mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.Intersectivity

Goal state before:
pp: case funProp.discharger
α : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ μ (s n)
M : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}
N : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)
hN₀ : μ N = 0
hN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)
f : ℕ → α → ℝ≥0∞ := fun n ↦ (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1
hfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a
n i : ℕ
a✝ : i ∈ Finset.range (n + 1)
⊢ MeasurableSet (s i)

Tactic:
exact hs _

Goal state after:
Module: Mathlib.Probability.Process.Filtration

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝¹ : LinearOrder ι
inst✝ : SuccOrder ι
𝓕 : Filtration ι m
⊢ 𝓕₊ = 𝓕

Tactic:
let := Preorder.topology ι; have : OrderTopology ι := ⟨rfl⟩
  ext _
  rw [rightCont_eq_of_nhdsGT_eq_bot _ SuccOrder.nhdsGT]

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Count

Goal state before:
pp: case neg
α : Type u_1
ε : Type u_2
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSingletonClass α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
p : ℝ≥0∞
f : α → ε
i : α
hp : p ≠ 0
h✝ : ¬p = ∞
⊢ eLpNorm' f p.toReal (dirac i) = ‖f i‖ₑ

Tactic:
simp [eLpNorm', ENNReal.toReal_eq_zero_iff, *]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : NormedAddCommGroup β
x✝ : MeasurableSpace α
f : ι → α → β
p : ℝ≥0∞
μ : Measure α
⊢ 0 < ∞ → ∃ s, μ s ≠ ∞ ∧ ∀ (i : ι), eLpNorm (sᶜ.indicator (f i)) p μ ≤ ∞

Tactic:
simp only [zero_lt_top, le_top, implies_true, and_true, true_implies]

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : NormedAddCommGroup β
x✝ : MeasurableSpace α
f : ι → α → β
p : ℝ≥0∞
μ : Measure α
⊢ ∃ s, μ s ≠ ∞
Module: Mathlib.MeasureTheory.Group.IntegralConvolution

Goal state before:
pp: M : Type u_1
F : Type u_2
inst✝⁵ : Monoid M
mM : MeasurableSpace M
inst✝⁴ : MeasurableMul₂ M
inst✝³ : NormedAddCommGroup F
μ ν : Measure M
f : M → F
inst✝² : NormedSpace ℝ F
inst✝¹ : SFinite μ
inst✝ : SFinite ν
hf : Integrable f (μ ∗ₘ ν)
⊢ ∫ (x : M), f x ∂μ ∗ₘ ν = ∫ (x : M), ∫ (y : M), f (x * y) ∂ν ∂μ

Tactic:
unfold Measure.mconv
  rw [integral_map (by fun_prop) hf.1, integral_prod]
  exact (integrable_map_measure hf.1 (by fun_prop)).mp hf

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

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : Subsingleton ι
hp_top : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), MemLp (f i) p μ
ε : ℝ≥0
hε : 0 < ε
hε_top : ¬↑ε = ∞
i : ι
s : Set α
left✝ : MeasurableSet s
hμs : μ s < ∞
hfε : eLpNorm (sᶜ.indicator (f i)) p μ < ↑ε
⊢ ∃ s, μ s ≠ ∞ ∧ ∀ (i : ι), eLpNorm (sᶜ.indicator (f i)) p μ ≤ ↑ε

Tactic:
refine ⟨s, ne_of_lt hμs, fun j => ?_⟩

Goal state after:
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : Subsingleton ι
hp_top : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), MemLp (f i) p μ
ε : ℝ≥0
hε : 0 < ε
hε_top : ¬↑ε = ∞
i : ι
s : Set α
left✝ : MeasurableSet s
hμs : μ s < ∞
hfε : eLpNorm (sᶜ.indicator (f i)) p μ < ↑ε
j : ι
⊢ eLpNorm (sᶜ.indicator (f j)) p μ ≤ ↑ε
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
⊢ ∃ B ε, 0 < ε ∧ ball x ε ⊆ ball z R ∧ ∀ (t : ℝ), ∀ y ∈ ball x ε, ‖circleTransformDeriv R z y f t‖ ≤ B

Tactic:
refine ⟨‖V b a‖ * ‖f X‖, ε', hε', H.trans (ball_subset_ball hr.le), fun y v hv ↦ ?_⟩

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'
⊢ ‖circleTransformDeriv R z v f y‖ ≤ ‖V b a‖ * ‖f X‖
Module: Mathlib.MeasureTheory.Group.GeometryOfNumbers

Goal state before:
pp: case refine_1
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
h_mes : μ s ≠ 0
h_nemp : s.Nonempty
u : ℕ → ℝ≥0 := ⋯.choose
K : ConvexBody E := { carrier := s, convex' := h_conv, isCompact' := h_cpt, nonempty' := h_nemp }
S : ℕ → ConvexBody E := fun n ↦ (1 + u n) • K
Z : ℕ → Set E := fun n ↦ ↑(S n) ∩ (↑L \ {0})
h_zero : 0 ∈ K
h_clos : IsClosed (↑L \ {0})
n : ℕ
⊢ Z (n + 1) ⊆ Z n

Tactic:
refine Set.inter_subset_inter_left _ (SetLike.coe_subset_coe.mpr ?_)
    refine ConvexBody.smul_le_of_le K h_zero ?_
    rw [add_le_add_iff_left]
    exact le_of_lt <| (exists_seq_strictAnti_tendsto (0 : ℝ≥0)).choose_spec.1 (Nat.lt_add_one n)

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

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝⁴ : TopologicalSpace α
inst✝³ : SemilatticeSup β
inst✝² : Zero β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousSup β
ι : Type u_5
s : Finset ι
H : s.Nonempty
f : ι → α →C_c β
⊢ ⇑(s.sup' H f) = s.sup' H fun i ↦ ⇑(f i)

Tactic:
ext; simp [finsetSup'_apply]

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)
⊢ ∃ c ∈ s, ∫ (x : α) in s, f x * g x ∂μ = f c * ∫ (x : α) in s, g x ∂μ

Tactic:
have hρ_top : ∀ᵐ x ∂μ.restrict s, ρ x < ⊤ := by simp [ρ]

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 < ⊤
⊢ ∃ c ∈ s, ∫ (x : α) in s, f x * g x ∂μ = f c * ∫ (x : α) in s, g x ∂μ
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:
integral_congr_ae h_toReal_ae

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
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 = ∫ (a : α) in s, g a ∂μ
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
a b : E
ω : E → E →L[𝕜] F
inst✝ : NormedSpace ℝ E
C : ℝ
h : ∀ z ∈ [a -[ℝ] b], ‖ω z‖ ≤ C
⊢ ‖∫ᶜ (x : E) in Path.segment a b, ω x‖ ≤ C * ‖b - a‖ * |1 - 0|

Tactic:
by
    let : NormedSpace ℝ F := .restrictScalars ℝ 𝕜 F
    rw [curveIntegral_segment]
    refine intervalIntegral.norm_integral_le_of_norm_le_const fun t ht ↦ ?_
    rw [segment_eq_image_lineMap] at h
    rw [uIoc_of_le zero_le_one] at ht
    apply_rules [(ω _).le_of_opNorm_le, mem_image_of_mem, Ioc_subset_Icc_self]

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

Goal state before:
pp: case refine_1
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
hfc : ContinuousOn f (segment ℝ a b)
hfd : ∀ t ∈ Ioo 0 1, LineDifferentiableAt ℝ f ((lineMap a b) t) (b - a)
hf' : ∀ᵐ (t : ℝ), t ∈ Ioo 0 1 → ‖lineDeriv ℝ f ((lineMap a b) t) (b - a)‖ ≤ C
g : ℝ → F := fun t ↦ f ((lineMap a b) t)
⊢ ContinuousOn (⇑(lineMap a b)) (Icc 0 1)

pp: case refine_2
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
hfc : ContinuousOn f (segment ℝ a b)
hfd : ∀ t ∈ Ioo 0 1, LineDifferentiableAt ℝ f ((lineMap a b) t) (b - a)
hf' : ∀ᵐ (t : ℝ), t ∈ Ioo 0 1 → ‖lineDeriv ℝ f ((lineMap a b) t) (b - a)‖ ≤ C
g : ℝ → F := fun t ↦ f ((lineMap a b) t)
⊢ MapsTo (⇑(lineMap a b)) (Icc 0 1) (segment ℝ a b)

Tactic:
· exact AffineMap.lineMap_continuous.continuousOn

Goal state after:
case refine_2
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
hfc : ContinuousOn f (segment ℝ a b)
hfd : ∀ t ∈ Ioo 0 1, LineDifferentiableAt ℝ f ((lineMap a b) t) (b - a)
hf' : ∀ᵐ (t : ℝ), t ∈ Ioo 0 1 → ‖lineDeriv ℝ f ((lineMap a b) t) (b - a)‖ ≤ C
g : ℝ → F := fun t ↦ f ((lineMap a b) t)
⊢ MapsTo (⇑(lineMap a b)) (Icc 0 1) (segment ℝ a b)
Module: Mathlib.MeasureTheory.Integral.Regular

Goal state before:
pp: case a.f_nonneg
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
⊢ 0 ≤ᵐ[μ] f

pp: case a.hs
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 ∈ k, 1 ≤ f x

Tactic:
· exact Eventually.of_forall f_nonneg

Goal state after:
case a.hs
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 ∈ k, 1 ≤ f x
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 ∈ Finset.range N, ∫ (x : ℝ) in a + ↑x * h..a + ↑(x + 1) * h, f x =
    trapezoidal_integral f N a (a + ↑N * h) - ∫ (x : ℝ) in a..a + ↑N * h, f x

Tactic:
rw [sum_integral_adjacent_intervals]

Goal state after:
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

f : ℝ → ℝ
N : ℕ
a h : ℝ
N_nonzero : 0 < N
h_f_int : IntervalIntegrable f volume a (a + ↑N * h)
⊢ ∀ k < N, IntervalIntegrable f volume (a + ↑k * h) (a + ↑(k + 1) * h)
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
g : ℝ → ℝ := fun t ↦ trapezoidal_error f 1 a t
dg : ℝ → ℝ := fun t ↦ 1 / 2 * (f a + f t) + (t - a) / 2 * _root_.derivWithin f (Set.Icc a b) t - f t
ddg : ℝ → ℝ := fun t ↦ (t - a) / 2 * iteratedDerivWithin 2 f (Set.Icc a b) t
h_dg : ∀ y ∈ Set.Icc a b, HasDerivWithinAt g (dg y) (Set.Icc a b) y
h_ddg : ∀ y ∈ Set.Icc a b, HasDerivWithinAt dg (ddg y) (Set.Icc a b) y
x : ℝ
hx : x ∈ Set.Icc a b
⊢ |ddg x| ≤ ζ / 2 * (x - a)

Tactic:
ddg,

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
g : ℝ → ℝ := fun t ↦ trapezoidal_error f 1 a t
dg : ℝ → ℝ := fun t ↦ 1 / 2 * (f a + f t) + (t - a) / 2 * _root_.derivWithin f (Set.Icc a b) t - f t
ddg : ℝ → ℝ := fun t ↦ (t - a) / 2 * iteratedDerivWithin 2 f (Set.Icc a b) t
h_dg : ∀ y ∈ Set.Icc a b, HasDerivWithinAt g (dg y) (Set.Icc a b) y
h_ddg : ∀ y ∈ Set.Icc a b, HasDerivWithinAt dg (ddg y) (Set.Icc a b) y
x : ℝ
hx : x ∈ Set.Icc a b
⊢ |(x - a) / 2 * iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ / 2 * (x - a)
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

Tactic:
intro m

Goal state after:
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
m : Fin N
⊢ m ∈ univ → ∀ ⦃y : Fin N⦄, y ∈ univ → m ≠ y → (Disjoint on E) m y
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
mF : MeasurableSpace F
μ : Measure E
ν : Measure F
p : ℝ≥0∞
inst✝² : Fact (1 ≤ p)
inst✝¹ : SFinite μ
inst✝ : SFinite ν
L : StrongDual ℝ (WithLp p (E × F))
⊢ ∫ (v : WithLp p (E × F)), cexp (↑(L v) * I) ∂Measure.map (toLp p) (μ.prod ν) =
    ∫ (z : E × F),
      cexp
        (↑(L
              ((↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inl ℝ E F) z.1 +
                (↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inr ℝ E F) z.2)) *
          I) ∂μ.prod ν

Tactic:
ContinuousLinearMap.comp_inl_add_comp_inr

Goal state after:
E : Type u_2
F : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
mF : MeasurableSpace F
μ : Measure E
ν : Measure F
p : ℝ≥0∞
inst✝² : Fact (1 ≤ p)
inst✝¹ : SFinite μ
inst✝ : SFinite ν
L : StrongDual ℝ (WithLp p (E × F))
⊢ ∫ (v : WithLp p (E × F)), cexp (↑(L v) * I) ∂Measure.map (toLp p) (μ.prod ν) =
    ∫ (z : E × F), cexp (↑(L (↑(prodContinuousLinearEquiv p ℝ E F).symm z)) * I) ∂μ.prod ν
Module: Mathlib.MeasureTheory.Measure.FiniteMeasurePi

Goal state before:
pp: case h
ι : Type u_1
α : ι → Type u_2
inst✝⁵ : Fintype ι
inst✝⁴ : (i : ι) → MeasurableSpace (α i)
inst✝³ : (i : ι) → TopologicalSpace (α i)
inst✝² : ∀ (i : ι), SecondCountableTopology (α i)
inst✝¹ : ∀ (i : ι), PseudoMetrizableSpace (α i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (α i)
μ : (i : ι) → ProbabilityMeasure (α i)
S : Set (Set ((i : ι) → α i)) :=
  {t | ∃ s, t = univ.pi s ∧ (∀ (i : ι), MeasurableSet (s i)) ∧ ∀ (i : ι), (μ i) (frontier (s i)) = 0}
this✝ : IsPiSystem S
this : (i : ι) → PseudoMetricSpace (α i) := fun i ↦ pseudoMetrizableSpacePseudoMetric (α i)
u : Set ((i : ι) → α i)
u_open : IsOpen[Pi.topologicalSpace] u
x : (i : ι) → α i
⊢ x ∈ u → ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u

Tactic:
xu

Goal state after:
case h
ι : Type u_1
α : ι → Type u_2
inst✝⁵ : Fintype ι
inst✝⁴ : (i : ι) → MeasurableSpace (α i)
inst✝³ : (i : ι) → TopologicalSpace (α i)
inst✝² : ∀ (i : ι), SecondCountableTopology (α i)
inst✝¹ : ∀ (i : ι), PseudoMetrizableSpace (α i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (α i)
μ : (i : ι) → ProbabilityMeasure (α i)
S : Set (Set ((i : ι) → α i)) :=
  {t | ∃ s, t = univ.pi s ∧ (∀ (i : ι), MeasurableSet (s i)) ∧ ∀ (i : ι), (μ i) (frontier (s i)) = 0}
this✝ : IsPiSystem S
this : (i : ι) → PseudoMetricSpace (α i) := fun i ↦ pseudoMetrizableSpacePseudoMetric (α i)
u : Set ((i : ι) → α i)
u_open : IsOpen[Pi.topologicalSpace] u
x : (i : ι) → α i
xu : x ∈ u
⊢ ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case calc_7
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)
h : ∑ n, y n ≤ ↑N * b
⊢ ∫ (x : X), f x ∂μ + 2 * ε' * μ.real K + ε' / ↑N * ∑ n, (|a| + y n + ε') ≤
    ∫ (x : X), f x ∂μ + ε' * (2 * μ.real K + |a| + b + ε')

Tactic:
simp only [Finset.sum_add_distrib, Finset.sum_add_distrib,
               Fin.sum_const, Fin.sum_const, nsmul_eq_mul, ← add_assoc, mul_add, ← mul_assoc]

Goal state after:
case calc_7
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)
h : ∑ n, y n ≤ ↑N * b
⊢ ∫ (x : X), f x ∂μ + 2 * ε' * μ.real K + ε' / ↑N * ↑N * |a| + ε' / ↑N * ∑ x, y x + ε' / ↑N * ↑N * ε' ≤
    ∫ (x : X), f x ∂μ + ε' * 2 * μ.real K + ε' * |a| + ε' * b + ε' * ε'
Module: Mathlib.MeasureTheory.Measure.FiniteMeasureProd

Goal state before:
pp: case h
α : Type u_1
inst✝⁹ : MeasurableSpace α
β : Type u_2
inst✝⁸ : MeasurableSpace β
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : SecondCountableTopology β
inst✝³ : PseudoMetrizableSpace α
inst✝² : PseudoMetrizableSpace β
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace β
μ : ProbabilityMeasure α × ProbabilityMeasure β
S : Set (Set (α × β)) :=
  {t | ∃ a b, MeasurableSet a ∧ μ.1 (frontier a) = 0 ∧ MeasurableSet b ∧ μ.2 (frontier b) = 0 ∧ t = a ×ˢ b}
this✝¹ : IsPiSystem S
this✝ : PseudoMetricSpace α := pseudoMetrizableSpacePseudoMetric α
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
u : Set (α × β)
u_open : IsOpen[instTopologicalSpaceProd] u
x : α × β
⊢ x ∈ u → ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u

Tactic:
xu

Goal state after:
case h
α : Type u_1
inst✝⁹ : MeasurableSpace α
β : Type u_2
inst✝⁸ : MeasurableSpace β
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : SecondCountableTopology β
inst✝³ : PseudoMetrizableSpace α
inst✝² : PseudoMetrizableSpace β
inst✝¹ : OpensMeasurableSpace α
inst✝ : OpensMeasurableSpace β
μ : ProbabilityMeasure α × ProbabilityMeasure β
S : Set (Set (α × β)) :=
  {t | ∃ a b, MeasurableSet a ∧ μ.1 (frontier a) = 0 ∧ MeasurableSet b ∧ μ.2 (frontier b) = 0 ∧ t = a ×ˢ b}
this✝¹ : IsPiSystem S
this✝ : PseudoMetricSpace α := pseudoMetrizableSpacePseudoMetric α
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
u : Set (α × β)
u_open : IsOpen[instTopologicalSpaceProd] u
x : α × β
xu : x ∈ u
⊢ ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u