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

Goal state before:
pp: case hcont
E✝ : Type u_1
inst✝³ : NormedAddCommGroup E✝
inst✝² : NormedSpace ℝ E✝
f✝ : ℝ → E✝
a b : ℝ
B : ℝ → ℝ
hab : a ≤ b
hBi : IntervalIntegrable B volume a b
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
hfc : ContinuousOn f (Icc a b)
hfd : DifferentiableOn ℝ f (Ioo a b)
hfB : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ B t
hE : CompleteSpace E
hfB' : (fun x ↦ ‖deriv f x‖) ≤ᵐ[volume.restrict (uIoc a b)] B
⊢ ContinuousOn f (uIcc a b)

Tactic:
rwa [uIcc_of_le hab]

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

Goal state before:
pp: case hBi.refine_1
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
⊢ volume (Ioo a b) ≠ ⊤

Tactic:
simp

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

Goal state before:
pp: case hBi.refine_2
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‖ₑ ≠ ⊤

Tactic:
simp

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

Goal state before:
pp: a : ℝ
f g : ℝ → ℝ
μ : Measure ℝ
hf : ContinuousOn f [[a, a]]
hg : IntervalIntegrable g μ a a
hg0 : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a a), 0 ≤ g x
⊢ a ∈ [[a, a]]

Tactic:
simp

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

Goal state before:
pp: a : ℝ
f g : ℝ → ℝ
μ : Measure ℝ
hf : ContinuousOn f [[a, a]]
hg : IntervalIntegrable g μ a a
hg0 : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a a), 0 ≤ g x
⊢ ∫ (x : ℝ) in a..a, f x * g x ∂μ = f a * ∫ (x : ℝ) in a..a, g x ∂μ

Tactic:
simp

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

Goal state before:
pp: case h₂
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
⊢ CurveIntegrable (fun x ↦ ω a) (Path.segment a b)

Tactic:
rw [curveIntegrable_segment]
    exact intervalIntegrable_const

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

Goal state before:
pp: case a
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,
      ⨅ (_ : Continuous f),
        ⨅ (_ : HasCompactSupport f), ⨅ (_ : EqOn f 1 k), ⨅ (_ : 0 ≤ f), ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤
    μ k

Tactic:
apply le_of_forall_gt (fun r hr ↦ ?_)

Goal state after:
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
r : ENNReal
hr : μ k < r
⊢ ⨅ f,
      ⨅ (_ : Continuous f),
        ⨅ (_ : HasCompactSupport f), ⨅ (_ : EqOn f 1 k), ⨅ (_ : 0 ≤ f), ENNReal.ofReal (∫ (x : X), f x ∂μ) <
    r
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‖
⊢ volume (Ioo 0 1) ≠ ⊤

Tactic:
simp

Goal state after:
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
inst✝³ : T2Space X
U : Set X
hU : IsOpen U
μ : Measure X
inst✝² : IsFiniteMeasure μ
inst✝¹ : μ.InnerRegularCompactLTTop
inst✝ : NormalSpace X
r : ENNReal
hr : r < μ U
⊢ μ U ≠ ⊤

Tactic:
simp

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

Goal state before:
pp: case xs
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 : ∀ᶠ (x : E) in 𝓝 a, ‖fderiv ℝ f x‖ ≤ C * ‖‖x - a‖ ^ r‖
ε : ℝ
hε₀ : ε > 0
hε : ∀ y ∈ ball a ε, DifferentiableAt ℝ f y ∧ ‖fderiv ℝ f y‖ ≤ C * ‖‖y - a‖ ^ r‖
y : E
hy : y ∈ ball a ε
hsub : closedBall a ‖y - a‖ ⊆ ball a ε
⊢ a ∈ closedBall a ‖y - a‖

Tactic:
simp

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

Goal state before:
pp: f : ℝ → ℝ
N : ℕ
a h : ℝ
N_nonzero : 0 < N
⊢ ∑ i ∈ Finset.range N, trapezoidal_integral f 1 (a + ↑i * h) (a + (↑i + 1) * h) =
    trapezoidal_integral f N a (a + ↑N * h)

Tactic:
simp_rw [trapezoidal_integral_one, add_sub_add_left_eq_sub, ← sub_mul, trapezoidal_integral,
    add_sub_cancel_left, one_mul, ← mul_sum, ← mul_div, show N * (h / N) = h by field]

Goal state after:
f : ℝ → ℝ
N : ℕ
a h : ℝ
N_nonzero : 0 < N
⊢ h / 2 * ∑ i ∈ Finset.range N, (f (a + ↑i * h) + f (a + (↑i + 1) * h)) =
    h * ((f a + f (a + ↑N * h)) / 2 + ∑ x ∈ Finset.range (N - 1), f (a + (↑x + 1) * h))
Module: Mathlib.Order.Interval.Set.Union

Goal state before:
pp: case zero
X : Type u_1
inst✝ : LinearOrder X
a : ℕ → X
⊢ Ioc (a 0) (a 0) ⊆ ⋃ i ∈ Finset.range 0, Ioc (a i) (a (i + 1))

Tactic:
simp

Goal state after:
Module: Mathlib.Order.Interval.Set.Union

Goal state before:
pp: case zero
X : Type u_1
inst✝ : LinearOrder X
a : ℕ → X
⊢ Ico (a 0) (a 0) ⊆ ⋃ i ∈ Finset.range 0, Ico (a i) (a (i + 1))

Tactic:
simp

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

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
f₁ f₂ : X →C_c ℝ≥0
h : f₁ ≤ f₂
g : X →C_c ℝ≥0
hg : f₁ + g = f₂
⊢ Λ f₁ ≤ Λ (f₁ + g)

Tactic:
simp

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

Goal state before:
pp: f : ℝ → ℝ
N : ℕ
a h : ℝ
N_nonzero : 0 < N
h_f_int : IntervalIntegrable f volume a (a + ↑N * h)
⊢ trapezoidal_integral f N a (a + ↑N * h) - ∫ (x : ℝ) in a + ↑0 * h..a + ↑N * h, f x =
    trapezoidal_integral f N a (a + ↑N * h) - ∫ (x : ℝ) in a..a + ↑N * h, f x

Tactic:
simp

Goal state after:
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 ∈ Ioo 0 1, ∀ b_1 ∈ Ioo 0 1, φ (a, b_1) ∈ t
hω : ∀ a ∈ Ioo 0 1, ∀ b_1 ∈ Ioo 0 1, (a, b_1) ∉ s → HasFDerivWithinAt ω (dω (φ (a, b_1))) t (φ (a, b_1))
hωc : ContinuousOn ω (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t)
hdω_symm :
  ∀ a ∈ Ioo 0 1,
    ∀ b_1 ∈ Ioo 0 1,
      (a, b_1) ∉ s →
        ∀ u ∈ tangentConeAt ℝ t (φ (a, b_1)),
          ∀ v ∈ tangentConeAt ℝ t (φ (a, b_1)), ((dω (φ (a, b_1))) u) v = ((dω (φ (a, 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 b_1 : ↑I), ψ (↑a, ↑b_1) = φ (a, 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
hmem_s' : ∀ (x y : ↑I), (↑x, ↑y) ∈ s' ↔ (x, y) ∈ s
hs'c : s'.Countable
hdψ : ∀ a ∈ U, HasFDerivAt ψ (dψ a) a
d2ψ : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ →L[ℝ] E := fderivWithin ℝ dψ (Icc 0 1)
hd2ψ : ∀ a ∈ U, HasFDerivAt dψ (d2ψ a) a
a : ℝ × ℝ
ha : a ∈ Icc 0 1
⊢ minSmoothness ℝ 2 ≤ 2

Tactic:
simp

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

Goal state before:
pp: 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)))
⊢ |trapezoidal_error f 1 (ak k) (ak (k + 1))| ≤ ζ / 12 * h ^ 3

Tactic:
have h7 (x : ℝ) : |iteratedDerivWithin 2 f (Set.Icc (ak k) (ak (k + 1))) x| ≤ ζ := by
    by_cases hx : x ∈ Icc (ak k) (ak (k + 1))
    · grw [← h6 hx, fpp_bound]
    · rw [iteratedDerivWithin_succ, derivWithin_zero_of_notMem_closure
        (by rwa [closure_Icc]), abs_zero]
      exact (abs_nonneg _).trans (fpp_bound 0)

Goal state after:
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
⊢ |trapezoidal_error f 1 (ak k) (ak (k + 1))| ≤ ζ / 12 * h ^ 3
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K : Compacts X
b : ℝ≥0∞
hb : ∀ (i : Compacts X), b ≤ ⨅ (_ : ↑K ⊆ interior ↑i), ↑(rieszContentAux Λ i)
this : b < ∞
ε : ℝ≥0
hε : 0 < ↑ε
f : X →C_c ℝ≥0
hfleoneonK : ∀ x ∈ K, 1 ≤ f x
hfle : ↑(Λ f) < ↑(sInf (⇑Λ '' {f | ∀ x ∈ K, 1 ≤ f x}) + NNReal.mk ↑ε ⋯)
α : ℝ≥0
hα : 1 < α
K' : Set X := ⇑f ⁻¹' Ici α⁻¹
hKK' : ↑K ⊆ interior K'
hK'cp : IsCompact K'
hb' : ↑K ⊆ interior K' → b ≤ ↑(rieszContentAux Λ { carrier := K', isCompact' := hK'cp })
⊢ ↑(rieszContentAux Λ { carrier := K', isCompact' := hK'cp }) ≠ ∞

Tactic:
simp

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

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K : Compacts X
b : ℝ≥0∞
hb : ∀ (i : Compacts X), b ≤ ⨅ (_ : ↑K ⊆ interior ↑i), ↑(rieszContentAux Λ i)
this : b < ∞
ε : ℝ≥0
hε : 0 < ↑ε
f : X →C_c ℝ≥0
hfleoneonK : ∀ x ∈ K, 1 ≤ f x
hfle : ↑(Λ f) < ↑(sInf (⇑Λ '' {f | ∀ x ∈ K, 1 ≤ f x}) + NNReal.mk ↑ε ⋯)
α : ℝ≥0
hα : 1 < α
K' : Set X := ⇑f ⁻¹' Ici α⁻¹
hKK' : ↑K ⊆ interior K'
hK'cp : IsCompact K'
hb' : ↑K ⊆ interior K' → b ≤ ↑(rieszContentAux Λ { carrier := K', isCompact' := hK'cp })
⊢ Λ (α • f) = Λ (α • f)

Tactic:
simp

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

Goal state before:
pp: X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
f : X →C_c ℝ≥0
a : X
⊢ 0 a ≤ (fun a ↦ ↑(f a)) a

Tactic:
simp

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

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

Tactic:
simp

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

Goal state before:
pp: X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
x✝ : X →C_c ℝ≥0
⊢ ↑((integralLinearMap (rieszMeasure Λ)) x✝) = ↑(Λ x✝)

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
f : α → α
⊢ f ⁻¹' ∅ = ∅

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
f : α → α
⊢ ∀ (s : Set α), f ⁻¹' s = s → f ⁻¹' sᶜ = sᶜ

Tactic:
simp

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

Goal state before:
pp: case refine_4
α : Type u
s : Set (Set α)
t : Set α
ht : t ∈ generateMeasurableRec s (ω_ 1)
f : ℕ → Set α
H : ∀ (n : ℕ), (∃ i, ∃ (_ : i < ω_ 1), f n ∈ generateMeasurableRec s i) ∧ ∃ j < ω_ 1, f n ∈ generateMeasurableRec s j
I : ℕ → Ordinal.{v}
hI : ∀ (n : ℕ), I n < ω_ 1 ∧ f n ∈ generateMeasurableRec s (I n)
⊢ Cardinal.lift.{v, 0} #ℕ < (Ordinal.lift.{0, v} (ω_ 1)).cof

Tactic:
simp

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

Goal state before:
pp: α : Type u
s : Set (Set α)
i✝¹ i✝ i : Ordinal.{v}
IH : ∀ y < i, y ≤ ω_ 1 → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀
hi : i ≤ ω_ 1
A : 𝔠 ≤ max (#↑s) 2 ^ ℵ₀
B : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
⊢ ∀ j < i, #↑(generateMeasurableRec s j) ≤ max (#↑s) 2 ^ ℵ₀

Tactic:
intro j

Goal state after:
α : Type u
s : Set (Set α)
i✝¹ i✝ i : Ordinal.{v}
IH : ∀ y < i, y ≤ ω_ 1 → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀
hi : i ≤ ω_ 1
A : 𝔠 ≤ max (#↑s) 2 ^ ℵ₀
B : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
j : Ordinal.{v}
⊢ j < i → #↑(generateMeasurableRec s j) ≤ max (#↑s) 2 ^ ℵ₀
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)
⊢ Λ (∑ n, g n • f) = ∑ n, Λ (g n • f)

Tactic:
simp

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, Λ ((y n + ε') • g n) = ∑ n, (y n + ε') * Λ (g n)

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion

Goal state before:
pp: μ : Measure ℝ
inst✝ : IsFiniteMeasure μ
n : ℕ
t : ℝ
hint : MemLp id (↑n) μ
⊢ I ^ n * ∫ (y : ℝ), ↑(∏ i, ⟪y, 1⟫) * cexp (↑⟪y, t⟫ * I) ∂μ = I ^ n * ∫ (x : ℝ), ↑x ^ n * cexp (↑t * ↑x * I) ∂μ

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: W : Type u_1
inst✝¹¹ : AddCommGroup W
inst✝¹⁰ : Module ℝ W
inst✝⁹ : TopologicalSpace W
e : AddChar ℝ Circle
V : Type u_2
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module ℝ V
inst✝⁶ : PseudoEMetricSpace V
inst✝⁵ : MeasurableSpace V
inst✝⁴ : BorelSpace V
inst✝³ : CompleteSpace V
inst✝² : SecondCountableTopology V
L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ
he : Continuous ⇑e
he' : e ≠ 1
hL' : ∀ (v : V), v ≠ 0 → L v ≠ 0
hL : Continuous fun p ↦ (L p.1) p.2
P P' : Measure V
inst✝¹ : IsFiniteMeasure P
inst✝ : IsFiniteMeasure P'
h : ∀ (w : W), ∫ (v : V), (char he hL w) v ∂P = ∫ (v : V), (char he hL w) v ∂P'
g✝ : V →ᵇ ℂ
w : AddMonoidAlgebra ℂ W
hw : ⇑g✝ = fun x ↦ w.coeff.sum fun a z ↦ z * ↑(e ((L x) a))
⊢ ∫ (x : V), g✝ x ∂P = ∫ (x : V), g✝ x ∂P'

Tactic:
hw

Goal state after:
W : Type u_1
inst✝¹¹ : AddCommGroup W
inst✝¹⁰ : Module ℝ W
inst✝⁹ : TopologicalSpace W
e : AddChar ℝ Circle
V : Type u_2
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module ℝ V
inst✝⁶ : PseudoEMetricSpace V
inst✝⁵ : MeasurableSpace V
inst✝⁴ : BorelSpace V
inst✝³ : CompleteSpace V
inst✝² : SecondCountableTopology V
L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ
he : Continuous ⇑e
he' : e ≠ 1
hL' : ∀ (v : V), v ≠ 0 → L v ≠ 0
hL : Continuous fun p ↦ (L p.1) p.2
P P' : Measure V
inst✝¹ : IsFiniteMeasure P
inst✝ : IsFiniteMeasure P'
h : ∀ (w : W), ∫ (v : V), (char he hL w) v ∂P = ∫ (v : V), (char he hL w) v ∂P'
g✝ : V →ᵇ ℂ
w : AddMonoidAlgebra ℂ W
hw : ⇑g✝ = fun x ↦ w.coeff.sum fun a z ↦ z * ↑(e ((L x) a))
⊢ ∫ (x : V), (fun x ↦ w.coeff.sum fun a z ↦ z * ↑(e ((L x) a))) x ∂P =
    ∫ (x : V), (fun x ↦ w.coeff.sum fun a z ↦ z * ↑(e ((L x) a))) x ∂P'
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝ : IsProbabilityMeasure P
X : Ω → ℝ
hX : AEMeasurable X P
h0 : ∫ (x : Ω), X x ∂P = 0
h1 : ∫ (x : Ω), (X ^ 2) x ∂P = 1
t : ℝ
⊢ 1 + ↑0 * ↑t * I - ↑1 * ↑t ^ 2 / 2 = 1 - ↑t ^ 2 / 2

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion

Goal state before:
pp: case e'_1
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝ : IsProbabilityMeasure P
X : Ω → ℝ
hX : AEMeasurable X P
h0 : ∫ (x : Ω), X x ∂P = 0
h1 : ∫ (x : Ω), (X ^ 2) x ∂P = 1
x✝ : ℝ
⊢ x✝ = x✝ - 0

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_2
mE : MeasurableSpace E
μ : Measure E
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
t : E
⊢ ∫ (v : E), ‖1 v‖ ∂μ = μ.real Set.univ

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_2
mE : MeasurableSpace E
μ : Measure E
inst✝² : SeminormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : IsProbabilityMeasure μ
t : E
⊢ μ.real Set.univ = 1

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: μ : Measure ℝ
inst✝ : IsFiniteMeasure μ
a b : ℝ
⊢ IntervalIntegrable (fun x ↦ μ.real Set.univ) volume a b

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_3
inst✝⁶ : MeasurableSpace E
μ ν : Measure E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : BorelSpace E
inst✝² : SecondCountableTopology E
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
t : E
⊢ ∀ᵐ (a : E) ∂μ ∗ ν, ‖cexp (↑⟪a, t⟫ * I)‖ ≤ ‖1‖

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.Separation.CompletelyRegular

Goal state before:
pp: ι : Type u_1
X : Type u_2
t : ι → TopologicalSpace X
ht : ∀ (i : ι), CompletelyRegularSpace X
this : TopologicalSpace X := ⨅ i, t i
⊢ ∀ (x : X) (K : Set X), IsOpen[⨅ i, t i] K → x ∈ K → ∃ f, Continuous[⨅ i, t i, _] f ∧ f x = 0 ∧ EqOn f 1 Kᶜ

Tactic:
intro x K hK hxK

Goal state after:
ι : Type u_1
X : Type u_2
t : ι → TopologicalSpace X
ht : ∀ (i : ι), CompletelyRegularSpace X
this : TopologicalSpace X := ⨅ i, t i
x : X
K : Set X
hK : IsOpen[⨅ i, t i] K
hxK : x ∈ K
⊢ ∃ f, Continuous[⨅ i, t i, _] f ∧ f x = 0 ∧ EqOn f 1 Kᶜ