mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.Order.Support

Goal state before:
pp: X : Type u_1
M : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : One M
inst✝ : SemilatticeInf M
f g : X → M
hf : HasCompactMulSupport f
hg : HasCompactMulSupport g
⊢ HasCompactMulSupport (f ⊓ g)

Tactic:
apply IsCompact.of_isClosed_subset (IsCompact.union hf hg) (isClosed_mulTSupport _)

Goal state after:
X : Type u_1
M : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : One M
inst✝ : SemilatticeInf M
f g : X → M
hf : HasCompactMulSupport f
hg : HasCompactMulSupport g
⊢ mulTSupport (f ⊓ g) ⊆ mulTSupport f ∪ mulTSupport g
Module: Mathlib.Topology.Algebra.Order.Support

Goal state before:
pp: X : Type u_1
M : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : One M
inst✝ : SemilatticeInf M
f g : X → M
hf : HasCompactMulSupport f
hg : HasCompactMulSupport g
⊢ HasCompactMulSupport (f ⊓ g)

Tactic:
apply IsCompact.of_isClosed_subset (IsCompact.union hf hg) (isClosed_mulTSupport _)
  rw [mulTSupport, mulTSupport, mulTSupport, ← closure_union]
  apply closure_mono
  exact Function.mulSupport_inf f g

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
hR : 0 < R
f : ℂ → E
z w : ℂ
hf : ContinuousOn f (sphere z R)
hw : w ∈ ball z R
⊢ Continuous (circleTransformDeriv R z w f)

Tactic:
circleTransformDeriv_eq

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
R : ℝ
hR : 0 < R
f : ℂ → E
z w : ℂ
hf : ContinuousOn f (sphere z R)
hw : w ∈ ball z R
⊢ Continuous fun θ ↦ (circleMap z R θ - w)⁻¹ • circleTransform R z w f θ
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))
ε : ℝ≥0∞
hε : ε > 0
hfinε : ε ≠ ∞
hμ : μ ≠ 0
hε' : 0 < ε / 3
Eg : Set α
hmEg : MeasurableSet Eg
hμEg : μ Eg < ∞
hgε : eLpNorm (Egᶜ.indicator g) p μ < ε / 3
Ef : Set α
hmEf : MeasurableSet Ef
hμEf : μ Ef < ∞
hfε : ∀ (i : ℕ), eLpNorm (Efᶜ.indicator (f i)) p μ ≤ ε / 3
E : Set α := Ef ∪ Eg
hmE : MeasurableSet E
hfmE : μ E < ∞
hgE' : MemLp g p (μ.restrict E)
huiE : UnifIntegrable f p (μ.restrict E)
hfgE : ∀ᵐ (x : α) ∂μ.restrict E, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ffmE : Fact (μ E < ∞)
hInner : ∀ ε > 0, ∃ N, ∀ n ≥ N, eLpNorm (f n - g) p (μ.restrict E) ≤ ε
N : ℕ
hfngε : ∀ n ≥ N, eLpNorm (f n - g) p (μ.restrict E) ≤ ε / 3
n : ℕ
hn : n ≥ N
hmfngE : AEStronglyMeasurable (E.indicator (f n - g)) μ
hfngEε : eLpNorm (E.indicator (f n - g)) p μ ≤ ε / 3
hmgEc : AEStronglyMeasurable (Eᶜ.indicator g) μ
hgEcε : eLpNorm (Eᶜ.indicator g) p μ ≤ ε / 3
hmfnEc : AEStronglyMeasurable (Eᶜ.indicator (f n)) μ
⊢ eLpNorm ((Ef ∪ Eg)ᶜ.indicator (f n)) p μ ≤ eLpNorm (Egᶜ.indicator (Efᶜ.indicator (f n))) p μ

Tactic:
rw [compl_union, inter_comm, ← indicator_indicator]

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

Goal state before:
pp: α : Type u_2
inst✝ : TopologicalSpace α
f : α →C_c ℝ
hf : 0 ≤ f
⊢ (-f).nnrealPart = 0

Tactic:
ext x
  simpa using hf x

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

Goal state before:
pp: case inl
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
ha : 0 ≤ a
⊢ ↑(Λ (a.toNNReal • f.nnrealPart)) - ↑(Λ (a • -f).nnrealPart) = a * (↑(Λ f.nnrealPart) - ↑(Λ (-f).nnrealPart))

Tactic:
nnrealPart_smul_pos (-f) ha

Goal state after:
case inl
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
ha : 0 ≤ a
⊢ ↑(Λ (a.toNNReal • f.nnrealPart)) - ↑(Λ (a.toNNReal • (-f).nnrealPart)) =
    a * (↑(Λ f.nnrealPart) - ↑(Λ (-f).nnrealPart))
Module: Mathlib.MeasureTheory.Integral.Gamma

Goal state before:
pp: p b : ℝ
hp : 1 ≤ p
hb : 0 < b
⊢ ∫ (x : ℂ), rexp (-b * ‖x‖ ^ p) = π * b ^ (-2 / p) * Real.Gamma (2 / p + 1)

Tactic:
convert! (integral_rpow_mul_exp_neg_mul_rpow hp (by linarith : (-2 : ℝ) < 0)) hb using 1

Goal state after:
case e'_2
p b : ℝ
hp : 1 ≤ p
hb : 0 < b
⊢ ∫ (x : ℂ), rexp (-b * ‖x‖ ^ p) = ∫ (x : ℂ), ‖x‖ ^ 0 * rexp (-b * ‖x‖ ^ p)

case e'_3
p b : ℝ
hp : 1 ≤ p
hb : 0 < b
⊢ π * b ^ (-2 / p) * Real.Gamma (2 / p + 1) = 2 * π / p * b ^ (-(0 + 2) / p) * Real.Gamma ((0 + 2) / p)
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
hg1 : ∫ (x : α) in s, g x ∂μ = ∫ (x : α) in s, 1 ∂ν
hν0 : ¬∫ (x : α) in s, 1 ∂ν = 0
hν0' : (ν s).toReal ≠ 0
hνfin : ν s ≠ ⊤
hν0'' : ν s ≠ 0
hmul_ae : (fun x ↦ (ρ x).toReal * f x) =ᵐ[μ.restrict s] fun x ↦ f x * g x
h_IntOn : IntegrableOn (fun x ↦ (ρ x).toReal * f x) s μ
h_Int : Integrable f ((μ.restrict s).withDensity ρ)
⊢ IntegrableOn f s ν

Tactic:
have hνrs : ν.restrict s = (μ.restrict s).withDensity ρ := by
        ext t ht
        simp [ht, ν, hs_meas]

Goal state after:
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
hg1 : ∫ (x : α) in s, g x ∂μ = ∫ (x : α) in s, 1 ∂ν
hν0 : ¬∫ (x : α) in s, 1 ∂ν = 0
hν0' : (ν s).toReal ≠ 0
hνfin : ν s ≠ ⊤
hν0'' : ν s ≠ 0
hmul_ae : (fun x ↦ (ρ x).toReal * f x) =ᵐ[μ.restrict s] fun x ↦ f x * g x
h_IntOn : IntegrableOn (fun x ↦ (ρ x).toReal * f x) s μ
h_Int : Integrable f ((μ.restrict s).withDensity ρ)
hνrs : ν.restrict s = (μ.restrict s).withDensity ρ
⊢ IntegrableOn f s ν
Module: Mathlib.MeasureTheory.Integral.MeanValue

Goal state before:
pp: case neg
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
hg1 : ∫ (x : α) in s, g x ∂μ = ∫ (x : α) in s, 1 ∂ν
hν0 : ¬∫ (x : α) in s, 1 ∂ν = 0
hν0' : (ν s).toReal ≠ 0
hνfin : ν s ≠ ⊤
hν0'' : ν s ≠ 0
hint : IntegrableOn f s ν
⊢ ∃ c ∈ s, ∫ (x : α) in s, f x * g x ∂μ = f c * ∫ (x : α) in s, g x ∂μ

Tactic:
obtain ⟨c, hc, h_ave⟩ := exists_eq_setAverage hs_conn hf hint hνfin hν0''

Goal state after:
case neg
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
hg1 : ∫ (x : α) in s, g x ∂μ = ∫ (x : α) in s, 1 ∂ν
hν0 : ¬∫ (x : α) in s, 1 ∂ν = 0
hν0' : (ν s).toReal ≠ 0
hνfin : ν s ≠ ⊤
hν0'' : ν s ≠ 0
hint : IntegrableOn f s ν
c : α
hc : c ∈ s
h_ave : f c = ⨍ (x : α) in s, f x ∂ν
⊢ ∃ c ∈ s, ∫ (x : α) in s, f x * g x ∂μ = f c * ∫ (x : α) in s, g x ∂μ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C : ℝ
hab : a ≤ b
hfc : ContinuousOn f (Icc a b)
hfd : DifferentiableOn ℝ f (Ioo a b)
hnorm : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ C
s : Set ℝ := toMeasurable volume {x | deriv f x ≠ 0}
hsm : MeasurableSet s
⊢ C * (volume ({x | deriv f x ≠ 0} ∩ Ioo a b)).toReal = C * (volume {x | x ∈ Ioo a b ∧ deriv f x ≠ 0}).toReal

Tactic:
simp only [inter_def, mem_ofPred_eq, and_comm]

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 : ∀ x ∈ s, 0 ≤ g x
⊢ ∀ᵐ (x : α) ∂μ.restrict s, 0 ≤ g x

Tactic:
rw [ae_restrict_iff' hs_meas]

Goal state after:
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 : ∀ x ∈ s, 0 ≤ g x
⊢ ∀ᵐ (x : α) ∂μ, x ∈ s → 0 ≤ g x
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Basic

Goal state before:
pp: case h₁.refine_2
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
a : E
s : Set E
ω : E → E →L[𝕜] F
hs : Convex ℝ s
hω : ∀ᶠ (x : E) in 𝓝[s] a, ContinuousWithinAt ω s x
ha : a ∈ s
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₀ : δ > 0
hδ : ball a δ ∩ s ⊆ {z | ContinuousWithinAt ω s z ∧ dist (ω z) (ω a) ≤ ε}
b : E
hb : b ∈ ball a δ
hbs : b ∈ s
hsub : [a -[ℝ] b] ⊆ ball a δ ∩ s
t : ℝ
ht : t ∈ Icc 0 1
⊢ ⇑(lineMap a b) '' Icc 0 1 ⊆ s

Tactic:
← segment_eq_image_lineMap

Goal state after:
case h₁.refine_2
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
a : E
s : Set E
ω : E → E →L[𝕜] F
hs : Convex ℝ s
hω : ∀ᶠ (x : E) in 𝓝[s] a, ContinuousWithinAt ω s x
ha : a ∈ s
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₀ : δ > 0
hδ : ball a δ ∩ s ⊆ {z | ContinuousWithinAt ω s z ∧ dist (ω z) (ω a) ≤ ε}
b : E
hb : b ∈ ball a δ
hbs : b ∈ s
hsub : [a -[ℝ] b] ⊆ ball a δ ∩ s
t : ℝ
ht : t ∈ Icc 0 1
⊢ [a -[ℝ] b] ⊆ s
Module: Mathlib.MeasureTheory.Integral.Regular

Goal state before:
pp: X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
k : Set X
hk : IsCompact k
μ : Measure X
inst✝³ : IsFiniteMeasureOnCompacts μ
inst✝² : μ.InnerRegularCompactLTTop
inst✝¹ : LocallyCompactSpace X
inst✝ : RegularSpace X
f : X → ℝ
f_cont : Continuous f
f_comp : HasCompactSupport f
fk : EqOn f 1 k
f_nonneg : 0 ≤ f
x : X
hx : x ∈ k
⊢ 1 ≤ f x

Tactic:
simp [fk hx]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral

Goal state before:
pp: 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 : E
r : ℝ
hr : 0 ≤ r
hdf : ∀ᶠ (x : E) in 𝓝 a, DifferentiableAt ℝ f x
hderiv : fderiv ℝ f =O[𝓝 a] fun x ↦ ‖x - a‖ ^ r
⊢ (fun x ↦ f x - f a) =O[𝓝 a] fun x ↦ ‖x - a‖ ^ (r + 1)

Tactic:
rcases hderiv.exists_pos with ⟨C, hC₀, hC⟩

Goal state after:
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 : E
r : ℝ
hr : 0 ≤ r
hdf : ∀ᶠ (x : E) in 𝓝 a, DifferentiableAt ℝ f x
hderiv : fderiv ℝ f =O[𝓝 a] fun x ↦ ‖x - a‖ ^ r
C : ℝ
hC₀ : C > 0
hC : Asymptotics.IsBigOWith C (𝓝 a) (fderiv ℝ f) fun x ↦ ‖x - a‖ ^ r
⊢ (fun x ↦ f x - f a) =O[𝓝 a] fun x ↦ ‖x - a‖ ^ (r + 1)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule

Goal state before:
pp: f : ℝ → ℝ
N : ℕ
N_nonzero : 0 < N
a b : ℝ
⊢ (b - a) / ↑N * ((f a + f b) / 2 + ∑ k ∈ Finset.range (N - 1), f (a + (↑k + 1) * (b - a) / ↑N)) =
    (b - a) / ↑N * ((f a + f b) / 2 + ∑ k ∈ Finset.range (N - 1), f (b + (↑k + 1) * (a - b) / ↑N))

Tactic:
← sum_range_reflect

Goal state after:
f : ℝ → ℝ
N : ℕ
N_nonzero : 0 < N
a b : ℝ
⊢ (b - a) / ↑N * ((f a + f b) / 2 + ∑ j ∈ Finset.range (N - 1), f (a + (↑(N - 1 - 1 - j) + 1) * (b - a) / ↑N)) =
    (b - a) / ↑N * ((f a + f b) / 2 + ∑ k ∈ Finset.range (N - 1), f (b + (↑k + 1) * (a - b) / ↑N))
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule

Goal state before:
pp: case inl
f : ℝ → ℝ
N : ℕ
a h : ℝ
N_nonzero : 0 < N
h_f_int : IntervalIntegrable f volume a (a + ↑N * h)
k✝ : ℕ
hk✝ : k✝ < N
h_neg : h ≤ 0
k : ℕ
hk : ↑k ≤ ↑N
⊢ a + ↑k * h ∈ [[a, a + ↑N * h]]

pp: case inr
f : ℝ → ℝ
N : ℕ
a h : ℝ
N_nonzero : 0 < N
h_f_int : IntervalIntegrable f volume a (a + ↑N * h)
k✝ : ℕ
hk✝ : k✝ < N
h_pos : 0 ≤ h
k : ℕ
hk : ↑k ≤ ↑N
⊢ a + ↑k * h ∈ [[a, a + ↑N * h]]

Tactic:
· simpa [Set.mem_uIcc] using .inr
        ⟨mul_le_mul_of_nonpos_right hk h_neg, mul_nonpos_of_nonneg_of_nonpos k.cast_nonneg h_neg⟩

Goal state after:
case inr
f : ℝ → ℝ
N : ℕ
a h : ℝ
N_nonzero : 0 < N
h_f_int : IntervalIntegrable f volume a (a + ↑N * h)
k✝ : ℕ
hk✝ : k✝ < N
h_pos : 0 ≤ h
k : ℕ
hk : ↑k ≤ ↑N
⊢ a + ↑k * h ∈ [[a, a + ↑N * h]]
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : Disjoint s₀ s₁
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
soopen : ∀ (j : Fin 2), IsOpen (so j)
hsot : t ⊆ ⋃ j, so j
⊢ ∃ f₀ f₁, EqOn (⇑f₀) 1 s₀ ∧ EqOn (⇑f₁) 1 s₁ ∧ EqOn (⇑(f₀ + f₁)) 1 t

Tactic:
obtain ⟨f, f_supp_in_so, sum_f_one_on_t, f_in_icc, f_hcs⟩ :=
    exists_continuous_sum_one_of_isOpen_isCompact soopen t_compact hsot

Goal state after:
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : Disjoint s₀ s₁
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
soopen : ∀ (j : Fin 2), IsOpen (so j)
hsot : t ⊆ ⋃ j, so j
f : Fin 2 → C(X, ℝ)
f_supp_in_so : ∀ (i : Fin 2), tsupport ⇑(f i) ⊆ so i
sum_f_one_on_t : EqOn (∑ i, ⇑(f i)) 1 t
f_in_icc : ∀ (i : Fin 2) (x : X), (f i) x ∈ Icc 0 1
f_hcs : ∀ (i : Fin 2), HasCompactSupport ⇑(f i)
⊢ ∃ f₀ f₁, EqOn (⇑f₀) 1 s₀ ∧ EqOn (⇑f₁) 1 s₁ ∧ EqOn (⇑(f₀ + f₁)) 1 t
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case neg
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
this :
  (rieszContent (toNNRealLinear Λ)).measure ↑{ carrier := tsupport ⇑f, isCompact' := ⋯ } =
    (rieszContent (toNNRealLinear Λ)) { carrier := tsupport ⇑f, isCompact' := ⋯ }
ε✝ : NNReal
hε : 0 < ε✝
g : X →C_c NNReal
hg :
  (∀ x ∈ { carrier := tsupport ⇑f, isCompact' := ⋯ }, 1 ≤ g x) ∧
    (toNNRealLinear Λ) g < rieszContentAux (toNNRealLinear Λ) { carrier := tsupport ⇑f, isCompact' := ⋯ } + ε✝
x : X
hx : x ∉ tsupport ⇑f
⊢ f x ≤ (toRealLinearMap g) x

Tactic:
simp [image_eq_zero_of_notMem_tsupport hx]

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

Goal state before:
pp: X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : T2Space X
inst✝⁴ : LocallyCompactSpace X
inst✝³ : MeasurableSpace X
inst✝² : BorelSpace X
μ ν : Measure X
inst✝¹ : μ.Regular
inst✝ : ν.Regular
hμν : integralLinearMap μ = integralLinearMap ν
f : X →C_c ℝ≥0
⊢ ∫ (x : X), ↑(f x) ∂μ = ∫ (x : X), ↑(f x) ∂ν

Tactic:
by simpa using congr(($hμν f).toReal)

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.Invariants

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f g : α → α
s : Set α
hs : MeasurableSet s
⊢ f ∘ g ⁻¹' s = s

Tactic:
rw [preimage_comp, hs.1.2, hs.2.2]

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

Goal state before:
pp: case h.refine_2
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
a ε : ℝ
hε : 0 < ε
N : ℕ
hf : range ⇑f ⊆ Ioo a (a + ↑N * ε)
b : ℝ := a + ↑N * ε
y : Fin N → ℝ := fun n ↦ a + ε * (↑↑n + 1)
hy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m
E : Fin N → Set X := fun n ↦ ⇑f ⁻¹' Ioc (y n - ε) (y n) ∩ tsupport ⇑f
⊢ univ.PairwiseDisjoint E

pp: case h.refine_3
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
a ε : ℝ
hε : 0 < ε
N : ℕ
hf : range ⇑f ⊆ Ioo a (a + ↑N * ε)
b : ℝ := a + ↑N * ε
y : Fin N → ℝ := fun n ↦ a + ε * (↑↑n + 1)
hy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m
E : Fin N → Set X := fun n ↦ ⇑f ⁻¹' Ioc (y n - ε) (y n) ∩ tsupport ⇑f
⊢ ∀ (n : Fin N), ∀ x ∈ E n, a + ε * ↑↑n < f x ∧ f x ≤ a + ε * (↑↑n + 1)

pp: case h.refine_4
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
a ε : ℝ
hε : 0 < ε
N : ℕ
hf : range ⇑f ⊆ Ioo a (a + ↑N * ε)
b : ℝ := a + ↑N * ε
y : Fin N → ℝ := fun n ↦ a + ε * (↑↑n + 1)
hy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m
E : Fin N → Set X := fun n ↦ ⇑f ⁻¹' Ioc (y n - ε) (y n) ∩ tsupport ⇑f
⊢ ∀ (n : Fin N), MeasurableSet (E n)

Tactic:
· -- The sets `E n` are pairwise disjoint.
    intro m _ n _ hmn
    apply Disjoint.preimage
    simp_rw [mem_preimage, mem_Ioc, disjoint_left]
    intro x hx
    rw [mem_ofPred_eq, and_assoc] at hx
    simp_rw [mem_ofPred_eq, not_and_or, not_lt, not_le, or_assoc]
    rcases (by lia : m < n ∨ n < m) with hc | hc
    · left
      exact le_trans hx.2.1 (le_tsub_of_add_le_right (hy hc))
    · right; left
      exact lt_of_le_of_lt (le_tsub_of_add_le_right (hy hc)) hx.1

Goal state after:
case h.refine_3
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
a ε : ℝ
hε : 0 < ε
N : ℕ
hf : range ⇑f ⊆ Ioo a (a + ↑N * ε)
b : ℝ := a + ↑N * ε
y : Fin N → ℝ := fun n ↦ a + ε * (↑↑n + 1)
hy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m
E : Fin N → Set X := fun n ↦ ⇑f ⁻¹' Ioc (y n - ε) (y n) ∩ tsupport ⇑f
⊢ ∀ (n : Fin N), ∀ x ∈ E n, a + ε * ↑↑n < f x ∧ f x ≤ a + ε * (↑↑n + 1)

case h.refine_4
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
a ε : ℝ
hε : 0 < ε
N : ℕ
hf : range ⇑f ⊆ Ioo a (a + ↑N * ε)
b : ℝ := a + ↑N * ε
y : Fin N → ℝ := fun n ↦ a + ε * (↑↑n + 1)
hy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m
E : Fin N → Set X := fun n ↦ ⇑f ⁻¹' Ioc (y n - ε) (y n) ∩ tsupport ⇑f
⊢ ∀ (n : Fin N), MeasurableSet (E n)
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
μ : Measure X := rieszMeasure Λ
K : Set X := tsupport ⇑f
ε : ℝ
hε : 0 < ε
a b : ℝ
hab : a < b ∧ range ⇑f ⊆ Ioo a b
N : ℕ
hN : 0 < N
ε' : ℝ := (b - a) / ↑N
hε' : 0 < ε' ∧ ε' * (2 * μ.real K + |a| + b + ε') ≤ ε
E : Fin N → Set X
hE :
  tsupport ⇑f = ⋃ j, E j ∧
    univ.PairwiseDisjoint E ∧
      (∀ (n : Fin N), ∀ x ∈ E n, a + ε' * ↑↑n < f x ∧ f x ≤ a + ε' * (↑↑n + 1)) ∧ ∀ (n : Fin N), MeasurableSet (E n)
y : Fin N → ℝ := fun n ↦ a + ε' * (↑↑n + 1)
n : Fin N
h : E n ⊆ tsupport ⇑f
⊢ (rieszContent (toNNRealLinear Λ)).measure (tsupport f.toFun) < ∞

Tactic:
Content.measure_apply _ f.2.measurableSet

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
μ : Measure X := rieszMeasure Λ
K : Set X := tsupport ⇑f
ε : ℝ
hε : 0 < ε
a b : ℝ
hab : a < b ∧ range ⇑f ⊆ Ioo a b
N : ℕ
hN : 0 < N
ε' : ℝ := (b - a) / ↑N
hε' : 0 < ε' ∧ ε' * (2 * μ.real K + |a| + b + ε') ≤ ε
E : Fin N → Set X
hE :
  tsupport ⇑f = ⋃ j, E j ∧
    univ.PairwiseDisjoint E ∧
      (∀ (n : Fin N), ∀ x ∈ E n, a + ε' * ↑↑n < f x ∧ f x ≤ a + ε' * (↑↑n + 1)) ∧ ∀ (n : Fin N), MeasurableSet (E n)
y : Fin N → ℝ := fun n ↦ a + ε' * (↑↑n + 1)
n : Fin N
h : E n ⊆ tsupport ⇑f
⊢ (rieszContent (toNNRealLinear Λ)).outerMeasure (tsupport f.toFun) < ∞
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case neg
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
μ : Measure X := rieszMeasure Λ
K : Set X := tsupport ⇑f
ε : ℝ
hε : 0 < ε
a b : ℝ
hab : a < b ∧ range ⇑f ⊆ Ioo a b
N : ℕ
hN : 0 < N
ε' : ℝ := (b - a) / ↑N
hε' : 0 < ε' ∧ ε' * (2 * μ.real K + |a| + b + ε') ≤ ε
E : Fin N → Set X
hE :
  tsupport ⇑f = ⋃ j, E j ∧
    univ.PairwiseDisjoint E ∧
      (∀ (n : Fin N), ∀ x ∈ E n, a + ε' * ↑↑n < f x ∧ f x ≤ a + ε' * (↑↑n + 1)) ∧ ∀ (n : Fin N), MeasurableSet (E n)
y : Fin N → ℝ := fun n ↦ a + ε' * (↑↑n + 1)
hE' : ∀ (n : Fin N), μ (E n) ≠ ∞
V : Fin N → Opens X
hV : ∀ (n : Fin N), E n ⊆ ↑(V n) ∧ (∀ x ∈ V n, f x < y n + ε') ∧ μ ↑(V n) ≤ μ (E n) + ENNReal.ofReal (ε' / ↑N)
g : Fin N → X →C_c ℝ
hg :
  (∀ (n : Fin N), tsupport ⇑(g n) ⊆ (V n).carrier) ∧
    EqOn (∑ n, ⇑(g n)) 1 (tsupport f.toFun) ∧
      (∀ (n : Fin N) (x : X), (g n) x ∈ Icc 0 1) ∧ ∀ (n : Fin N), HasCompactSupport ⇑(g n)
x : X
hx : x ∉ tsupport ⇑f
⊢ f x = (∑ i, ⇑(g i)) x * f x

Tactic:
simp [image_eq_zero_of_notMem_tsupport hx]

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

Goal state before:
pp: case neg
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
μ : Measure X := rieszMeasure Λ
K : Set X := tsupport ⇑f
ε : ℝ
hε : 0 < ε
a b : ℝ
hab : a < b ∧ range ⇑f ⊆ Ioo a b
N : ℕ
hN : 0 < N
ε' : ℝ := (b - a) / ↑N
hε' : 0 < ε' ∧ ε' * (2 * μ.real K + |a| + b + ε') ≤ ε
E : Fin N → Set X
hE :
  tsupport ⇑f = ⋃ j, E j ∧
    univ.PairwiseDisjoint E ∧
      (∀ (n : Fin N), ∀ x ∈ E n, a + ε' * ↑↑n < f x ∧ f x ≤ a + ε' * (↑↑n + 1)) ∧ ∀ (n : Fin N), MeasurableSet (E n)
y : Fin N → ℝ := fun n ↦ a + ε' * (↑↑n + 1)
hE' : ∀ (n : Fin N), μ (E n) ≠ ∞
V : Fin N → Opens X
hV : ∀ (n : Fin N), E n ⊆ ↑(V n) ∧ (∀ x ∈ V n, f x < y n + ε') ∧ μ ↑(V n) ≤ μ (E n) + ENNReal.ofReal (ε' / ↑N)
g : Fin N → X →C_c ℝ
hg :
  (∀ (n : Fin N), tsupport ⇑(g n) ⊆ (V n).carrier) ∧
    EqOn (∑ n, ⇑(g n)) 1 (tsupport f.toFun) ∧
      (∀ (n : Fin N) (x : X), (g n) x ∈ Icc 0 1) ∧ ∀ (n : Fin N), HasCompactSupport ⇑(g n)
n : Fin N
hn : n ∈ Finset.univ
x : X
hx : x ∉ tsupport ⇑(g n)
⊢ (g n • f) x ≤ ((y n + ε') • g n) x

Tactic:
simp [image_eq_zero_of_notMem_tsupport hx]

Goal state after: