mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.CircleTransform

Goal state before:
pp: R : ℝ
hR : 0 < R
z x : ℂ
f : ℂ → ℂ
hx : x ∈ ball z R
hf : ContinuousOn f (sphere z R)
r : ℝ
hr : r < R
hrx : x ∈ ball z r
ε' : ℝ
hε' : ε' > 0
H : ball x ε' ⊆ ball z r
a : ℂ
b : ℝ
ha : (a, b).1 ∈ closedBall z r
hb : (a, b).2 ∈ [[0, 2 * π]]
hab :
  ∀ (y : ↑(closedBall z r ×ˢ [[0, 2 * π]])),
    ‖circleTransformBoundingFunction R z ↑y‖ ≤ ‖circleTransformBoundingFunction R z ↑⟨(a, b), ⋯⟩‖
V : ℝ → ℂ → ℂ := fun θ w ↦ circleTransformDeriv R z w (fun x ↦ 1) θ
X : ℂ
HX2 : IsMaxOn (fun x ↦ ‖f x‖) (sphere z R) X
y : ℝ
v : ℂ
hv : v ∈ ball x ε'
⊢ ‖circleTransformDeriv R z v f y‖ ≤ ‖V b a‖ * ‖f X‖

Tactic:
obtain ⟨y1, hy1, hfun⟩ :=
    Periodic.exists_mem_Ico₀ (circleTransformDeriv_periodic R z v f) Real.two_pi_pos y

Goal state after:
R : ℝ
hR : 0 < R
z x : ℂ
f : ℂ → ℂ
hx : x ∈ ball z R
hf : ContinuousOn f (sphere z R)
r : ℝ
hr : r < R
hrx : x ∈ ball z r
ε' : ℝ
hε' : ε' > 0
H : ball x ε' ⊆ ball z r
a : ℂ
b : ℝ
ha : (a, b).1 ∈ closedBall z r
hb : (a, b).2 ∈ [[0, 2 * π]]
hab :
  ∀ (y : ↑(closedBall z r ×ˢ [[0, 2 * π]])),
    ‖circleTransformBoundingFunction R z ↑y‖ ≤ ‖circleTransformBoundingFunction R z ↑⟨(a, b), ⋯⟩‖
V : ℝ → ℂ → ℂ := fun θ w ↦ circleTransformDeriv R z w (fun x ↦ 1) θ
X : ℂ
HX2 : IsMaxOn (fun x ↦ ‖f x‖) (sphere z R) X
y : ℝ
v : ℂ
hv : v ∈ ball x ε'
y1 : ℝ
hy1 : y1 ∈ Ico 0 (2 * π)
hfun : circleTransformDeriv R z v f y = circleTransformDeriv R z v f y1
⊢ ‖circleTransformDeriv R z v f y‖ ≤ ‖V b a‖ * ‖f X‖
Module: Mathlib.MeasureTheory.Integral.Gamma

Goal state before:
pp: p q : ℝ
hp : 1 ≤ p
hq : -2 < q
⊢ ∫ (p_1 : ℝ × ℝ) in Ioi 0 ×ˢ Ioo (-π) π,
      p_1.1 • (‖↑Complex.polarCoord.symm p_1‖ ^ q * rexp (-‖↑Complex.polarCoord.symm p_1‖ ^ p)) =
    ∫ (x : ℝ × ℝ) in Ioi 0 ×ˢ Ioo (-π) π, x.1 * (|x.1| ^ q * rexp (-|x.1| ^ p))

Tactic:
Complex.norm_polarCoord_symm,

Goal state after:
p q : ℝ
hp : 1 ≤ p
hq : -2 < q
⊢ ∫ (p_1 : ℝ × ℝ) in Ioi 0 ×ˢ Ioo (-π) π, p_1.1 • (|p_1.1| ^ q * rexp (-|p_1.1| ^ p)) =
    ∫ (x : ℝ × ℝ) in Ioi 0 ×ˢ Ioo (-π) π, x.1 * (|x.1| ^ q * rexp (-|x.1| ^ p))
Module: Mathlib.MeasureTheory.Integral.Gamma

Goal state before:
pp: p q : ℝ
hp : 1 ≤ p
hq : -2 < q
⊢ ∫ (p_1 : ℝ × ℝ) in Ioi 0 ×ˢ Ioo (-π) π,
      p_1.1 • (‖↑Complex.polarCoord.symm p_1‖ ^ q * rexp (-‖↑Complex.polarCoord.symm p_1‖ ^ p)) =
    ∫ (x : ℝ × ℝ) in Ioi 0 ×ˢ Ioo (-π) π, x.1 * (|x.1| ^ q * rexp (-|x.1| ^ p))

Tactic:
simp_rw [Complex.norm_polarCoord_symm, smul_eq_mul]

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

Goal state before:
pp: p q : ℝ
hp : 1 ≤ p
hq : -2 < q
⊢ 2 * π * ∫ (x : ℝ) in Ioi 0, x ^ (q + 1) * rexp (-x ^ p) = 2 * π / p * Real.Gamma ((q + 2) / p)

Tactic:
rw [_root_.integral_rpow_mul_exp_neg_rpow (by linarith) (by linarith), add_assoc,
        one_add_one_eq_two]
      ring

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 : E
ω : E → E →L[𝕜] F
γ : Path a b
h : CurveIntegrable ω γ.symm
⊢ CurveIntegrable ω γ

Tactic:
simpa using h.symm

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

Goal state before:
pp: p q b : ℝ
hp : 1 ≤ p
hq : -2 < q
hb : 0 < b
⊢ ∫ (p_1 : ℝ × ℝ) in Ioi 0 ×ˢ Ioo (-π) π,
      p_1.1 • (‖↑Complex.polarCoord.symm p_1‖ ^ q * rexp (-b * ‖↑Complex.polarCoord.symm p_1‖ ^ p)) =
    ∫ (x : ℝ × ℝ) in Ioi 0 ×ˢ Ioo (-π) π, x.1 * (|x.1| ^ q * rexp (-b * |x.1| ^ p))

Tactic:
Complex.norm_polarCoord_symm,

Goal state after:
p q b : ℝ
hp : 1 ≤ p
hq : -2 < q
hb : 0 < b
⊢ ∫ (p_1 : ℝ × ℝ) in Ioi 0 ×ˢ Ioo (-π) π, p_1.1 • (|p_1.1| ^ q * rexp (-b * |p_1.1| ^ p)) =
    ∫ (x : ℝ × ℝ) in Ioi 0 ×ˢ Ioo (-π) π, x.1 * (|x.1| ^ q * rexp (-b * |x.1| ^ p))
Module: Mathlib.MeasureTheory.Integral.Gamma

Goal state before:
pp: p q b : ℝ
hp : 1 ≤ p
hq : -2 < q
hb : 0 < b
⊢ ∫ (p_1 : ℝ × ℝ) in Ioi 0 ×ˢ Ioo (-π) π,
      p_1.1 • (‖↑Complex.polarCoord.symm p_1‖ ^ q * rexp (-b * ‖↑Complex.polarCoord.symm p_1‖ ^ p)) =
    ∫ (x : ℝ × ℝ) in Ioi 0 ×ˢ Ioo (-π) π, x.1 * (|x.1| ^ q * rexp (-b * |x.1| ^ p))

Tactic:
simp_rw [Complex.norm_polarCoord_symm, smul_eq_mul]

Goal state after:
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 ≠ ⊤
⊢ ∃ c ∈ s, ∫ (x : α) in s, f x * g x ∂μ = f c * ∫ (x : α) in s, g x ∂μ

Tactic:
have hν0'' : ν s ≠ 0 := by intro this; apply hν0'; simp [this]

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
⊢ ∃ 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.MeanValue

Goal state before:
pp: a b : ℝ
f g : ℝ → ℝ
μ : Measure ℝ
hf : ContinuousOn f [[a, b]]
hg : IntervalIntegrable g μ a b
hg0 : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), 0 ≤ g x
h : ¬a = b
this :
  ∀ {a b : ℝ},
    ContinuousOn f [[a, b]] →
      IntervalIntegrable g μ a b →
        (∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), 0 ≤ g x) →
          ¬a = b → a < b → ∃ c ∈ [[a, b]], ∫ (x : ℝ) in a..b, f x * g x ∂μ = f c * ∫ (x : ℝ) in a..b, g x ∂μ
hab : b ≤ a
⊢ ¬b = a

Tactic:
lia

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

Goal state before:
pp: case refine_2
f : ℝ → ℝ
ζ a b : ℝ
a_lt_b : a < b
h_df : DifferentiableOn ℝ f (Set.Icc a b)
h_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)
fpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ
N : ℕ
N_nonzero : 0 < N
h : ℝ := (b - a) / ↑N
ak : ℕ → ℝ := fun k ↦ a + ↑k * h
h0 : ∀ (k : ℕ), ak (k + 1) - ak k = h
hab : 0 < b - a
hpos : 0 < h
hb : b = a + ↑N * h
k : ℕ
hk : k < N
h1 : a ≤ ak k
h2 : ak (k + 1) ≤ b
h3 : Set.Icc (ak k) (ak (k + 1)) ⊆ Set.Icc a b
h4 : ak k < ak (k + 1)
h5 :
  EqOn (_root_.derivWithin f (Set.Icc a b)) (_root_.derivWithin f (Set.Icc (ak k) (ak (k + 1))))
    (Set.Icc (ak k) (ak (k + 1)))
h6 :
  EqOn (iteratedDerivWithin 2 f (Set.Icc a b)) (iteratedDerivWithin 2 f (Set.Icc (ak k) (ak (k + 1))))
    (Set.Icc (ak k) (ak (k + 1)))
h7 : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc (ak k) (ak (k + 1))) x| ≤ ζ
⊢ (ak (k + 1) - ak k) ^ 3 * ζ / 12 = ζ / 12 * h ^ 3

Tactic:
· rw [h0, mul_div_assoc, mul_comm]

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

Goal state before:
pp: case inr
α : Type u
s : Set (Set α)
i j : Ordinal.{u_1}
h✝ : i ≤ j
x : Set α
hx : x ∈ generateMeasurableRec s i
h : i < j
⊢ x ∈ generateMeasurableRec s j

Tactic:
convert! iUnion_mem_generateMeasurableRec fun _ => ⟨i, h, hx⟩
    exact (iUnion_const x).symm

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

Goal state before:
pp: 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
m : Fin N
a✝¹ : m ∈ univ
n : Fin N
a✝ : n ∈ univ
hmn : m ≠ n
x : X
hx : y m - ε < f x ∧ f x ≤ y m ∧ x ∈ tsupport ⇑f
⊢ m < n ∨ n < m

Tactic:
lia

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
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
⊢ μ.real ↑(V n) ≤ μ.real (E n) + (ENNReal.ofReal (ε' / ↑N)).toReal

Tactic:
apply ENNReal.toReal_le_add (hV n).2.2 (hE' n)

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 := ⋯
K : Set X := ⋯
ε : ℝ
hε : 0 < ε
a b : ℝ
hab : a < b ∧ range ⇑f ⊆ Ioo a b
N : ℕ
hN : 0 < N
ε' : ℝ := ⋯
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 := ⋯
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
⊢ ENNReal.ofReal (ε' / ↑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 := ⋯
K : Set X := ⋯
ε : ℝ
hε : 0 < ε
a b : ℝ
hab : a < b ∧ range ⇑f ⊆ Ioo a b
N : ℕ
hN : 0 < N
ε' : ℝ := ⋯
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 := ⋯
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
⊢ ENNReal.ofReal (ε' / ↑N) ≠ ∞

Tactic:
· finiteness

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
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)
this : ∀ (n : Fin N), (|a| + y n + ε') * μ.real (E n) = (|a| + 2 * ε') * μ.real (E n) + (y n - ε') * μ.real (E n)
⊢ μ (⋃ j, E j) = ∑ i, μ (E i)

Tactic:
measure_iUnion (fun m n hmn ↦ hE.2.1 trivial trivial hmn) hE.2.2.2

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)
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)
this : ∀ (n : Fin N), (|a| + y n + ε') * μ.real (E n) = (|a| + 2 * ε') * μ.real (E n) + (y n - ε') * μ.real (E n)
⊢ ∑' (i : Fin N), μ (E i) = ∑ i, μ (E i)
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoEMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
LPμν_finite : ¬levyProkhorovEDist μ ν = ∞
LPνκ_finite : ¬levyProkhorovEDist ν κ = ∞
ε : ℝ≥0∞
B : Set Ω
ε_pos : 0 < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
half_ε_pos : 0 < ε / 2
r : ℝ≥0∞ := levyProkhorovEDist μ ν + ε / 2
s : ℝ≥0∞ := levyProkhorovEDist ν κ + ε / 2
lt_r : levyProkhorovEDist μ ν < r
lt_s : levyProkhorovEDist ν κ < s
⊢ levyProkhorovEDist ν κ + ε / 2 + r = levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε

Tactic:
r,

Goal state after:
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoEMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
LPμν_finite : ¬levyProkhorovEDist μ ν = ∞
LPνκ_finite : ¬levyProkhorovEDist ν κ = ∞
ε : ℝ≥0∞
B : Set Ω
ε_pos : 0 < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
half_ε_pos : 0 < ε / 2
r : ℝ≥0∞ := levyProkhorovEDist μ ν + ε / 2
s : ℝ≥0∞ := levyProkhorovEDist ν κ + ε / 2
lt_r : levyProkhorovEDist μ ν < r
lt_s : levyProkhorovEDist ν κ < s
⊢ levyProkhorovEDist ν κ + ε / 2 + (levyProkhorovEDist μ ν + ε / 2) =
    levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: case ha
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoEMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
LPμν_finite : ¬levyProkhorovEDist μ ν = ∞
LPνκ_finite : ¬levyProkhorovEDist ν κ = ∞
ε : ℝ≥0∞
B : Set Ω
ε_pos : 0 < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
half_ε_pos : 0 < ε / 2
r : ℝ≥0∞ := levyProkhorovEDist μ ν + ε / 2
s : ℝ≥0∞ := levyProkhorovEDist ν κ + ε / 2
lt_r : levyProkhorovEDist μ ν < r
lt_s : levyProkhorovEDist ν κ < s
hs_add_r : s + r = levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε
⊢ s ≠ ∞

pp: case hb
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoEMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
LPμν_finite : ¬levyProkhorovEDist μ ν = ∞
LPνκ_finite : ¬levyProkhorovEDist ν κ = ∞
ε : ℝ≥0∞
B : Set Ω
ε_pos : 0 < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
half_ε_pos : 0 < ε / 2
r : ℝ≥0∞ := levyProkhorovEDist μ ν + ε / 2
s : ℝ≥0∞ := levyProkhorovEDist ν κ + ε / 2
lt_r : levyProkhorovEDist μ ν < r
lt_s : levyProkhorovEDist ν κ < s
hs_add_r : s + r = levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε
⊢ r ≠ ∞

Tactic:
· finiteness

Goal state after:
case hb
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoEMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
LPμν_finite : ¬levyProkhorovEDist μ ν = ∞
LPνκ_finite : ¬levyProkhorovEDist ν κ = ∞
ε : ℝ≥0∞
B : Set Ω
ε_pos : 0 < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
half_ε_pos : 0 < ε / 2
r : ℝ≥0∞ := levyProkhorovEDist μ ν + ε / 2
s : ℝ≥0∞ := levyProkhorovEDist ν κ + ε / 2
lt_r : levyProkhorovEDist μ ν < r
lt_s : levyProkhorovEDist ν κ < s
hs_add_r : s + r = levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε
⊢ r ≠ ∞
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: case hb
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoEMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
LPμν_finite : ¬levyProkhorovEDist μ ν = ∞
LPνκ_finite : ¬levyProkhorovEDist ν κ = ∞
ε : ℝ≥0∞
B : Set Ω
ε_pos : 0 < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
half_ε_pos : 0 < ε / 2
r : ℝ≥0∞ := levyProkhorovEDist μ ν + ε / 2
s : ℝ≥0∞ := levyProkhorovEDist ν κ + ε / 2
lt_r : levyProkhorovEDist μ ν < r
lt_s : levyProkhorovEDist ν κ < s
hs_add_r : s + r = levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε
⊢ r ≠ ∞

Tactic:
· finiteness

Goal state after:
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: Ω : Type u_1
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : PseudoEMetricSpace Ω
inst✝³ : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
inst✝² : IsFiniteMeasure μ
inst✝¹ : IsFiniteMeasure ν
inst✝ : IsFiniteMeasure κ
dμν_finite : levyProkhorovEDist μ ν ≠ ∞
dνκ_finite : levyProkhorovEDist ν κ ≠ ∞
⊢ levyProkhorovEDist μ ν + levyProkhorovEDist ν κ ≠ ∞

Tactic:
exact ENNReal.add_ne_top.mpr ⟨dμν_finite, dνκ_finite⟩

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

Goal state before:
pp: case hfint
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 := ⋯
K : Set X := ⋯
ε : ℝ
hε : 0 < ε
a b : ℝ
hab : a < b ∧ range ⇑f ⊆ Ioo a b
N : ℕ
hN : 0 < N
ε' : ℝ := ⋯
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 := ⋯
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
⊢ IntegrableOn (fun x ↦ f x) (E n) μ

Tactic:
apply Integrable.integrableOn
        dsimp [μ, rieszMeasure]
        exact Continuous.integrable_of_hasCompactSupport f.1.2 f.2

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
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
⊢ ↑n + 1 ≤ N

Tactic:
lia

Goal state after:
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
t : ℝ
⊢ ↑μ {a | t ≤ f a} ≠ ∞

Tactic:
· finiteness

Goal state after:
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: case neg
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)
ν : LevyProkhorov (ProbabilityMeasure Ω)
hμs : Tendsto μs atTop (𝓝 ν)
P : ProbabilityMeasure Ω := ν.toMeasure
Ps : ℕ → ProbabilityMeasure Ω := toMeasure ∘ μs
f✝ f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
f_zero : ¬‖f‖ = 0
norm_f_pos : 0 < ‖f‖
δ : ℝ
⊢ 0 < δ → limsup (fun i ↦ ∫ (x : Ω), f x ∂↑(Ps i)) atTop ≤ ∫ (x : Ω), f x ∂↑P + δ

Tactic:
δ_pos

Goal state after:
case neg
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)
ν : LevyProkhorov (ProbabilityMeasure Ω)
hμs : Tendsto μs atTop (𝓝 ν)
P : ProbabilityMeasure Ω := ν.toMeasure
Ps : ℕ → ProbabilityMeasure Ω := toMeasure ∘ μs
f✝ f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
f_zero : ¬‖f‖ = 0
norm_f_pos : 0 < ‖f‖
δ : ℝ
δ_pos : 0 < δ
⊢ limsup (fun i ↦ ∫ (x : Ω), f x ∂↑(Ps i)) atTop ≤ ∫ (x : Ω), f x ∂↑P + δ
Module: Mathlib.MeasureTheory.Measure.Haar.DistribChar

Goal state before:
pp: G : Type u_1
A : Type u_2
inst✝¹⁰ : Group G
inst✝⁹ : AddCommGroup A
inst✝⁸ : DistribMulAction G A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : IsTopologicalAddGroup A
inst✝⁵ : LocallyCompactSpace A
inst✝⁴ : ContinuousConstSMul G A
inst✝³ : MeasurableSpace A
inst✝² : BorelSpace A
μ : Measure A
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : μ.Regular
g : G
s : Set A
this : (DomMulAct.mk g • μ) s = μ (g • s)
⊢ (DomMulAct.mk g • μ) s = (DomMulAct.mk g • μ).addHaarScalarFactor μ • μ s

Tactic:
← Measure.smul_apply,

Goal state after:
G : Type u_1
A : Type u_2
inst✝¹⁰ : Group G
inst✝⁹ : AddCommGroup A
inst✝⁸ : DistribMulAction G A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : IsTopologicalAddGroup A
inst✝⁵ : LocallyCompactSpace A
inst✝⁴ : ContinuousConstSMul G A
inst✝³ : MeasurableSpace A
inst✝² : BorelSpace A
μ : Measure A
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : μ.Regular
g : G
s : Set A
this : (DomMulAct.mk g • μ) s = μ (g • s)
⊢ (DomMulAct.mk g • μ) s = ((DomMulAct.mk g • μ).addHaarScalarFactor μ • μ) s
Module: Mathlib.MeasureTheory.Measure.Haar.MulEquivHaarChar

Goal state before:
pp: G : Type u_1
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : IsTopologicalGroup G
inst✝² : LocallyCompactSpace G
μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ.Regular
f : G → ℝ
φ : G ≃ₜ* G
e : G ≃ᵐ G := φ.toHomeomorph.toMeasurableEquiv
this✝ : (map (⇑e.symm) μ).IsHaarMeasure
this : (map (⇑e.symm) μ).Regular
⊢ ⇑φ ∘ ⇑e.symm = id

Tactic:
ext; simp [e]

Goal state after: