mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Process.Filtration

Goal state before:
pp: case pos
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝ : PartialOrder ι
𝓕 : Filtration ι m
this✝ : TopologicalSpace ι := Preorder.topology ι
this : OrderTopology ι
i : ι
hne : (𝓝[>] i).NeBot
⊢ ↑𝓕₊₊ i ≤ ↑𝓕₊ i

Tactic:
have hineq : (⨅ j > i, 𝓕₊ j) ≤ ⨅ j > i, 𝓕 j := by
      apply le_iInf₂ fun u hu => ?_
      have hiou : Set.Ioo i u ∈ 𝓝[>] i := by
        rw [mem_nhdsWithin_iff_exists_mem_nhds_inter]
        exact ⟨Set.Iio u, (isOpen_Iio' u).mem_nhds hu, fun _ hx ↦ ⟨hx.2, hx.1⟩⟩
      obtain ⟨v, hv⟩ := hne.nonempty_of_mem hiou
      have hle₁ : (⨅ j > i, 𝓕₊ j) ≤ 𝓕₊ v := iInf₂_le_of_le v hv.1 le_rfl
      have hle₂ : 𝓕₊ v ≤ 𝓕 u := by
        by_cases hnv : (𝓝[>] v).NeBot
        · simpa [rightCont_eq_of_neBot_nhdsGT] using iInf₂_le_of_le u hv.2 le_rfl
        · simpa [rightCont_apply, hnv] using 𝓕.mono hv.2.le
      exact hle₁.trans hle₂

Goal state after:
case pos
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝ : PartialOrder ι
𝓕 : Filtration ι m
this✝ : TopologicalSpace ι := Preorder.topology ι
this : OrderTopology ι
i : ι
hne : (𝓝[>] i).NeBot
hineq : ⨅ j, ⨅ (_ : j > i), ↑𝓕₊ j ≤ ⨅ j, ⨅ (_ : j > i), ↑𝓕 j
⊢ ↑𝓕₊₊ i ≤ ↑𝓕₊ i
Module: Mathlib.MeasureTheory.Function.LpSpace.ContinuousCompMeasurePreserving

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : BorelSpace X
inst✝⁸ : R1Space X
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : BorelSpace Y
inst✝⁴ : R1Space Y
μ : Measure X
ν : Measure Y
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure ν
E : Type u_3
inst✝¹ : NormedAddCommGroup E
p : ℝ≥0∞
inst✝ : Fact (1 ≤ p)
hp : p ≠ ∞
hp₀ : p ≠ 0
c : E
s : Set Y
hs : MeasurableSet s
hνs : ν s < ∞
f : { f // MeasurePreserving (⇑f) μ ν }
⊢ ∀ᶠ (a : { f // MeasurePreserving (⇑f) μ ν }) in 𝓝 f, MeasurePreserving (⇑↑a) μ ν

Tactic:
exact .of_forall Subtype.property

Goal state after:
Module: Mathlib.MeasureTheory.Function.UnifTight

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp_top : p ≠ ∞
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), MemLp (f i) p μ) → UnifTight f p μ
f : Fin (n + 1) → α → β
hfLp : ∀ (i : Fin (n + 1)), MemLp (f i) p μ
ε : ℝ≥0
hε : 0 < ε
hε_top : ¬↑ε = ∞
g : Fin n → α → β := fun k ↦ f k.castSucc
hgLp : ∀ (i : Fin n), MemLp (g i) p μ
S : Set α
hμS : μ S ≠ ∞
hFε : ∀ (i : Fin n), eLpNorm (Sᶜ.indicator (g i)) p μ ≤ ↑ε
⊢ ∃ s, μ s ≠ ∞ ∧ ∀ (i : Fin (n + 1)), eLpNorm (sᶜ.indicator (f i)) p μ ≤ ↑ε

Tactic:
obtain ⟨s, _, hμs, hfε⟩ :=
      (hfLp (Fin.last n)).exists_eLpNorm_indicator_compl_lt hp_top (coe_ne_zero.2 hε.ne')

Goal state after:
case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp_top : p ≠ ∞
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), MemLp (f i) p μ) → UnifTight f p μ
f : Fin (n + 1) → α → β
hfLp : ∀ (i : Fin (n + 1)), MemLp (f i) p μ
ε : ℝ≥0
hε : 0 < ε
hε_top : ¬↑ε = ∞
g : Fin n → α → β := fun k ↦ f k.castSucc
hgLp : ∀ (i : Fin n), MemLp (g i) p μ
S : Set α
hμS : μ S ≠ ∞
hFε : ∀ (i : Fin n), eLpNorm (Sᶜ.indicator (g i)) p μ ≤ ↑ε
s : Set α
left✝ : MeasurableSet s
hμs : μ s < ∞
hfε : eLpNorm (sᶜ.indicator (f (Fin.last n))) p μ < ↑ε
⊢ ∃ s, μ s ≠ ∞ ∧ ∀ (i : Fin (n + 1)), eLpNorm (sᶜ.indicator (f i)) p μ ≤ ↑ε
Module: Mathlib.MeasureTheory.Function.UnifTight

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp' : p ≠ ∞
hf : ∀ (n : ℕ), MemLp (f n) p μ
hg : MemLp g p μ
hfg : Tendsto (fun n ↦ eLpNorm (f n - g) p μ) atTop (𝓝 0)
⊢ UnifTight f p μ

Tactic:
have : f = (fun _ => g) + fun n => f n - g := by ext1 n; simp

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp' : p ≠ ∞
hf : ∀ (n : ℕ), MemLp (f n) p μ
hg : MemLp g p μ
hfg : Tendsto (fun n ↦ eLpNorm (f n - g) p μ) atTop (𝓝 0)
this : f = (fun x ↦ g) + fun n ↦ f n - g
⊢ UnifTight f p μ
Module: Mathlib.MeasureTheory.Function.UnifTight

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
⊢ Tendsto (fun n ↦ eLpNorm (f n - g) p μ) atTop (𝓝 0)

Tactic:
rw [ENNReal.tendsto_atTop_zero]

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, eLpNorm (f n - g) p μ ≤ ε
Module: Mathlib.MeasureTheory.Group.GeometryOfNumbers

Goal state before:
pp: E : Type u_1
inst✝⁸ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : Nontrivial E
inst✝² : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝¹ : Countable ↥L
inst✝ : DiscreteTopology ↥L
fund : IsAddFundamentalDomain (↥L) F μ
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h_cpt : IsCompact s
h : μ F * 2 ^ finrank ℝ E ≤ μ s
hμ : μ s = 0
⊢ μ F = 0

Tactic:
by simpa [hμ] using h

Goal state after:
Module: Mathlib.MeasureTheory.Integral.CircleTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
z w : ℂ
f : ℂ → E
⊢ ∫ (θ : ℝ) in 0..2 * π, circleTransform R z w f θ = (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(z, R), (z - w)⁻¹ • f z

Tactic:
simp_rw [circleTransform, circleIntegral, deriv_circleMap, circleMap]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
z w : ℂ
f : ℂ → E
⊢ ∫ (θ : ℝ) in 0..2 * π,
      (2 * ↑π * I)⁻¹ • ((0 + ↑R * cexp (↑θ * I)) * I) • (z + ↑R * cexp (↑θ * I) - w)⁻¹ • f (z + ↑R * cexp (↑θ * I)) =
    (2 * ↑π * I)⁻¹ •
      ∫ (θ : ℝ) in 0..2 * π,
        ((0 + ↑R * cexp (↑θ * I)) * I) • (z + ↑R * cexp (↑θ * I) - w)⁻¹ • f (z + ↑R * cexp (↑θ * I))
Module: Mathlib.MeasureTheory.Integral.CircleTransform

Goal state before:
pp: case h₀
R r : ℝ
hr : r < R
z : ℂ
⊢ ∀ x ∈ closedBall z r ×ˢ univ, circleMap z R x.2 - x.1 ≠ 0

Tactic:
rintro ⟨a, b⟩ ⟨ha, -⟩
    have ha2 : a ∈ ball z R := closedBall_subset_ball hr ha
    exact sub_ne_zero.2 (circleMap_ne_mem_ball ha2 b)

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : Zero β
f : α →C_c β
f' : α → β
h : f' = ⇑f
⊢ HasCompactSupport f'

Tactic:
simp_rw [h]
    exact f.hasCompactSupport'

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: case neg
F : Type u_1
α : Type u_2
β : Type u_3
γ✝ : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : TopologicalSpace β
inst✝² : Zero β
γ : Type u_5
inst✝¹ : TopologicalSpace γ
inst✝ : Zero γ
g : C(β, γ)
f : α →C_c β
hg : ¬g 0 = 0
⊢ HasCompactSupport (ContinuousMap.toFun 0)

Tactic:
· exact .zero

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: case inl
α : Type u_2
inst✝ : TopologicalSpace α
f : α →C_c ℝ
a : ℝ
ha : 0 ≤ a
x : α
hfx : 0 ≤ f x
⊢ max (a * f x) 0 = a * max (f x) 0

Tactic:
simp [ha, hfx, mul_nonneg]

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : TopologicalSpace α
Λ : (α →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
a : ℝ
f : α →C_c ℝ
⊢ ↑(Λ (a • f).nnrealPart) - ↑(Λ (-(a • f)).nnrealPart) = (RingHom.id ℝ) a • (↑(Λ f.nnrealPart) - ↑(Λ (-f).nnrealPart))

Tactic:
by
        rcases le_total 0 a with ha | ha
        · rw [RingHom.id_apply, smul_eq_mul, ← (smul_neg a f), nnrealPart_smul_pos f ha,
            nnrealPart_smul_pos (-f) ha]
          simp [sup_of_le_left ha, mul_sub]
        · simp only [RingHom.id_apply, smul_eq_mul, ← (smul_neg a f),
            nnrealPart_smul_neg f ha, nnrealPart_smul_neg (-f) ha, map_smul,
            NNReal.coe_mul, Real.coe_toNNReal', neg_neg, sup_of_le_left (neg_nonneg.mpr ha)]
          ring

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁷ : TopologicalSpace α
inst✝⁶ : R1Space α
inst✝⁵ : Group α
inst✝⁴ : TopologicalSpace β
inst✝³ : R1Space β
inst✝² : Group β
inst✝¹ : ContinuousMul β
inst✝ : NormedAddCommGroup γ
φ : α →* β
hφ : Topology.IsClosedEmbedding ⇑φ
f : β →C_c γ
b : β
K : Set β
hK : IsCompact K
hf : ∀ x ∉ K, f x = 0
⊢ HasCompactSupport fun a ↦ f (b * φ a)

Tactic:
refine exists_compact_iff_hasCompactSupport.mp ⟨φ ⁻¹' (b⁻¹ • K),
      hφ.isCompact_preimage (hK.smul b⁻¹), fun x hx ↦ hf _ ?_⟩

Goal state after:
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁷ : TopologicalSpace α
inst✝⁶ : R1Space α
inst✝⁵ : Group α
inst✝⁴ : TopologicalSpace β
inst✝³ : R1Space β
inst✝² : Group β
inst✝¹ : ContinuousMul β
inst✝ : NormedAddCommGroup γ
φ : α →* β
hφ : Topology.IsClosedEmbedding ⇑φ
f : β →C_c γ
b : β
K : Set β
hK : IsCompact K
hf : ∀ x ∉ K, f x = 0
x : α
hx : x ∉ ⇑φ ⁻¹' (b⁻¹ • K)
⊢ b * φ x ∉ K
Module: Mathlib.MeasureTheory.Integral.Indicator

Goal state before:
pp: case refine_4
α : Type u_1
inst✝¹ : MeasurableSpace α
A : Set α
ι : Type u_2
L : Filter ι
inst✝ : L.IsCountablyGenerated
As : ι → Set α
μ : Measure α
A_mble : MeasurableSet A
As_mble : ∀ (i : ι), MeasurableSet (As i)
B : Set α
B_mble : MeasurableSet B
B_finmeas : μ B ≠ ∞
As_le_B : ∀ᶠ (i : ι) in L, As i ⊆ B
h_lim : ∀ᵐ (x : α) ∂μ, ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ (As n).indicator 1 a) L (𝓝 (A.indicator 1 a))

Tactic:
simpa only [Pi.one_def, tendsto_indicator_const_apply_iff_eventually] using h_lim

Goal state after:
Module: Mathlib.MeasureTheory.Integral.MeanValue

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
s : Set α
f g : α → ℝ
μ : Measure α
hs_conn : IsConnected s
hs_meas : MeasurableSet s
hf : ContinuousOn f s
hg : IntegrableOn g s μ
hfg : IntegrableOn (fun x ↦ f x * g x) s μ
hg0 : ∀ᵐ (x : α) ∂μ.restrict s, 0 ≤ g x
ρ : α → ENNReal := fun x ↦ ENNReal.ofReal (g x)
ν : Measure α := μ.withDensity ρ
hρ_ae : AEMeasurable ρ (μ.restrict s)
hρ_top : ∀ᵐ (x : α) ∂μ.restrict s, ρ x < ⊤
h_toReal_ae : (fun x ↦ (ρ x).toReal) =ᵐ[μ.restrict s] g
heq : ∫ (x : α) in s, f x * g x ∂μ = ∫ (x : α) in s, f x ∂ν
h : (∫ (x : α) in s, 1 ∂μ.withDensity fun x ↦ ρ x) = ∫ (x : α) in s, (ρ x).toReal • 1 ∂μ
⊢ ∫ (x : α) in s, g x ∂μ = ∫ (x : α) in s, (ρ x).toReal ∂μ

Tactic:
rw [integral_congr_ae h_toReal_ae]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
a b c : E
ω : E → E →L[𝕜] F
γab : Path a b
γbc : Path b c
h₁ : CurveIntegrable ω γab
h₂ : CurveIntegrable ω γbc
instF : NormedSpace ℝ F := NormedSpace.restrictScalars ℝ 𝕜 F
⊢ (∫ (x : ℝ) in 2 * 0..2 * (1 / 2), curveIntegralFun ω γab x) +
      ∫ (x : ℝ) in 2 * (1 / 2)..2 * 1, curveIntegralFun ω γbc (x - 1) =
    ∫ᶜ (x : E) in γab, ω x + ∫ᶜ (x : E) in γbc, ω x

Tactic:
intervalIntegral.integral_comp_sub_right

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
a b c : E
ω : E → E →L[𝕜] F
γab : Path a b
γbc : Path b c
h₁ : CurveIntegrable ω γab
h₂ : CurveIntegrable ω γbc
instF : NormedSpace ℝ F := NormedSpace.restrictScalars ℝ 𝕜 F
⊢ (∫ (x : ℝ) in 2 * 0..2 * (1 / 2), curveIntegralFun ω γab x) +
      ∫ (x : ℝ) in 2 * (1 / 2) - 1..2 * 1 - 1, curveIntegralFun ω γbc x =
    ∫ᶜ (x : E) in γab, ω x + ∫ᶜ (x : E) in γbc, ω x
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral

Goal state before:
pp: case h₂
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
a b : E
C : ℝ
s : Set E
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hf : DiffContOnCl ℝ f s
hab : openSegment ℝ a b ⊆ s
hC : ∀ x ∈ s, ‖fderiv ℝ f x‖ ≤ C
hmem_s : ∀ t ∈ Ioo 0 1, (lineMap a b) t ∈ s
hC₀ : 0 ≤ C
hfc : ContinuousOn f (segment ℝ a b)
hfd : ∀ t ∈ Ioo 0 1, LineDifferentiableAt ℝ f ((lineMap a b) t) (b - a)
hfC : ∀ t ∈ Ioo 0 1, ‖lineDeriv ℝ f ((lineMap a b) t) (b - a)‖ ≤ C * ‖b - a‖
⊢ autoParam (volume {t | t ∈ Ioo 0 1 ∧ fderiv ℝ f ((lineMap a b) t) ≠ 0} ≠ ⊤) measureReal_mono._auto_1

pp: case h₁
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
a b : E
C : ℝ
s : Set E
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hf : DiffContOnCl ℝ f s
hab : openSegment ℝ a b ⊆ s
hC : ∀ x ∈ s, ‖fderiv ℝ f x‖ ≤ C
hmem_s : ∀ t ∈ Ioo 0 1, (lineMap a b) t ∈ s
hC₀ : 0 ≤ C
hfc : ContinuousOn f (segment ℝ a b)
hfd : ∀ t ∈ Ioo 0 1, LineDifferentiableAt ℝ f ((lineMap a b) t) (b - a)
hfC : ∀ t ∈ Ioo 0 1, ‖lineDeriv ℝ f ((lineMap a b) t) (b - a)‖ ≤ C * ‖b - a‖
t : ℝ
a✝ : t ∈ Ioo 0 1
⊢ fderiv ℝ f ((lineMap a b) t) = 0 → lineDeriv ℝ f ((lineMap a b) t) (b - a) = 0

Tactic:
· refine ne_top_of_le_ne_top ?_ (measure_mono inter_subset_left)
    simp

Goal state after:
case h₁
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
a b : E
C : ℝ
s : Set E
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hf : DiffContOnCl ℝ f s
hab : openSegment ℝ a b ⊆ s
hC : ∀ x ∈ s, ‖fderiv ℝ f x‖ ≤ C
hmem_s : ∀ t ∈ Ioo 0 1, (lineMap a b) t ∈ s
hC₀ : 0 ≤ C
hfc : ContinuousOn f (segment ℝ a b)
hfd : ∀ t ∈ Ioo 0 1, LineDifferentiableAt ℝ f ((lineMap a b) t) (b - a)
hfC : ∀ t ∈ Ioo 0 1, ‖lineDeriv ℝ f ((lineMap a b) t) (b - a)‖ ≤ C * ‖b - a‖
t : ℝ
a✝ : t ∈ Ioo 0 1
⊢ fderiv ℝ f ((lineMap a b) t) = 0 → lineDeriv ℝ f ((lineMap a b) t) (b - a) = 0
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
a b c d : E
γ₁ : Path a b
γ₂ : Path c d
s : Set (↑I × ↑I)
t : Set E
ω : E → E →L[ℝ] F
dω : E → E →L[ℝ] E →L[ℝ] F
φ : (↑γ₁).Homotopy ↑γ₂
hs : s.Countable
hφt : ∀ a_1 ∈ Ioo 0 1, ∀ b_1 ∈ Ioo 0 1, φ (a_1, b_1) ∈ t
hω : ∀ a_1 ∈ Ioo 0 1, ∀ b_1 ∈ Ioo 0 1, (a_1, b_1) ∉ s → HasFDerivWithinAt ω (dω (φ (a_1, b_1))) t (φ (a_1, b_1))
hωc : ContinuousOn ω (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t)
hdω_symm :
  ∀ a_1 ∈ Ioo 0 1,
    ∀ b_1 ∈ Ioo 0 1,
      (a_1, b_1) ∉ s →
        ∀ u ∈ tangentConeAt ℝ t (φ (a_1, b_1)),
          ∀ v ∈ tangentConeAt ℝ t (φ (a_1, b_1)), ((dω (φ (a_1, b_1))) u) v = ((dω (φ (a_1, b_1))) v) u
U : Set (ℝ × ℝ) := Ioo 0 1 ×ˢ Ioo 0 1
hU : U = Ioo 0 1 ×ˢ Ioo 0 1
hinterior : interior (Icc 0 1) = U
hunique : UniqueDiffOn ℝ (Icc 0 1)
hUopen : IsOpen U
hU_subset : U ⊆ Icc 0 1
hclosure : closure U = Icc 0 1
ψ : ℝ × ℝ → E := fun xy ↦ Set.IccExtend ⋯ (⇑(φ.extend xy.1)) xy.2
hcontdiff : ContDiffOn ℝ 2 ψ (Icc 0 1)
hψ : ψ = fun xy ↦ Set.IccExtend ⋯ (⇑(φ.extend xy.1)) xy.2
hψφ : ∀ (a_1 b_1 : ↑I), ψ (↑a_1, ↑b_1) = φ (a_1, b_1)
hψ_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ψ
hψUt : MapsTo ψ U t
dψ : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fderivWithin ℝ ψ (Icc 0 1)
s' : Set (ℝ × ℝ) := Prod.map Subtype.val Subtype.val '' s
hs' : s' = Prod.map Subtype.val Subtype.val '' s
x y : ↑I
⊢ (↑x, ↑y) ∈ s' ↔ (x, y) ∈ s

Tactic:
rw [hs', ← Prod.map_apply, Injective.mem_set_image]

Goal state after:
case hf
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
a b c d : E
γ₁ : Path a b
γ₂ : Path c d
s : Set (↑I × ↑I)
t : Set E
ω : E → E →L[ℝ] F
dω : E → E →L[ℝ] E →L[ℝ] F
φ : (↑γ₁).Homotopy ↑γ₂
hs : s.Countable
hφt : ∀ a_1 ∈ Ioo 0 1, ∀ b_1 ∈ Ioo 0 1, φ (a_1, b_1) ∈ t
hω : ∀ a_1 ∈ Ioo 0 1, ∀ b_1 ∈ Ioo 0 1, (a_1, b_1) ∉ s → HasFDerivWithinAt ω (dω (φ (a_1, b_1))) t (φ (a_1, b_1))
hωc : ContinuousOn ω (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t)
hdω_symm :
  ∀ a_1 ∈ Ioo 0 1,
    ∀ b_1 ∈ Ioo 0 1,
      (a_1, b_1) ∉ s →
        ∀ u ∈ tangentConeAt ℝ t (φ (a_1, b_1)),
          ∀ v ∈ tangentConeAt ℝ t (φ (a_1, b_1)), ((dω (φ (a_1, b_1))) u) v = ((dω (φ (a_1, b_1))) v) u
U : Set (ℝ × ℝ) := Ioo 0 1 ×ˢ Ioo 0 1
hU : U = Ioo 0 1 ×ˢ Ioo 0 1
hinterior : interior (Icc 0 1) = U
hunique : UniqueDiffOn ℝ (Icc 0 1)
hUopen : IsOpen U
hU_subset : U ⊆ Icc 0 1
hclosure : closure U = Icc 0 1
ψ : ℝ × ℝ → E := fun xy ↦ Set.IccExtend ⋯ (⇑(φ.extend xy.1)) xy.2
hcontdiff : ContDiffOn ℝ 2 ψ (Icc 0 1)
hψ : ψ = fun xy ↦ Set.IccExtend ⋯ (⇑(φ.extend xy.1)) xy.2
hψφ : ∀ (a_1 b_1 : ↑I), ψ (↑a_1, ↑b_1) = φ (a_1, b_1)
hψ_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ψ
hψUt : MapsTo ψ U t
dψ : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fderivWithin ℝ ψ (Icc 0 1)
s' : Set (ℝ × ℝ) := Prod.map Subtype.val Subtype.val '' s
hs' : s' = Prod.map Subtype.val Subtype.val '' s
x y : ↑I
⊢ Injective (Prod.map Subtype.val Subtype.val)
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
Λ : (X →C_c ℝ) →ₚ[ℝ] ℝ
inst✝ : LocallyCompactSpace X
f : X →C_c ℝ
hf : ∀ (x : X), 0 ≤ f x ∧ f x ≤ 1
V : Set X
hV : tsupport ⇑f ⊆ V
⊢ ENNReal.ofReal (Λ f) ≤ (rieszMeasure Λ) V

Tactic:
by
  apply le_trans _ (measure_mono hV)
  have := Content.measure_eq_content_of_regular (rieszContent (toNNRealLinear Λ))
    (contentRegular_rieszContent (toNNRealLinear Λ)) (⟨tsupport f, f.hasCompactSupport⟩)
  rw [← Compacts.coe_mk (tsupport f) f.hasCompactSupport, rieszMeasure, this, rieszContent,
    ENNReal.ofReal_eq_coe_nnreal (Λ.map_nonneg fun x ↦ (hf x).1), Content.mk_apply,
    ENNReal.coe_le_coe]
  apply le_iff_forall_pos_le_add.mpr
  intro _ hε
  obtain ⟨g, hg⟩ := exists_lt_rieszContentAux_add_pos (toNNRealLinear Λ)
    ⟨tsupport f, f.hasCompactSupport⟩ (Real.toNNReal_pos.mpr hε)
  simp_rw [NNReal.val_eq_coe, Real.toNNReal_coe] at hg
  refine (Λ.mono ?_).trans hg.2.le
  intro x
  by_cases hx : x ∈ tsupport f
  · simpa using le_trans (hf x).2 (hg.1 x hx)
  · simp [image_eq_zero_of_notMem_tsupport hx]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
Λ : (X →C_c ℝ) →ₚ[ℝ] ℝ
inst✝ : LocallyCompactSpace X
f : X →C_c ℝ
hf : ∀ (x : X), 0 ≤ f x
K : Set X
hK : IsCompact K
hfK : ∀ x ∈ K, f x = 1
p : NNReal
⊢ ENNReal.ofReal (Λ f) = ↑p → (rieszContent (toNNRealLinear Λ)).toFun { carrier := K, isCompact' := hK } ≤ p

Tactic:
hp

Goal state after:
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
Λ : (X →C_c ℝ) →ₚ[ℝ] ℝ
inst✝ : LocallyCompactSpace X
f : X →C_c ℝ
hf : ∀ (x : X), 0 ≤ f x
K : Set X
hK : IsCompact K
hfK : ∀ x ∈ K, f x = 1
p : NNReal
hp : ENNReal.ofReal (Λ f) = ↑p
⊢ (rieszContent (toNNRealLinear Λ)).toFun { carrier := K, isCompact' := hK } ≤ p
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case h
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
Λ : (X →C_c ℝ) →ₚ[ℝ] ℝ
inst✝ : LocallyCompactSpace X
f : X →C_c ℝ
hf : ∀ (x : X), 0 ≤ f x
K : Set X
hK : IsCompact K
hfK : ∀ x ∈ K, f x = 1
p : NNReal
hp : Λ f = ↑p
⊢ Λ f = ↑p

Tactic:
hp

Goal state after:
case h
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
Λ : (X →C_c ℝ) →ₚ[ℝ] ℝ
inst✝ : LocallyCompactSpace X
f : X →C_c ℝ
hf : ∀ (x : X), 0 ≤ f x
K : Set X
hK : IsCompact K
hfK : ∀ x ∈ K, f x = 1
p : NNReal
hp : Λ f = ↑p
⊢ ↑p = ↑p
Module: Mathlib.Topology.Separation.CompletelyRegular

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
⊢ (∀ (x : X) (x_1 : Set X), IsOpen[inst✝] x_1 → ¬x ∉ x_1 → ∃ f, Continuous[inst✝, _] f ∧ f x = 0 ∧ EqOn f 1 x_1ᶜ) =
    ?m.25

Tactic:
not_not

Goal state after:
X : Type u
inst✝ : TopologicalSpace X
⊢ (∀ (x : X) (x_1 : Set X), IsOpen[inst✝] x_1 → x ∈ x_1 → ∃ f, Continuous[inst✝, _] f ∧ f x = 0 ∧ EqOn f 1 x_1ᶜ) = ?m.25
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_2
mE : MeasurableSpace E
inst✝² : SeminormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
X : Type u_3
mX : MeasurableSpace X
μ : Measure X
inst✝ : BorelSpace E
f : X → E
hf : AEMeasurable f μ
r : ℝ
t : E
⊢ charFun (Measure.map ((fun x ↦ r • x) ∘ f) μ) t = charFun (Measure.map f μ) (r • t)

Tactic:
← AEMeasurable.map_map_of_aemeasurable,

Goal state after:
E : Type u_2
mE : MeasurableSpace E
inst✝² : SeminormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
X : Type u_3
mX : MeasurableSpace X
μ : Measure X
inst✝ : BorelSpace E
f : X → E
hf : AEMeasurable f μ
r : ℝ
t : E
⊢ charFun (Measure.map (fun x ↦ r • x) (Measure.map f μ)) t = charFun (Measure.map f μ) (r • t)

case hg
E : Type u_2
mE : MeasurableSpace E
inst✝² : SeminormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
X : Type u_3
mX : MeasurableSpace X
μ : Measure X
inst✝ : BorelSpace E
f : X → E
hf : AEMeasurable f μ
r : ℝ
t : E
⊢ AEMeasurable (fun x ↦ r • x) (Measure.map f μ)

case hf
E : Type u_2
mE : MeasurableSpace E
inst✝² : SeminormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
X : Type u_3
mX : MeasurableSpace X
μ : Measure X
inst✝ : BorelSpace E
f : X → E
hf : AEMeasurable f μ
r : ℝ
t : E
⊢ AEMeasurable f μ
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
f : Ω →ᵇ ℝ
A : Set ℝ
A_finmeas : volume A ≠ ∞
μ : ProbabilityMeasure Ω
i : ℝ
⊢ ∀ᵐ (a : ℝ) ∂volume.restrict A, ‖↑(μ (thickening i {a_1 | a ≤ f a_1}))‖ ≤ 1 a

Tactic:
apply Eventually.of_forall (fun t ↦ ?_)

Goal state after:
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
f : Ω →ᵇ ℝ
A : Set ℝ
A_finmeas : volume A ≠ ∞
μ : ProbabilityMeasure Ω
i t : ℝ
⊢ ‖↑(μ (thickening i {a | t ≤ f a}))‖ ≤ 1 t
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: case h_lim
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
f : Ω →ᵇ ℝ
A : Set ℝ
A_finmeas : volume A ≠ ∞
μ : ProbabilityMeasure Ω
⊢ ∀ᵐ (a : ℝ) ∂volume.restrict A,
    Tendsto (fun n ↦ ↑(μ (thickening n {a_1 | a ≤ f a_1}))) (𝓝[>] 0) (𝓝 ↑(μ {a_1 | a ≤ f a_1}))

Tactic:
apply Eventually.of_forall (fun t ↦ ?_)

Goal state after:
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
f : Ω →ᵇ ℝ
A : Set ℝ
A_finmeas : volume A ≠ ∞
μ : ProbabilityMeasure Ω
t : ℝ
⊢ Tendsto (fun n ↦ ↑(μ (thickening n {a | t ≤ f a}))) (𝓝[>] 0) (𝓝 ↑(μ {a | t ≤ f a}))