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

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

Tactic:
hx

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

Goal state before:
pp: case h.refine_1
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
ε : ℝ
hε : 0 < ε
E : Set X
μ : Content X
hμ : μ.outerMeasure E ≠ ∞
hμ' : MeasurableSet E
c : ℝ
hfE : ∀ x ∈ E, f x < c
hε' : ε.toNNReal ≠ 0
V₁ : Opens X
hV₁ : E ⊆ ↑V₁ ∧ μ.outerMeasure ↑V₁ ≤ μ.outerMeasure E + ↑ε.toNNReal
V₂ : Opens X := { carrier := ⇑f ⁻¹' Iio c, is_open' := ⋯ }
x : X
⊢ x ∈ V₁ ⊓ V₂ → f x < c

Tactic:
hx

Goal state after:
case h.refine_1
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
ε : ℝ
hε : 0 < ε
E : Set X
μ : Content X
hμ : μ.outerMeasure E ≠ ∞
hμ' : MeasurableSet E
c : ℝ
hfE : ∀ x ∈ E, f x < c
hε' : ε.toNNReal ≠ 0
V₁ : Opens X
hV₁ : E ⊆ ↑V₁ ∧ μ.outerMeasure ↑V₁ ≤ μ.outerMeasure E + ↑ε.toNNReal
V₂ : Opens X := { carrier := ⇑f ⁻¹' Iio c, is_open' := ⋯ }
x : X
hx : x ∈ V₁ ⊓ V₂
⊢ f x < c
Module: Mathlib.MeasureTheory.MeasurableSpace.Card

Goal state before:
pp: case h1.h1.h1
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
C : #↑(⋃ j, ⋃ (_ : j < i), generateMeasurableRec s j) ≤ max (#↑s) 2 ^ ℵ₀
⊢ #↑s ≤ max (#↑s) 2 ^ ℵ₀

Tactic:
exact (self_le_power _ one_le_aleph0).trans (power_le_power_right (le_max_left _ _))

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

Goal state before:
pp: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : NormalSpace X
inst✝ : R0Space X
x : X
K : Set X
hK : IsClosed[inst✝²] K
⊢ x ∉ K → ∃ f, Continuous[inst✝², _] f ∧ f x = 0 ∧ EqOn f 1 K

Tactic:
hx

Goal state after:
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : NormalSpace X
inst✝ : R0Space X
x : X
K : Set X
hK : IsClosed[inst✝²] K
hx : x ∉ K
⊢ ∃ f, Continuous[inst✝², _] f ∧ f x = 0 ∧ EqOn f 1 K
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
x : E
L : StrongDual ℝ E
⊢ charFunDual (Measure.dirac x) L = cexp (↑(L x) * I)

Tactic:
charFunDual_apply,

Goal state after:
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
x : E
L : StrongDual ℝ E
⊢ ∫ (v : E), cexp (↑(L v) * I) ∂Measure.dirac x = cexp (↑(L x) * I)
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
mE : MeasurableSpace E
μ : Measure E
inst✝ : BorelSpace E
r : E
L : StrongDual ℝ E
⊢ charFunDual (Measure.map (fun x ↦ x + r) μ) L = charFunDual μ L * cexp (↑(L r) * I)

Tactic:
charFunDual_apply,

Goal state after:
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
mE : MeasurableSpace E
μ : Measure E
inst✝ : BorelSpace E
r : E
L : StrongDual ℝ E
⊢ ∫ (v : E), cexp (↑(L v) * I) ∂Measure.map (fun x ↦ x + r) μ = charFunDual μ L * cexp (↑(L r) * I)
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
mE : MeasurableSpace E
μ : Measure E
inst✝ : BorelSpace E
r : E
L : StrongDual ℝ E
⊢ ∫ (v : E), cexp (↑(L v) * I) ∂Measure.map (fun x ↦ x + r) μ = charFunDual μ L * cexp (↑(L r) * I)

Tactic:
charFunDual_apply,

Goal state after:
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
mE : MeasurableSpace E
μ : Measure E
inst✝ : BorelSpace E
r : E
L : StrongDual ℝ E
⊢ ∫ (v : E), cexp (↑(L v) * I) ∂Measure.map (fun x ↦ x + r) μ = (∫ (v : E), cexp (↑(L v) * I) ∂μ) * cexp (↑(L r) * I)
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
mF : MeasurableSpace F
μ : Measure E
ν : Measure F
inst✝¹ : SFinite μ
inst✝ : SFinite ν
L : StrongDual ℝ (E × F)
⊢ charFunDual (μ.prod ν) L =
    charFunDual μ (L ∘SL ContinuousLinearMap.inl ℝ E F) * charFunDual ν (L ∘SL ContinuousLinearMap.inr ℝ E F)

Tactic:
charFunDual_apply,

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

Tactic:
charFunDual_apply,

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) =
    (∫ (v : E),
        cexp (↑((L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inl ℝ E F) v) * I) ∂μ) *
      ∫ (v : F), cexp (↑((L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inr ℝ E F) v) * I) ∂ν
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: ι : Type u_4
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
E : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace ℝ (E i)
mE : (i : ι) → MeasurableSpace (E i)
μ : (i : ι) → Measure (E i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
L : StrongDual ℝ ((i : ι) → E i)
⊢ charFunDual (Measure.pi μ) L = ∏ i, charFunDual (μ i) (L ∘SL ContinuousLinearMap.single ℝ E i)

Tactic:
charFunDual_apply,

Goal state after:
ι : Type u_4
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
E : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace ℝ (E i)
mE : (i : ι) → MeasurableSpace (E i)
μ : (i : ι) → Measure (E i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
L : StrongDual ℝ ((i : ι) → E i)
⊢ ∫ (v : (i : ι) → E i), cexp (↑(L v) * I) ∂Measure.pi μ =
    ∏ i, ∫ (v : E i), cexp (↑((L ∘SL ContinuousLinearMap.single ℝ E i) v) * I) ∂μ i
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: p : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ p)
ι : Type u_4
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
E : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace ℝ (E i)
mE : (i : ι) → MeasurableSpace (E i)
μ : (i : ι) → Measure (E i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
L : StrongDual ℝ (PiLp p E)
⊢ charFunDual (Measure.map (toLp p) (Measure.pi μ)) L =
    ∏ i, charFunDual (μ i) (L ∘SL ↑(PiLp.continuousLinearEquiv p ℝ E).symm ∘SL ContinuousLinearMap.single ℝ E i)

Tactic:
charFunDual_apply,

Goal state after:
p : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ p)
ι : Type u_4
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
E : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace ℝ (E i)
mE : (i : ι) → MeasurableSpace (E i)
μ : (i : ι) → Measure (E i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
L : StrongDual ℝ (PiLp p E)
⊢ ∫ (v : WithLp p ((i : ι) → E i)), cexp (↑(L v) * I) ∂Measure.map (toLp p) (Measure.pi μ) =
    ∏ i,
      ∫ (v : E i),
        cexp (↑((L ∘SL ↑(PiLp.continuousLinearEquiv p ℝ E).symm ∘SL ContinuousLinearMap.single ℝ E i) v) * I) ∂μ i
Module: Mathlib.MeasureTheory.Measure.DiracProba

Goal state before:
pp: X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : T0Space X
x : X
μ : ↑(range diracProba)
h : ↑μ = diracProba x
⊢ diracProbaInverse μ = x

Tactic:
apply injective_diracProba (X := X)
  simp only [← h]
  exact (mem_range.mp μ.prop).choose_spec

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

Goal state before:
pp: case hf
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
x : X
⊢ x ∈ E n → y n - ε' ≤ f x

Tactic:
hx

Goal state after:
case hf
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
x : X
hx : x ∈ E n
⊢ y n - ε' ≤ f x
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)), IsOpen[Pi.topologicalSpace] u → ∀ x ∈ u, ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u

Tactic:
intro u u_open x 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.Measure.LevyProkhorovMetric

Goal state before:
pp: Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoEMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ ν : Measure Ω
hLP : levyProkhorovEDist μ ν = 0
s : Set Ω
s_mble : MeasurableSet s
h_finite : ∃ δ > 0, ν (thickening δ s) ≠ ∞
x : ℝ≥0∞
⊢ x ∈ Ioo 0 1 → x.toReal ∈ Ioi 0

Tactic:
hx

Goal state after:
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoEMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ ν : Measure Ω
hLP : levyProkhorovEDist μ ν = 0
s : Set Ω
s_mble : MeasurableSet s
h_finite : ∃ δ > 0, ν (thickening δ s) ≠ ∞
x : ℝ≥0∞
hx : x ∈ Ioo 0 1
⊢ x.toReal ∈ Ioi 0
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
key : (fun t ↦ μ.real {a | t ≤ f a}) ≤ fun t ↦ ν.real (thickening ε {a | t ≤ f a}) + ε
intble₁ : IntegrableOn (fun t ↦ μ.real {a | t ≤ f a}) (Ioc 0 ‖f‖) volume
⊢ Antitone fun x ↦ ν (thickening ε {a | x ≤ f a})

Tactic:
exact fun _ _ hst ↦ measure_mono <| thickening_subset_of_subset ε (fun _ h ↦ hst.trans h)

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
⊢ SeqContinuous toMeasure

Tactic:
intro μs

Goal state after:
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)
⊢ ∀ ⦃p : LevyProkhorov (ProbabilityMeasure Ω)⦄, Tendsto μs atTop (𝓝 p) → Tendsto (toMeasure ∘ μs) atTop (𝓝 p.toMeasure)
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)), IsOpen[instTopologicalSpaceProd] u → ∀ x ∈ u, ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u

Tactic:
intro u u_open x 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
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: 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 < ∞
⊢ μ.real K ≤ ν.real K + ↑ε

Tactic:
obtain ⟨f, pf1, pf2, pf3⟩ :
      ∃ f : C_c(X, ℝ), Set.EqOn (⇑f) 1 K ∧ tsupport ⇑f ⊆ V ∧ ∀ (x : X), f x ∈ Set.Icc 0 1 := by
    obtain ⟨f, hf1, hf2, hf3⟩ := exists_continuousMap_one_of_isCompact_subset_isOpen hK pV2 pV1
    exact ⟨⟨f, hasCompactSupport_def.mpr hf2⟩, hf1, hf3⟩

Goal state after:
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
⊢ μ.real K ≤ ν.real K + ↑ε
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: case inl
Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
ε : ℝ
ε_pos : 0 < ε
h✝ : IsEmpty Ω
⊢ ∃ As,
    (∀ (n : ℕ), MeasurableSet (As n)) ∧
      (∀ (n : ℕ), Bornology.IsBounded (As n)) ∧
        (∀ (n : ℕ), diam (As n) ≤ ε) ∧ ⋃ n, As n = univ ∧ Pairwise fun n m ↦ Disjoint (As n) (As m)

Tactic:
refine ⟨fun _ ↦ ∅, fun _ ↦ MeasurableSet.empty, fun _ ↦ Bornology.isBounded_empty, ?_, ?_,
            fun _ _ _ ↦ disjoint_of_subsingleton⟩

Goal state after:
case inl.refine_1
Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
ε : ℝ
ε_pos : 0 < ε
h✝ : IsEmpty Ω
⊢ ∀ (n : ℕ), diam ((fun x ↦ ∅) n) ≤ ε

case inl.refine_2
Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
ε : ℝ
ε_pos : 0 < ε
h✝ : IsEmpty Ω
⊢ ⋃ n, (fun x ↦ ∅) n = univ
Module: Mathlib.MeasureTheory.Measure.IntegralCharFun

Goal state before:
pp: case neg.refine_1
μ : Measure ℝ
r : ℝ
inst✝ : IsFiniteMeasure μ
hr : 0 < r
h_int : Integrable (Function.uncurry fun x y ↦ cexp (↑x * ↑y * I)) ((volume.restrict (Set.uIoc (-r) r)).prod μ)
y : ℝ
hy : ¬y = 0
x : ℝ
⊢ x ∈ Set.uIcc (-r) r → HasDerivAt (fun x ↦ y * x) y x

Tactic:
hx

Goal state after:
case neg.refine_1
μ : Measure ℝ
r : ℝ
inst✝ : IsFiniteMeasure μ
hr : 0 < r
h_int : Integrable (Function.uncurry fun x y ↦ cexp (↑x * ↑y * I)) ((volume.restrict (Set.uIoc (-r) r)).prod μ)
y : ℝ
hy : ¬y = 0
x : ℝ
hx : x ∈ Set.uIcc (-r) r
⊢ HasDerivAt (fun x ↦ y * x) y x
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: ι : Type u_1
inst✝¹ : Nonempty ι
inst✝ : Fintype ι
x : EuclideanSpace ℝ ι
r : ℝ
⊢ volume (Metric.ball x r) = ENNReal.ofReal r ^ card ι * ENNReal.ofReal (√π ^ card ι / Gamma (↑(card ι) / 2 + 1))

Tactic:
obtain hr | hr := le_total r 0

Goal state after:
case inl
ι : Type u_1
inst✝¹ : Nonempty ι
inst✝ : Fintype ι
x : EuclideanSpace ℝ ι
r : ℝ
hr : r ≤ 0
⊢ volume (Metric.ball x r) = ENNReal.ofReal r ^ card ι * ENNReal.ofReal (√π ^ card ι / Gamma (↑(card ι) / 2 + 1))

case inr
ι : Type u_1
inst✝¹ : Nonempty ι
inst✝ : Fintype ι
x : EuclideanSpace ℝ ι
r : ℝ
hr : 0 ≤ r
⊢ volume (Metric.ball x r) = ENNReal.ofReal r ^ card ι * ENNReal.ofReal (√π ^ card ι / Gamma (↑(card ι) / 2 + 1))
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare

Goal state before:
pp: case refine_4
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
a b c : E
s : Set E
ω : E → E →L[𝕜] F
dω : E → E →L[ℝ] E →L[𝕜] F
hs : Convex ℝ s
hω : ∀ x ∈ s, HasFDerivWithinAt ω (dω x) s x
hdω : ∀ a ∈ s, ∀ x ∈ tangentConeAt ℝ s a, ∀ y ∈ tangentConeAt ℝ s a, ((dω a) x) y = ((dω a) y) x
ha : a ∈ s
hb : b ∈ s
hc : c ∈ s
φ : (↑(Path.segment a b)).Homotopy ↑(Path.segment a c) :=
  ContinuousMap.Homotopy.affine ↑(Path.segment a b) ↑(Path.segment a c)
hφs : range ⇑φ ⊆ s
x : E
⊢ x ∈ range ⇑φ → ∀ u ∈ tangentConeAt ℝ (range ⇑φ) x, ∀ v ∈ tangentConeAt ℝ (range ⇑φ) x, ((dω x) u) v = ((dω x) v) u

Tactic:
hx

Goal state after:
case refine_4
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
a b c : E
s : Set E
ω : E → E →L[𝕜] F
dω : E → E →L[ℝ] E →L[𝕜] F
hs : Convex ℝ s
hω : ∀ x ∈ s, HasFDerivWithinAt ω (dω x) s x
hdω : ∀ a ∈ s, ∀ x ∈ tangentConeAt ℝ s a, ∀ y ∈ tangentConeAt ℝ s a, ((dω a) x) y = ((dω a) y) x
ha : a ∈ s
hb : b ∈ s
hc : c ∈ s
φ : (↑(Path.segment a b)).Homotopy ↑(Path.segment a c) :=
  ContinuousMap.Homotopy.affine ↑(Path.segment a b) ↑(Path.segment a c)
hφs : range ⇑φ ⊆ s
x : E
hx : x ∈ range ⇑φ
⊢ ∀ u ∈ tangentConeAt ℝ (range ⇑φ) x, ∀ v ∈ tangentConeAt ℝ (range ⇑φ) x, ((dω x) u) v = ((dω x) v) u
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: case h
E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_2
ι : Type u_3
inst✝³ : RCLike 𝕜
inst✝² : Fintype ι
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
b : OrthonormalBasis ι 𝕜 E
h : ∀ (i : ι), Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖⟪b i, x⟫_𝕜‖}) atTop (𝓝 0)
hE : Nontrivial E
h_rank : 0 < ↑(Fintype.card ι)
this✝ : Nonempty ι
this : ProperSpace E
r : ℝ
μ : Measure E
hμS : μ ∈ S
x : E
⊢ x ∈ {x | r < ‖x‖} → x ∈ ⋃ i, {x | r / √↑(Fintype.card ι) < ‖⟪b i, x⟫_𝕜‖}

Tactic:
hx

Goal state after:
case h
E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_2
ι : Type u_3
inst✝³ : RCLike 𝕜
inst✝² : Fintype ι
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
b : OrthonormalBasis ι 𝕜 E
h : ∀ (i : ι), Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖⟪b i, x⟫_𝕜‖}) atTop (𝓝 0)
hE : Nontrivial E
h_rank : 0 < ↑(Fintype.card ι)
this✝ : Nonempty ι
this : ProperSpace E
r : ℝ
μ : Measure E
hμS : μ ∈ S
x : E
hx : x ∈ {x | r < ‖x‖}
⊢ x ∈ ⋃ i, {x | r / √↑(Fintype.card ι) < ‖⟪b i, x⟫_𝕜‖}