairwiseDisjoint fun p => Metric.closedBall p.1 p.2
-- Besicovitch.exists_disjoint_closedBall_covering_ae
theorem exists_disjoint_closedBall_covering_ae (μ : Measure α) [SFinite μ] (f : α → Set ℝ)
    (s : Set α) (hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty) (R : α → ℝ)
    (hR : ∀ x ∈ s, 0 < R x) :
    ∃ (t : Set α) (r : α → ℝ), t.Countable ∧ t ⊆ s ∧
      (∀ x ∈ t, r x ∈ f x ∩ Ioo 0 (R x)) ∧ μ (s \ ⋃ x ∈ t, closedBall x (r x)) = 0 ∧
        t.PairwiseDisjoint fun x => closedBall x (r x) 
Arguments: {α : Type u_1} [MetricSpace α] [SecondCountableTopology α] [MeasurableSpace α] [OpensMeasurableSpace α] [HasBesicovitchCovering α] (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ] (f : α → Set ℝ) (s : Set α) (hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Set.Ioo 0 δ).Nonempty) (R : α → ℝ) (hR : ∀ x ∈ s, 0 < R x)
Goal: ∃ t r,
  t.Countable ∧
    t ⊆ s ∧
      (∀ x ∈ t, r x ∈ f x ∩ Set.Ioo 0 (R x)) ∧
        μ (s \ ⋃ x ∈ t, Metric.closedBall x (r x)) = 0 ∧ t.PairwiseDisjoint fun x => Metric.closedBall x (r x)
-- Besicovitch.exists_closedBall_covering_tsum_measure_le
theorem exists_closedBall_covering_tsum_measure_le (μ : Measure α) [SFinite μ]
    [Measure.OuterRegular μ] {ε : ℝ≥0∞} (hε : ε ≠ 0) (f : α → Set ℝ) (s : Set α)
    (hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty) :
    ∃ (t : Set α) (r : α → ℝ), t.Countable ∧ t ⊆ s ∧ (∀ x ∈ t, r x ∈ f x) ∧
      (s ⊆ ⋃ x ∈ t, closedBall x (r x)) ∧ (∑' x : t, μ (closedBall x (r x))) ≤ μ s + ε 
Arguments: {α : Type u_1} [MetricSpace α] [SecondCountableTopology α] [MeasurableSpace α] [OpensMeasurableSpace α] [HasBesicovitchCovering α] (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ] [μ.OuterRegular] {ε : ℝ≥0∞} (hε : ε ≠ 0) (f : α → Set ℝ) (s : Set α) (hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Set.Ioo 0 δ).Nonempty)
Goal: ∃ t r,
  t.Countable ∧
    t ⊆ s ∧
      (∀ x ∈ t, r x ∈ f x) ∧
        s ⊆ ⋃ x ∈ t, Metric.closedBall x (r x) ∧ ∑' (x : ↑t), μ (Metric.closedBall (↑x) (r ↑x)) ≤ μ s + ε
-- Besicovitch.tendsto_filterAt
theorem tendsto_filterAt (μ : Measure α) [SFinite μ] (x : α) :
    Tendsto (fun r => closedBall x r) (𝓝[>] 0) ((Besicovitch.vitaliFamily μ).filterAt x) 
Arguments: {α : Type u_1} [MetricSpace α] [SecondCountableTopology α] [MeasurableSpace α] [OpensMeasurableSpace α] [HasBesicovitchCovering α] (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ] (x : α)
Goal: Filter.Tendsto (fun r => Metric.closedBall x r) (𝓝[>] 0) ((Besicovitch.vitaliFamily μ).filterAt x)
-- Besicovitch.ae_tendsto_rnDeriv
theorem ae_tendsto_rnDeriv (ρ μ : Measure β) [IsLocallyFiniteMeasure μ] [IsLocallyFiniteMeasure ρ] :
    ∀ᵐ x ∂μ,
      Tendsto (fun r => ρ (closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 (ρ.rnDeriv μ x)) 
Arguments: {β : Type u} [MetricSpace β] [MeasurableSpace β] [BorelSpace β] [SecondCountableTopology β] [HasBesicovitchCovering β] (ρ μ : MeasureTheory.Measure β) [MeasureTheory.IsLocallyFiniteMeasure μ] [MeasureTheory.IsLocallyFiniteMeasure ρ]
Goal: ∀ᵐ (x : β) ∂μ,
  Filter.Tendsto (fun r => ρ (Metric.closedBall x r) / μ (Metric.closedBall x r)) (𝓝[>] 0) (𝓝 (ρ.rnDeriv μ x))
-- Besicovitch.ae_tendsto_measure_inter_div_of_measurableSet
theorem ae_tendsto_measure_inter_div_of_measurableSet (μ : Measure β) [IsLocallyFiniteMeasure μ]
    {s : Set β} (hs : MeasurableSet s) :
    ∀ᵐ x ∂μ,
      Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0)
        (𝓝 (s.indicator 1 x)) 
Arguments: {β : Type u} [MetricSpace β] [MeasurableSpace β] [BorelSpace β] [SecondCountableTopology β] [HasBesicovitchCovering β] (μ : MeasureTheory.Measure β) [MeasureTheory.IsLocallyFiniteMeasure μ] {s : Set β} (hs : MeasurableSet s)
Goal: ∀ᵐ (x : β) ∂μ,
  Filter.Tendsto (fun r => μ (s ∩ Metric.closedBall x r) / μ (Metric.closedBall x r)) (𝓝[>] 0) (𝓝 (s.indicator 1 x))
-- Besicovitch.ae_tendsto_measure_inter_div
theorem ae_tendsto_measure_inter_div (μ : Measure β) [IsLocallyFiniteMeasure μ] (s : Set β) :
    ∀ᵐ x ∂μ.restrict s,
      Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1) 
Arguments: {β : Type u} [MetricSpace β] [MeasurableSpace β] [BorelSpace β] [SecondCountableTopology β] [HasBesicovitchCovering β] (μ : MeasureTheory.Measure β) [MeasureTheory.IsLocallyFiniteMeasure μ] (s : Set β)
Goal: ∀ᵐ (x : β) ∂μ.restrict s,
  Filter.Tendsto (fun r => μ (s ∩ Metric.closedBall x r) / μ (Metric.closedBall x r)) (𝓝[>] 0) (𝓝 1)
-- MeasureTheory.tilted_of_not_integrable
@[simp]
lemma tilted_of_not_integrable (hf : ¬ Integrable (fun x ↦ exp (f x)) μ) : μ.tilted f = 0 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : ¬MeasureTheory.Integrable (fun x => rexp (f x)) μ)
Goal: μ.tilted f = 0
-- MeasureTheory.tilted_of_not_aemeasurable
@[simp]
lemma tilted_of_not_aemeasurable (hf : ¬ AEMeasurable f μ) : μ.tilted f = 0 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : ¬AEMeasurable f μ)
Goal: μ.tilted f = 0
-- MeasureTheory.tilted_zero_measure
@[simp]
lemma tilted_zero_measure (f : α → ℝ) : (0 : Measure α).tilted f = 0 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (f : α → ℝ)
Goal: MeasureTheory.Measure.tilted 0 f = 0
-- MeasureTheory.tilted_const'
@[simp]
lemma tilted_const' (μ : Measure α) (c : ℝ) :
    μ.tilted (fun _ ↦ c) = (μ Set.univ)⁻¹ • μ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ : MeasureTheory.Measure α) (c : ℝ)
Goal: (μ.tilted fun x => c) = (μ Set.univ)⁻¹ • μ
-- MeasureTheory.tilted_const
lemma tilted_const (μ : Measure α) [IsProbabilityMeasure μ] (c : ℝ) :
    μ.tilted (fun _ ↦ c) = μ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ] (c : ℝ)
Goal: (μ.tilted fun x => c) = μ
-- MeasureTheory.tilted_zero'
@[simp]
lemma tilted_zero' (μ : Measure α) : μ.tilted 0 = (μ Set.univ)⁻¹ • μ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ : MeasureTheory.Measure α)
Goal: μ.tilted 0 = (μ Set.univ)⁻¹ • μ
-- MeasureTheory.tilted_zero
lemma tilted_zero (μ : Measure α) [IsProbabilityMeasure μ] : μ.tilted 0 = μ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ]
Goal: μ.tilted 0 = μ
-- MeasureTheory.tilted_congr
lemma tilted_congr {g : α → ℝ} (hfg : f =ᵐ[μ] g) :
    μ.tilted f = μ.tilted g 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ} (hfg : f =ᵐ[μ] g)
Goal: μ.tilted f = μ.tilted g
-- MeasureTheory.tilted_eq_withDensity_nnreal
lemma tilted_eq_withDensity_nnreal (μ : Measure α) (f : α → ℝ) :
    μ.tilted f = μ.withDensity (fun x ↦ ((↑) : ℝ≥0 → ℝ≥0∞)
      (⟨exp (f x) / ∫ x, exp (f x) ∂μ, by positivity⟩ : ℝ≥0)) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ : MeasureTheory.Measure α) (f : α → ℝ)
Goal: μ.tilted f = μ.withDensity fun x => ↑⟨rexp (f x) / ∫ (x : α), rexp (f x) ∂μ, ⋯⟩
-- MeasureTheory.tilted_apply'
lemma tilted_apply' (μ : Measure α) (f : α → ℝ) {s : Set α} (hs : MeasurableSet s) :
    μ.tilted f s = ∫⁻ a in s, ENNReal.ofReal (exp (f a) / ∫ x, exp (f x) ∂μ) ∂μ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ : MeasureTheory.Measure α) (f : α → ℝ) {s : Set α} (hs : MeasurableSet s)
Goal: (μ.tilted f) s = ∫⁻ (a : α) in s, ENNReal.ofReal (rexp (f a) / ∫ (x : α), rexp (f x) ∂μ) ∂μ
-- MeasureTheory.tilted_apply
lemma tilted_apply (μ : Measure α) [SFinite μ] (f : α → ℝ) (s : Set α) :
    μ.tilted f s = ∫⁻ a in s, ENNReal.ofReal (exp (f a) / ∫ x, exp (f x) ∂μ) ∂μ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ] (f : α → ℝ) (s : Set α)
Goal: (μ.tilted f) s = ∫⁻ (a : α) in s, ENNReal.ofReal (rexp (f a) / ∫ (x : α), rexp (f x) ∂μ) ∂μ
-- MeasureTheory.tilted_apply_eq_ofReal_integral'
lemma tilted_apply_eq_ofReal_integral' {s : Set α} (f : α → ℝ) (hs : MeasurableSet s) :
    μ.tilted f s = ENNReal.ofReal (∫ a in s, exp (f a) / ∫ x, exp (f x) ∂μ ∂μ) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (f : α → ℝ) (hs : MeasurableSet s)
Goal: (μ.tilted f) s = ENNReal.ofReal (∫ (a : α) in s, rexp (f a) / ∫ (x : α), rexp (f x) ∂μ ∂μ)
-- MeasureTheory.tilted_apply_eq_ofReal_integral
lemma tilted_apply_eq_ofReal_integral [SFinite μ] (f : α → ℝ) (s : Set α) :
    μ.tilted f s = ENNReal.ofReal (∫ a in s, exp (f a) / ∫ x, exp (f x) ∂μ ∂μ) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SFinite μ] (f : α → ℝ) (s : Set α)
Goal: (μ.tilted f) s = ENNReal.ofReal (∫ (a : α) in s, rexp (f a) / ∫ (x : α), rexp (f x) ∂μ ∂μ)
-- MeasureTheory.isProbabilityMeasure_tilted
lemma isProbabilityMeasure_tilted [NeZero μ] (hf : Integrable (fun x ↦ exp (f x)) μ) :
    IsProbabilityMeasure (μ.tilted f) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} [NeZero μ] (hf : MeasureTheory.Integrable (fun x => rexp (f x)) μ)
Goal: MeasureTheory.IsProbabilityMeasure (μ.tilted f)
-- MeasureTheory.setLIntegral_tilted'
lemma setLIntegral_tilted' (f : α → ℝ) (g : α → ℝ≥0∞) {s : Set α} (hs : MeasurableSet s) :
    ∫⁻ x in s, g x ∂(μ.tilted f)
      = ∫⁻ x in s, ENNReal.ofReal (exp (f x) / ∫ x, exp (f x) ∂μ) * g x ∂μ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ℝ) (g : α → ℝ≥0∞) {s : Set α} (hs : MeasurableSet s)
Goal: ∫⁻ (x : α) in s, g x ∂μ.tilted f = ∫⁻ (x : α) in s, ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) * g x ∂μ
-- MeasureTheory.setLIntegral_tilted
lemma setLIntegral_tilted [SFinite μ] (f : α → ℝ) (g : α → ℝ≥0∞) (s : Set α) :
    ∫⁻ x in s, g x ∂(μ.tilted f)
      = ∫⁻ x in s, ENNReal.ofReal (exp (f x) / ∫ x, exp (f x) ∂μ) * g x ∂μ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SFinite μ] (f : α → ℝ) (g : α → ℝ≥0∞) (s : Set α)
Goal: ∫⁻ (x : α) in s, g x ∂μ.tilted f = ∫⁻ (x : α) in s, ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) * g x ∂μ
-- MeasureTheory.lintegral_tilted
lemma lintegral_tilted (f : α → ℝ) (g : α → ℝ≥0∞) :
    ∫⁻ x, g x ∂(μ.tilted f)
      = ∫⁻ x, ENNReal.ofReal (exp (f x) / ∫ x, exp (f x) ∂μ) * (g x) ∂μ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ℝ) (g : α → ℝ≥0∞)
Goal: ∫⁻ (x : α), g x ∂μ.tilted f = ∫⁻ (x : α), ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) * g x ∂μ
-- MeasureTheory.setIntegral_tilted'
lemma setIntegral_tilted' (f : α → ℝ) (g : α → E) {s : Set α} (hs : MeasurableSet s) :
    ∫ x in s, g x ∂(μ.tilted f) = ∫ x in s, (exp (f x) / ∫ x, exp (f x) ∂μ) • (g x) ∂μ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (f : α → ℝ) (g : α → E) {s : Set α} (hs : MeasurableSet s)
Goal: ∫ (x : α) in s, g x ∂μ.tilted f = ∫ (x : α) in s, (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) • g x ∂μ
-- MeasureTheory.setIntegral_tilted
lemma setIntegral_tilted [SFinite μ] (f : α → ℝ) (g : α → E) (s : Set α) :
    ∫ x in s, g x ∂(μ.tilted f) = ∫ x in s, (exp (f x) / ∫ x, exp (f x) ∂μ) • (g x) ∂μ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasureTheory.SFinite μ] (f : α → ℝ) (g : α → E) (s : Set α)
Goal: ∫ (x : α) in s, g x ∂μ.tilted f = ∫ (x : α) in s, (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) • g x ∂μ
-- MeasureTheory.integral_tilted
lemma integral_tilted (f : α → ℝ) (g : α → E) :
    ∫ x, g x ∂(μ.tilted f) = ∫ x, (exp (f x) / ∫ x, exp (f x) ∂μ) • (g x) ∂μ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] (f : α → ℝ) (g : α → E)
Goal: ∫ (x : α), g x ∂μ.tilted f = ∫ (x : α), (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) • g x ∂μ
-- MeasureTheory.integral_exp_tilted
lemma integral_exp_tilted (f g : α → ℝ) :
    ∫ x, exp (g x) ∂(μ.tilted f) = (∫ x, exp ((f + g) x) ∂μ) / ∫ x, exp (f x) ∂μ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f g : α → ℝ)
Goal: ∫ (x : α), rexp (g x) ∂μ.tilted f = (∫ (x : α), rexp ((f + g) x) ∂μ) / ∫ (x : α), rexp (f x) ∂μ
-- MeasureTheory.tilted_tilted
lemma tilted_tilted (hf : Integrable (fun x ↦ exp (f x)) μ) (g : α → ℝ) :
    (μ.tilted f).tilted g = μ.tilted (f + g) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.Integrable (fun x => rexp (f x)) μ) (g : α → ℝ)
Goal: (μ.tilted f).tilted g = μ.tilted (f + g)
-- MeasureTheory.tilted_comm
lemma tilted_comm (hf : Integrable (fun x ↦ exp (f x)) μ) {g : α → ℝ}
    (hg : Integrable (fun x ↦ exp (g x)) μ) :
    (μ.tilted f).tilted g = (μ.tilted g).tilted f 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.Integrable (fun x => rexp (f x)) μ) {g : α → ℝ} (hg : MeasureTheory.Integrable (fun x => rexp (g x)) μ)
Goal: (μ.tilted f).tilted g = (μ.tilted g).tilted f
-- MeasureTheory.tilted_neg_same'
@[simp]
lemma tilted_neg_same' (hf : Integrable (fun x ↦ exp (f x)) μ) :
    (μ.tilted f).tilted (-f) = (μ Set.univ)⁻¹ • μ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.Integrable (fun x => rexp (f x)) μ)
Goal: (μ.tilted f).tilted (-f) = (μ Set.univ)⁻¹ • μ
-- MeasureTheory.tilted_neg_same
@[simp]
lemma tilted_neg_same [IsProbabilityMeasure μ] (hf : Integrable (fun x ↦ exp (f x)) μ) :
    (μ.tilted f).tilted (-f) = μ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.IsProbabilityMeasure μ] (hf : MeasureTheory.Integrable (fun x => rexp (f x)) μ)
Goal: (μ.tilted f).tilted (-f) = μ
-- MeasureTheory.tilted_absolutelyContinuous
lemma tilted_absolutelyContinuous (μ : Measure α) (f : α → ℝ) : μ.tilted f ≪ μ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ : MeasureTheory.Measure α) (f : α → ℝ)
Goal: (μ.tilted f).AbsolutelyContinuous μ
-- MeasureTheory.absolutelyContinuous_tilted
lemma absolutelyContinuous_tilted (hf : Integrable (fun x ↦ exp (f x)) μ) : μ ≪ μ.tilted f 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.Integrable (fun x => rexp (f x)) μ)
Goal: μ.AbsolutelyContinuous (μ.tilted f)
-- MeasureTheory.rnDeriv_tilted_right
lemma rnDeriv_tilted_right (μ ν : Measure α) [SigmaFinite μ] [SigmaFinite ν]
    (hf : Integrable (fun x ↦ exp (f x)) ν) :
    μ.rnDeriv (ν.tilted f)
      =ᵐ[ν] fun x ↦ ENNReal.ofReal (exp (- f x) * ∫ x, exp (f x) ∂ν) * μ.rnDeriv ν x 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {f : α → ℝ} (μ ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] (hf : MeasureTheory.Integrable (fun x => rexp (f x)) ν)
Goal: μ.rnDeriv (ν.tilted f) =ᵐ[ν] fun x => ENNReal.ofReal (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * μ.rnDeriv ν x
-- MeasureTheory.toReal_rnDeriv_tilted_right
lemma toReal_rnDeriv_tilted_right (μ ν : Measure α) [SigmaFinite μ] [SigmaFinite ν]
    (hf : Integrable (fun x ↦ exp (f x)) ν) :
    (fun x ↦ (μ.rnDeriv (ν.tilted f) x).toReal)
      =ᵐ[ν] fun x ↦ exp (- f x) * (∫ x, exp (f x) ∂ν) * (μ.rnDeriv ν x).toReal 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {f : α → ℝ} (μ ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] (hf : MeasureTheory.Integrable (fun x => rexp (f x)) ν)
Goal: (fun x => (μ.rnDeriv (ν.tilted f) x).toReal) =ᵐ[ν] fun x =>
  (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * (μ.rnDeriv ν x).toReal
-- MeasureTheory.rnDeriv_tilted_left
lemma rnDeriv_tilted_left {ν : Measure α} [SigmaFinite μ] [SigmaFinite ν] (hfν : AEMeasurable f ν) :
    (μ.tilted f).rnDeriv ν
      =ᵐ[ν] fun x ↦ ENNReal.ofReal (exp (f x) / (∫ x, exp (f x) ∂μ)) * μ.rnDeriv ν x 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ : MeasureTheory.Measure α) {f : α → ℝ} {ν : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] (hfν : AEMeasurable f ν)
Goal: (μ.tilted f).rnDeriv ν =ᵐ[ν] fun x => ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) * μ.rnDeriv ν x
-- MeasureTheory.toReal_rnDeriv_tilted_left
lemma toReal_rnDeriv_tilted_left {ν : Measure α} [SigmaFinite μ] [SigmaFinite ν]
    (hfν : AEMeasurable f ν) :
    (fun x ↦ ((μ.tilted f).rnDeriv ν x).toReal)
      =ᵐ[ν] fun x ↦ exp (f x) / (∫ x, exp (f x) ∂μ) * (μ.rnDeriv ν x).toReal 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ : MeasureTheory.Measure α) {f : α → ℝ} {ν : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] (hfν : AEMeasurable f ν)
Goal: (fun x => ((μ.tilted f).rnDeriv ν x).toReal) =ᵐ[ν] fun x =>
  (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) * (μ.rnDeriv ν x).toReal