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

Goal state before:
pp: case e'_3
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)
hgc : ContinuousOn g (Icc 0 1)
hdg : ∀ t ∈ Ioo 0 1, HasDerivAt g (lineDeriv ℝ f ((lineMap a b) t) (b - a)) t
this : ‖g 1 - g 0‖ ≤ C * volume.real {t | t ∈ Ioo 0 1 ∧ deriv g t ≠ 0}
⊢ ‖f b - f a‖ = ‖g 1 - g 0‖

pp: case e'_4
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)
hgc : ContinuousOn g (Icc 0 1)
hdg : ∀ t ∈ Ioo 0 1, HasDerivAt g (lineDeriv ℝ f ((lineMap a b) t) (b - a)) t
this : ‖g 1 - g 0‖ ≤ C * volume.real {t | t ∈ Ioo 0 1 ∧ deriv g t ≠ 0}
⊢ C * volume.real {t | t ∈ Ioo 0 1 ∧ lineDeriv ℝ f ((lineMap a b) t) (b - a) ≠ 0} =
    C * volume.real {t | t ∈ Ioo 0 1 ∧ deriv g t ≠ 0}

Tactic:
· simp [g]

Goal state after:
case e'_4
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)
hgc : ContinuousOn g (Icc 0 1)
hdg : ∀ t ∈ Ioo 0 1, HasDerivAt g (lineDeriv ℝ f ((lineMap a b) t) (b - a)) t
this : ‖g 1 - g 0‖ ≤ C * volume.real {t | t ∈ Ioo 0 1 ∧ deriv g t ≠ 0}
⊢ C * volume.real {t | t ∈ Ioo 0 1 ∧ lineDeriv ℝ f ((lineMap a b) t) (b - a) ≠ 0} =
    C * volume.real {t | t ∈ Ioo 0 1 ∧ deriv g t ≠ 0}
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 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)
t : ℝ
⊢ t ∈ Ioo 0 1 → ‖lineDeriv ℝ f ((lineMap a b) t) (b - a)‖ ≤ C * ‖b - a‖

Tactic:
ht

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 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)
t : ℝ
ht : t ∈ Ioo 0 1
⊢ ‖lineDeriv ℝ f ((lineMap a b) t) (b - a)‖ ≤ C * ‖b - a‖
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: case h
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
V : Set X
hVcp : IsCompact V
hKsubintV : K.carrier ⊆ interior V
hIsCompact_closure_interior : IsCompact (closure (interior V))
f : C(X, ℝ)
hsuppfsubV : tsupport ⇑f ⊆ interior V
hfeq1onK : EqOn (⇑f) 1 K.carrier
hfinicc : ∀ (x : X), f x ∈ Icc 0 1
hfHasCompactSupport : HasCompactSupport ⇑f
x : X
hx : x ∈ K
⊢ 1 = { toContinuousMap := f, hasCompactSupport' := hfHasCompactSupport }.nnrealPart x

Tactic:
simp only [nnrealPart_apply, CompactlySupportedContinuousMap.coe_mk]

Goal state after:
case h
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
V : Set X
hVcp : IsCompact V
hKsubintV : K.carrier ⊆ interior V
hIsCompact_closure_interior : IsCompact (closure (interior V))
f : C(X, ℝ)
hsuppfsubV : tsupport ⇑f ⊆ interior V
hfeq1onK : EqOn (⇑f) 1 K.carrier
hfinicc : ∀ (x : X), f x ∈ Icc 0 1
hfHasCompactSupport : HasCompactSupport ⇑f
x : X
hx : x ∈ K
⊢ 1 = (f x).toNNReal
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
y : ℝ
hx : y ∈ Set.Icc a b
dfaky : ℝ := _root_.derivWithin f (Set.Icc a b) y
⊢ HasDerivWithinAt (_root_.derivWithin f (Set.Icc a b)) (iteratedDerivWithin 2 f (Set.Icc a b) y) (Set.Icc a b) y

Tactic:
iteratedDerivWithin_eq_iterate

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
y : ℝ
hx : y ∈ Set.Icc a b
dfaky : ℝ := _root_.derivWithin f (Set.Icc a b) y
⊢ HasDerivWithinAt (_root_.derivWithin f (Set.Icc a b)) ((fun g ↦ _root_.derivWithin g (Set.Icc a b))^[2] f y)
    (Set.Icc a b) y
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 : s₀ᶜ ∪ s₁ᶜ = univ
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)
x : X
hx : x ∈ t
⊢ x ∈ s₀ᶜ ∨ x ∈ s₁ᶜ

Tactic:
rw [← mem_union, disj]

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 : s₀ᶜ ∪ s₁ᶜ = univ
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)
x : X
hx : x ∈ t
⊢ x ∈ univ
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
bound_ddg : ∀ x ∈ Set.Icc a b, |ddg x| ≤ ζ / 2 * (x - a) ^ 1
φ φ' : ℝ → ℝ
h : ∀ x ∈ Set.Icc a b, HasDerivWithinAt φ (φ' x) (Set.Icc a b) x
h0 : φ a = 0
c : ℝ
n : ℕ
h_bound : ∀ t ∈ Set.Icc a b, |φ' t| ≤ c * (t - a) ^ n
t : ℝ
⊢ t ∈ Set.Icc a b → |φ t| ≤ c / (↑n + 1) * (t - a) ^ (n + 1)

Tactic:
ht

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
bound_ddg : ∀ x ∈ Set.Icc a b, |ddg x| ≤ ζ / 2 * (x - a) ^ 1
φ φ' : ℝ → ℝ
h : ∀ x ∈ Set.Icc a b, HasDerivWithinAt φ (φ' x) (Set.Icc a b) x
h0 : φ a = 0
c : ℝ
n : ℕ
h_bound : ∀ t ∈ Set.Icc a b, |φ' t| ≤ c * (t - a) ^ n
t : ℝ
ht : t ∈ Set.Icc a b
⊢ |φ t| ≤ c / (↑n + 1) * (t - a) ^ (n + 1)
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
Λ : (X →C_c ℝ) →ₚ[ℝ] ℝ
inst✝ : LocallyCompactSpace X
f : X →C_c ℝ
hf : ∀ (x : X), 0 ≤ f x ∧ f x ≤ 1
V : Set X
hV : tsupport ⇑f ⊆ V
this :
  (rieszContent (toNNRealLinear Λ)).measure ↑{ carrier := tsupport ⇑f, isCompact' := ⋯ } =
    (rieszContent (toNNRealLinear Λ)) { carrier := tsupport ⇑f, isCompact' := ⋯ }
⊢ ENNReal.ofReal (Λ f) ≤ (rieszContent (toNNRealLinear Λ)) { carrier := tsupport ⇑f, isCompact' := ⋯ }

Tactic:
rieszContent,

Goal state after:
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
Λ : (X →C_c ℝ) →ₚ[ℝ] ℝ
inst✝ : LocallyCompactSpace X
f : X →C_c ℝ
hf : ∀ (x : X), 0 ≤ f x ∧ 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' := ⋯ }
⊢ ENNReal.ofReal (Λ f) ≤
    { toFun := rieszContentAux (toNNRealLinear Λ), mono' := ⋯, sup_disjoint' := ⋯, sup_le' := ⋯ }
      { carrier := tsupport ⇑f, isCompact' := ⋯ }
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

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

Tactic:
intro p hp

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

Goal state before:
pp: case neg
X : Type u_1
inst✝⁴ : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ≥0
hf : ∀ (x : X), f x ≤ 1
K : Set X
hK : IsCompact K
h : tsupport ⇑f ⊆ K
ε : ℝ≥0
hε : 0 < ε
g : X →C_c ℝ≥0
hg :
  (∀ x ∈ { carrier := K, isCompact' := hK }, 1 ≤ g x) ∧ Λ g < rieszContentAux Λ { carrier := K, isCompact' := hK } + ε
x : X
hx : x ∉ tsupport ⇑f
⊢ f x ≤ g x

Tactic:
image_eq_zero_of_notMem_tsupport hx

Goal state after:
case neg
X : Type u_1
inst✝⁴ : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ≥0
hf : ∀ (x : X), f x ≤ 1
K : Set X
hK : IsCompact K
h : tsupport ⇑f ⊆ K
ε : ℝ≥0
hε : 0 < ε
g : X →C_c ℝ≥0
hg :
  (∀ x ∈ { carrier := K, isCompact' := hK }, 1 ≤ g x) ∧ Λ g < rieszContentAux Λ { carrier := K, isCompact' := hK } + ε
x : X
hx : x ∉ tsupport ⇑f
⊢ 0 ≤ g x
Module: Mathlib.MeasureTheory.MeasurableSpace.Card

Goal state before:
pp: α : Type u
s : Set (Set α)
t : Set α
⊢ t ∈ generateMeasurableRec s (ω_ 1) → t ∈ ⋃ i, ⋃ (_ : i < ω_ 1), generateMeasurableRec s i

Tactic:
ht

Goal state after:
α : Type u
s : Set (Set α)
t : Set α
ht : t ∈ generateMeasurableRec s (ω_ 1)
⊢ t ∈ ⋃ i, ⋃ (_ : i < ω_ 1), generateMeasurableRec s i
Module: Mathlib.MeasureTheory.MeasurableSpace.Card

Goal state before:
pp: case refine_1
α : Type u
s : Set (Set α)
t✝ : Set α
ht : t✝ ∈ generateMeasurableRec s (ω_ 1)
t : Set α
⊢ t ∈ s → ∃ i, ∃ (_ : i < ω_ 1), t ∈ generateMeasurableRec s i

Tactic:
ht

Goal state after:
case refine_1
α : Type u
s : Set (Set α)
t✝ : Set α
ht✝ : t✝ ∈ generateMeasurableRec s (ω_ 1)
t : Set α
ht : t ∈ s
⊢ ∃ i, ∃ (_ : i < ω_ 1), t ∈ generateMeasurableRec s i
Module: Mathlib.MeasureTheory.MeasurableSpace.Card

Goal state before:
pp: case refine_1
α : Type u
s : Set (Set α)
t : Set α
ht : t ∈ generateMeasurableRec s (ω_ 1)
⊢ ∀ t ∈ s, ∃ i, ∃ (_ : i < ω_ 1), t ∈ generateMeasurableRec s i

Tactic:
intro t ht
    exact ⟨0, omega_pos 1, self_subset_generateMeasurableRec s 0 ht⟩

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)
⊢ ∃ i, ∃ (_ : i < ω_ 1), ⋃ n, f n ∈ generateMeasurableRec s i

Tactic:
refine ⟨_, Ordinal.lift_iSup_add_one_lt_of_lt_cof ?_ fun n => (hI n).1,
      iUnion_mem_generateMeasurableRec fun n => ⟨_, Ordinal.lt_iSup_add_one I n, (hI n).2⟩⟩

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

Goal state before:
pp: α : Type u
s : Set (Set α)
t : Set α
⊢ t ∈ {t | GenerateMeasurable s t} → t ∈ generateMeasurableRec s (ω_ 1)

Tactic:
ht

Goal state after:
α : Type u
s : Set (Set α)
t : Set α
ht : t ∈ {t | GenerateMeasurable s t}
⊢ t ∈ generateMeasurableRec s (ω_ 1)
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
⊢ μ (E n) + ENNReal.ofReal (ε' / ↑N) < ∞

Tactic:
WithTop.add_lt_top

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)
n : Fin N
hn : n ∈ Finset.univ
⊢ μ (E n) < ⊤ ∧ ENNReal.ofReal (ε' / ↑N) < ⊤
Module: Mathlib.MeasureTheory.Measure.DiracProba

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
K : Set X
K_closed : IsClosed[inst✝¹] K
x : X
x_notin_K : x ∉ K
g : X → ↑unitInterval
g_cont : Continuous[inst✝¹, _] g
gx_zero : g x = 0
g_one_on_K : EqOn g 1 K
⊢ ∀ (x y : X), dist (↑(g x)).toNNReal (↑(g y)).toNNReal ≤ 1

Tactic:
refine fun x y ↦ ((Real.lipschitzWith_toNNReal).dist_le_mul (g x) (g y)).trans ?_

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
K : Set X
K_closed : IsClosed[inst✝¹] K
x✝ : X
x_notin_K : x✝ ∉ K
g : X → ↑unitInterval
g_cont : Continuous[inst✝¹, _] g
gx_zero : g x✝ = 0
g_one_on_K : EqOn g 1 K
x y : X
⊢ ↑1 * dist ↑(g x) ↑(g y) ≤ 1
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 ∈ Ioo 0 1, φ (a, b) ∈ t
hω : ∀ a ∈ Ioo 0 1, ∀ b ∈ Ioo 0 1, (a, b) ∉ s → HasFDerivWithinAt ω (dω (φ (a, b))) t (φ (a, b))
hωc : ContinuousOn ω (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t)
hdω_symm :
  ∀ a ∈ Ioo 0 1,
    ∀ b ∈ Ioo 0 1,
      (a, b) ∉ s →
        ∀ u ∈ tangentConeAt ℝ t (φ (a, b)),
          ∀ v ∈ tangentConeAt ℝ t (φ (a, b)), ((dω (φ (a, b))) u) v = ((dω (φ (a, b))) 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 : ↑I), ψ (↑a, ↑b) = φ (a, b)
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
hd2ψ_symm : ∀ a ∈ Icc 0 1, ∀ (x y : ℝ × ℝ), ((d2ψ a) x) y = ((d2ψ a) y) x
η : ℝ × ℝ → ℝ × ℝ →L[ℝ] F := fun a ↦ ω (ψ a) ∘SL dψ a
f : ℝ × ℝ → F := fun a ↦ (η a) (0, 1)
hf : ∀ a ∈ Icc 0 1, f a = (ω (ψ a)) (derivWithin (ψ ∘ fun x ↦ (a.1, x)) I a.2)
g : ℝ × ℝ → F := fun a ↦ -(η a) (1, 0)
hg : ∀ a ∈ Icc 0 1, g a = (ω (ψ a)) (-derivWithin (ψ ∘ fun x ↦ (x, a.2)) I a.1)
dη : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ →L[ℝ] F :=
  fun a ↦ (ContinuousLinearMap.compL ℝ (ℝ × ℝ) E F) (ω (ψ a)) ∘SL d2ψ a + (dω (ψ a)).bilinearComp (dψ a) (dψ a)
hdη : ∀ a ∈ U \ s', HasFDerivAt η (dη a) a
u v : ℝ × ℝ
a b : { x // x ∈ I }
hU : (↑a, ↑b) ∈ U
hs' : (↑a, ↑b) ∉ s'
hdψ_mem : ∀ (u : ℝ × ℝ), (dψ (↑a, ↑b)) u ∈ tangentConeAt ℝ t (φ (a, b))
⊢ (a, b) ∉ s

Tactic:
simpa [hmem_s'] using hs'

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

Goal state before:
pp: Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
ε : ℝ
ε_pos : 0 < ε
hμν : levyProkhorovEDist μ ν < ENNReal.ofReal ε
f : Ω →ᵇ ℝ
f_nn : 0 ≤ᵐ[μ] ⇑f
⊢ ∫ (ω : Ω), f ω ∂μ ≤ (∫ (t : ℝ) in Ioc 0 ‖f‖, ν.real (thickening ε {a | t ≤ f a})) + ε * ‖f‖

Tactic:
BoundedContinuousFunction.integral_eq_integral_meas_le f μ f_nn

Goal state after:
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
ε : ℝ
ε_pos : 0 < ε
hμν : levyProkhorovEDist μ ν < ENNReal.ofReal ε
f : Ω →ᵇ ℝ
f_nn : 0 ≤ᵐ[μ] ⇑f
⊢ ∫ (t : ℝ) in Ioc 0 ‖f‖, μ.real {a | t ≤ f a} ≤ (∫ (t : ℝ) in Ioc 0 ‖f‖, ν.real (thickening ε {a | t ≤ f a})) + ε * ‖f‖
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case a
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
⊢ ∫ (x : X), f x ∂rieszMeasure Λ ≤ Λ f

pp: case a
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
⊢ Λ f ≤ ∫ (x : X), f x ∂rieszMeasure Λ

Tactic:
· calc
      _ = - ∫ x, (-f) x ∂(rieszMeasure Λ) := by simpa using integral_neg' (-f)
      _ ≤ - Λ (-f) := neg_le_neg (integral_riesz_aux Λ (-f))
      _ = _ := by simp

Goal state after:
case a
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
⊢ Λ f ≤ ∫ (x : X), f x ∂rieszMeasure Λ
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case neg
X : Type u_1
inst✝⁷ : TopologicalSpace X
inst✝⁶ : T2Space X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
μ ν : Measure X
inst✝³ : LocallyCompactSpace X
inst✝² : ν.OuterRegular
inst✝¹ : IsFiniteMeasureOnCompacts ν
inst✝ : IsFiniteMeasureOnCompacts μ
hμν : ∀ (f : X →C_c ℝ), ∫ (x : X), f x ∂μ ≤ ∫ (x : X), f x ∂ν
K : Set X
hK : IsCompact K
ε : NNReal
hε : 0 < ε
hν : ν K < ∞
hνK : ν K ≠ ∞
hμK : μ K ≠ ∞
V : Set X
pV1 : V ⊇ K
pV2 : IsOpen[inst✝⁷] V
pV3 : ν V ≤ ν K + ↑ε
VltTop : ν V < ∞
f : X →C_c ℝ
pf1 : EqOn (⇑f) 1 K
pf2 : tsupport ⇑f ⊆ V
pf3 : ∀ (x : X), f x ∈ Icc 0 1
x : X
hx : x ∉ tsupport ⇑f
⊢ f x ≤ V.indicator 1 x

Tactic:
· simp [image_eq_zero_of_notMem_tsupport hx, Set.indicator_nonneg]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
this✝ : (ENNReal.ofReal p).toReal = p
h₂ : ∀ (x : ι → ℝ), 0 ≤ ∑ i, |x i| ^ p
eq_norm : ∀ (x : ι → ℝ), ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, |x i| ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
nm_zero : ‖toLp (ENNReal.ofReal p) 0‖ = 0
eq_zero : ∀ (x : ι → ℝ), ‖toLp (ENNReal.ofReal p) x‖ = 0 ↔ x = 0
⊢ volume {x | ∑ i, |x i| ^ p < 1} = ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (↑(card ι) / p + 1))

Tactic:
have nm_neg := fun x : ι → ℝ => norm_neg (toLp (.ofReal p) x)

Goal state after:
ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
this✝ : (ENNReal.ofReal p).toReal = p
h₂ : ∀ (x : ι → ℝ), 0 ≤ ∑ i, |x i| ^ p
eq_norm : ∀ (x : ι → ℝ), ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, |x i| ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
nm_zero : ‖toLp (ENNReal.ofReal p) 0‖ = 0
eq_zero : ∀ (x : ι → ℝ), ‖toLp (ENNReal.ofReal p) x‖ = 0 ↔ x = 0
nm_neg : ∀ (x : ι → ℝ), ‖-toLp (ENNReal.ofReal p) x‖ = ‖toLp (ENNReal.ofReal p) x‖
⊢ volume {x | ∑ i, |x i| ^ p < 1} = ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (↑(card ι) / p + 1))