mathlib-initiative/mathlib-tactics huggingface dataset:
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
K₁ K₂ : Compacts X
h : K₁ ≤ K₂
⊢ sInf (⇑Λ '' {f | ∀ x ∈ K₁, 1 ≤ f x}) ≤ sInf (⇑Λ '' {f | ∀ x ∈ K₂, 1 ≤ f x})

Tactic:
gcongr

Goal state after:
case h₁
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
K₁ K₂ : Compacts X
h : K₁ ≤ K₂
⊢ (⇑Λ '' {f | ∀ x ∈ K₂, 1 ≤ f x}).Nonempty
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 : Disjoint s₀ s₁
hst : s₀ ∪ s₁ ⊆ t
⊢ ∃ f₀ f₁, EqOn (⇑f₀) 1 s₀ ∧ EqOn (⇑f₁) 1 s₁ ∧ EqOn (⇑(f₀ + f₁)) 1 t

Tactic:
set so : Fin 2 → Set X := fun j => if j = 0 then s₀ᶜ else s₁ᶜ with hso

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 : Disjoint s₀ s₁
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
⊢ ∃ f₀ f₁, EqOn (⇑f₀) 1 s₀ ∧ EqOn (⇑f₁) 1 s₁ ∧ EqOn (⇑(f₀ + f₁)) 1 t
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
hp : Λ f = ↑p
⊢ ∃ x ∈ {f | ∀ x ∈ { carrier := K, isCompact' := hK }, 1 ≤ f x}, (toNNRealLinear Λ) x = p

Tactic:
use f.nnrealPart

Goal state after:
case h
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 : Λ f = ↑p
⊢ f.nnrealPart ∈ {f | ∀ x ∈ { carrier := K, isCompact' := hK }, 1 ≤ f x} ∧ (toNNRealLinear Λ) f.nnrealPart = p
Module: Mathlib.MeasureTheory.MeasurableSpace.Card

Goal state before:
pp: α : Type u
s : Set (Set α)
i j : Ordinal.{u_1}
h : j < i
t : Set α
ht : t ∈ generateMeasurableRec s j
⊢ tᶜ ∈
    have S := ⋃ j, ⋃ (_ : j < i), generateMeasurableRec s j;
    s ∪ {∅} ∪ compl '' S ∪ range fun f ↦ ⋃ n, ↑(f n)

Tactic:
exact mem_union_left _ (mem_union_right _ ⟨t, mem_iUnion₂.2 ⟨j, h, ht⟩, rfl⟩)

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

Goal state before:
pp: case h.refine_2
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' := ⋯ }
⊢ μ.measure ↑(V₁ ⊓ V₂) ≤ μ.measure E + ENNReal.ofReal ε

Tactic:
calc
      _ ≤ μ.measure V₁ := by simp [measure_mono]
      _ = μ.outerMeasure V₁ := Content.measure_apply μ (V₁.2.measurableSet)
      _ ≤ μ.outerMeasure E + ε.toNNReal := hV₁.2
      _ = _ := by rw [Content.measure_apply μ hμ', ENNReal.ofNNReal_toNNReal]

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) ≠ ∞
⊢ Λ f ≤ ∫ (x : X), f x ∂rieszMeasure Λ + ε

Tactic:
obtain ⟨V, hV⟩ : ∃ V : Fin N → Opens X, ∀ n, E n ⊆ (V n) ∧ (∀ x ∈ V n, f x < y n + ε') ∧
      μ (V n) ≤ μ (E n) + ENNReal.ofReal (ε' / N) := by
    have h_ε' := (div_pos hε'.1 (Nat.cast_pos'.mpr hN))
    have h n x (hx : x ∈ E n) := lt_add_of_le_of_pos ((hE.2.2.1 n x hx).right) hε'.1
    have h' n := Eq.trans_ne
      (Content.measure_apply (rieszContent (toNNRealLinear Λ)) (hE.2.2.2 n)).symm (hE' n)
    choose V hV using fun n ↦ exists_open_approx f h_ε' (E n) (h' n) (hE.2.2.2 n) (h n)
    exact ⟨V, hV⟩

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)
⊢ Λ f ≤ ∫ (x : X), f x ∂rieszMeasure Λ + ε
Module: Mathlib.Topology.Separation.CompletelyRegular

Goal state before:
pp: X : Type u
inst✝² : TopologicalSpace X
Y : Type v
inst✝¹ : TopologicalSpace Y
inst✝ : CompletelyRegularSpace Y
f : X → Y
hf : IsInducing f
x : X
K : Set X
hK : ∃ t, IsClosed[inst✝¹] t ∧ f ⁻¹' t = K
hxK : x ∉ K
⊢ ∃ f, Continuous[inst✝², _] f ∧ f x = 0 ∧ EqOn f 1 K

Tactic:
obtain ⟨K, hK, rfl⟩ := hK

Goal state after:
X : Type u
inst✝² : TopologicalSpace X
Y : Type v
inst✝¹ : TopologicalSpace Y
inst✝ : CompletelyRegularSpace Y
f : X → Y
hf : IsInducing f
x : X
K : Set Y
hK : IsClosed[inst✝¹] K
hxK : x ∉ f ⁻¹' K
⊢ ∃ f_1, Continuous[inst✝², _] f_1 ∧ f_1 x = 0 ∧ EqOn f_1 1 (f ⁻¹' K)
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_4
F : Type u_5
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ E
inst✝² : InnerProductSpace ℝ F
mE : MeasurableSpace E
mF : MeasurableSpace F
μ : Measure E
ν : Measure F
inst✝¹ : SFinite μ
inst✝ : SFinite ν
t : WithLp 2 (E × F)
⊢ ∫ (x : E × F),
      cexp
        (↑(⟪((MeasurableEquiv.toLp 2 (E × F)) x).ofLp.1, t.ofLp.1⟫ +
              ⟪((MeasurableEquiv.toLp 2 (E × F)) x).ofLp.2, t.ofLp.2⟫) *
          I) ∂μ.prod ν =
    ∫ (z : E × F), cexp (↑⟪z.1, t.ofLp.1⟫ * I) * cexp (↑⟪z.2, t.ofLp.2⟫ * I) ∂μ.prod ν

Tactic:
simp [ofReal_add, add_mul, Complex.exp_add]

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

Goal state before:
pp: case calc_4
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, (|a| + y n + ε') * (μ.real (E n) + ε' / ↑N) - |a| * ∑ n, Λ (g n) ≤
    ∑ n, (|a| + y n + ε') * (μ.real (E n) + ε' / ↑N) - |a| * μ.real K

Tactic:
gcongr

Goal state after:
case hbc
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)
⊢ μ.real K ≤ ∑ n, Λ (g n)
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case calc_6
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, (y n - ε') * μ.real (E n) + 2 * ε' * μ.real K + ε' / ↑N * ∑ n, (|a| + y n + ε') ≤
    ∫ (x : X), f x ∂μ + 2 * ε' * μ.real K + ε' / ↑N * ∑ n, (|a| + y n + ε')

Tactic:
gcongr

Goal state after:
case h₁
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, (y n - ε') * μ.real (E n) ≤ ∫ (x : X), f x ∂μ
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoEMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : IsProbabilityMeasure ν
δ ε : ℝ≥0∞
B : Set Ω
h : 1 - μ (thickening ε.toReal B) ≤ ν (thickening ε.toReal (thickening ε.toReal B)ᶜ) + ε
ε_gt : δ < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
B_subset : B ⊆ (thickening ε.toReal (thickening ε.toReal B)ᶜ)ᶜ
Tc_mble : MeasurableSet (thickening ε.toReal B)ᶜ
almost : 1 ≤ ν (thickening ε.toReal (thickening ε.toReal B)ᶜ) + (ε + μ (thickening ε.toReal B))
⊢ ε + μ (thickening ε.toReal B) ≤ μ (thickening ε.toReal B) + ε

Tactic:
add_comm ε

Goal state after:
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoEMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : IsProbabilityMeasure ν
δ ε : ℝ≥0∞
B : Set Ω
h : 1 - μ (thickening ε.toReal B) ≤ ν (thickening ε.toReal (thickening ε.toReal B)ᶜ) + ε
ε_gt : δ < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
B_subset : B ⊆ (thickening ε.toReal (thickening ε.toReal B)ᶜ)ᶜ
Tc_mble : MeasurableSet (thickening ε.toReal B)ᶜ
almost : 1 ≤ ν (thickening ε.toReal (thickening ε.toReal B)ᶜ) + (ε + μ (thickening ε.toReal B))
⊢ μ (thickening ε.toReal B) + ε ≤ μ (thickening ε.toReal B) + ε
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
t : ℝ
⊢ ‖μ.real {a | t ≤ f a}‖ ≤ μ.real univ

Tactic:
simp only [Real.norm_eq_abs, abs_of_nonneg measureReal_nonneg]

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
key : (fun t ↦ μ.real {a | t ≤ f a}) ≤ fun t ↦ ν.real (thickening ε {a | t ≤ f a}) + ε
t : ℝ
⊢ μ.real {a | t ≤ f a} ≤ μ.real univ
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
t : ℝ
⊢ ‖ν.real (thickening ε {a | t ≤ f a})‖ ≤ ν.real univ

Tactic:
simp only [Real.norm_eq_abs, abs_of_nonneg measureReal_nonneg]

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
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
t : ℝ
⊢ ν.real (thickening ε {a | t ≤ f a}) ≤ ν.real univ
Module: Mathlib.MeasureTheory.Measure.FiniteMeasureProd

Goal state before:
pp: 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
ε : ℝ
εpos : ε > 0
hε : ball x ε ⊆ u
r : ℝ
hr : r ∈ Ioo 0 ε
r' : ℝ
hr' : r' ∈ Ioo 0 ε
μr : ↑μ.1 (frontier (ball x.1 r)) = 0
μr' : ↑μ.2 (frontier (ball x.2 r')) = 0
⊢ ball x.1 r ×ˢ ball x.2 r' ⊆ ball x.1 ε ×ˢ ball x.2 ε

Tactic:
gcongr

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)) := ⋯
this✝¹ : IsPiSystem S
this✝ : PseudoMetricSpace α := ⋯
this : PseudoMetricSpace β := ⋯
u : Set (α × β)
u_open : IsOpen[instTopologicalSpaceProd] u
x : α × β
xu : x ∈ u
ε : ℝ
εpos : ε > 0
hε : ball x ε ⊆ u
r : ℝ
hr : r ∈ Ioo 0 ε
r' : ℝ
hr' : r' ∈ Ioo 0 ε
μr : ↑μ.1 (frontier (ball x.1 r)) = 0
μr' : ↑μ.2 (frontier (ball x.2 r')) = 0
⊢ r ≤ ε

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)) := ⋯
this✝¹ : IsPiSystem S
this✝ : PseudoMetricSpace α := ⋯
this : PseudoMetricSpace β := ⋯
u : Set (α × β)
u_open : IsOpen[instTopologicalSpaceProd] u
x : α × β
xu : x ∈ u
ε : ℝ
εpos : ε > 0
hε : ball x ε ⊆ u
r : ℝ
hr : r ∈ Ioo 0 ε
r' : ℝ
hr' : r' ∈ Ioo 0 ε
μr : ↑μ.1 (frontier (ball x.1 r)) = 0
μr' : ↑μ.2 (frontier (ball x.2 r')) = 0
⊢ r' ≤ ε
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: case refine_4
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)
ν : LevyProkhorov (ProbabilityMeasure Ω)
hμs : Tendsto μs atTop (𝓝 ν)
P : ProbabilityMeasure Ω := ν.toMeasure
Ps : ℕ → ProbabilityMeasure Ω := toMeasure ∘ μs
f✝ f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
f_zero : ¬‖f‖ = 0
norm_f_pos : 0 < ‖f‖
δ : ℝ
δ_pos : 0 < δ
εs : ℕ → ℝ
left✝ : StrictAnti εs
εs_lim : Tendsto εs atTop (𝓝 0)
ε_of_room : Tendsto (fun x ↦ dist (μs x) ν + εs x) atTop (𝓝 0)
ε_of_room' : Tendsto (fun n ↦ dist (μs n) ν + εs n) atTop (𝓝[>] 0)
key :
  Tendsto (fun x ↦ ∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P).real (thickening (dist (μs x) ν + εs x) {a | t ≤ f a})) atTop
    (𝓝 (∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P).real {a | t ≤ f a}))
aux : ∀ (z : ℝ), Iio (z + δ / 2) ∈ 𝓝 z
n : ℕ
εs_pos : 0 < εs n
hn :
  ∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P).real (thickening (dist (μs n) ν + εs n) {a | t ≤ f a}) <
    (∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P).real {a | t ≤ f a}) + δ / 2
hn' : dist (μs n) ν + εs n < ‖f‖⁻¹ * δ / 2
bound :
  ∫ (ω : Ω), f ω ∂↑(Ps n) ≤
    (∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P).real (thickening (dist (μs n) ν + εs n) {a | t ≤ f a})) + (dist (μs n) ν + εs n) * ‖f‖
⊢ (∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P).real {a | t ≤ f a}) + (δ / 2 + ‖f‖ * (dist (μs n) ν + εs n)) ≤
    (∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P).real {a | t ≤ f a}) + δ

Tactic:
gcongr

Goal state after:
case h₂
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)
ν : LevyProkhorov (ProbabilityMeasure Ω)
hμs : Tendsto μs atTop (𝓝 ν)
P : ProbabilityMeasure Ω := ⋯
Ps : ℕ → ProbabilityMeasure Ω := ⋯
f✝ f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
f_zero : ¬‖f‖ = 0
norm_f_pos : 0 < ‖f‖
δ : ℝ
δ_pos : 0 < δ
εs : ℕ → ℝ
left✝ : StrictAnti εs
εs_lim : Tendsto εs atTop (𝓝 0)
ε_of_room : Tendsto (fun x ↦ dist (μs x) ν + εs x) atTop (𝓝 0)
ε_of_room' : Tendsto (fun n ↦ dist (μs n) ν + εs n) atTop (𝓝[>] 0)
key :
  Tendsto (fun x ↦ ∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P).real (thickening (dist (μs x) ν + εs x) {a | t ≤ f a})) atTop
    (𝓝 (∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P).real {a | t ≤ f a}))
aux : ∀ (z : ℝ), Iio (z + δ / 2) ∈ 𝓝 z
n : ℕ
εs_pos : 0 < εs n
hn :
  ∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P).real (thickening (dist (μs n) ν + εs n) {a | t ≤ f a}) <
    (∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P).real {a | t ≤ f a}) + δ / 2
hn' : dist (μs n) ν + εs n < ‖f‖⁻¹ * δ / 2
bound :
  ∫ (ω : Ω), f ω ∂↑(Ps n) ≤
    (∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P).real (thickening (dist (μs n) ν + εs n) {a | t ≤ f a})) + (dist (μs n) ν + εs n) * ‖f‖
⊢ δ / 2 + ‖f‖ * (dist (μs n) ν + εs n) ≤ δ
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)
ν : LevyProkhorov (ProbabilityMeasure Ω)
hμs : Tendsto μs atTop (𝓝 ν)
P : ProbabilityMeasure Ω := ν.toMeasure
Ps : ℕ → ProbabilityMeasure Ω := toMeasure ∘ μs
f✝ f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
f_zero : ¬‖f‖ = 0
norm_f_pos : 0 < ‖f‖
δ : ℝ
δ_pos : 0 < δ
εs : ℕ → ℝ
left✝ : StrictAnti εs
εs_lim : Tendsto εs atTop (𝓝 0)
ε_of_room : Tendsto (fun x ↦ dist (μs x) ν + εs x) atTop (𝓝 0)
ε_of_room' : Tendsto (fun n ↦ dist (μs n) ν + εs n) atTop (𝓝[>] 0)
key :
  Tendsto (fun x ↦ ∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P).real (thickening (dist (μs x) ν + εs x) {a | t ≤ f a})) atTop
    (𝓝 (∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P).real {a | t ≤ f a}))
aux : ∀ (z : ℝ), Iio (z + δ / 2) ∈ 𝓝 z
n : ℕ
εs_pos : 0 < εs n
hn :
  ∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P).real (thickening (dist (μs n) ν + εs n) {a | t ≤ f a}) <
    (∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P).real {a | t ≤ f a}) + δ / 2
hn' : dist (μs n) ν + εs n < ‖f‖⁻¹ * δ / 2
bound :
  ∫ (ω : Ω), f ω ∂↑(Ps n) ≤
    (∫ (t : ℝ) in Ioc 0 ‖f‖, (↑P).real (thickening (dist (μs n) ν + εs n) {a | t ≤ f a})) + (dist (μs n) ν + εs n) * ‖f‖
⊢ δ / 2 + ‖f‖ * (dist (μs n) ν + εs n) ≤ δ / 2 + ‖f‖ * (‖f‖⁻¹ * δ / 2)

Tactic:
gcongr

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

Goal state before:
pp: μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
⊢ 2 * ∫ (x : ℝ), 1 - sinc (2 * r⁻¹ * x) ∂μ ≤ 2 * ‖∫ (x : ℝ), 1 - sinc (2 * r⁻¹ * x) ∂μ‖

Tactic:
gcongr

Goal state after:
case hbc
μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
⊢ ∫ (x : ℝ), 1 - sinc (2 * r⁻¹ * x) ∂μ ≤ ‖∫ (x : ℝ), 1 - sinc (2 * r⁻¹ * x) ∂μ‖
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: E : Type u_1
inst✝⁴ : MeasurableSpace E
inst✝³ : TopologicalSpace E
inst✝² : T2Space E
inst✝¹ : BorelSpace E
inst✝ : CompactSpace E
C : ℝ≥0
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C} ∈ f
g : E →C_c ℝ
μ : FiniteMeasure E
hμ : μ.mass ≤ C
⊢ (↑μ).real univ * ‖g.toBoundedContinuousFunction‖ ≤ ↑C * ‖g.toBoundedContinuousFunction‖

Tactic:
gcongr

Goal state after:
case hbc
E : Type u_1
inst✝⁴ : MeasurableSpace E
inst✝³ : TopologicalSpace E
inst✝² : T2Space E
inst✝¹ : BorelSpace E
inst✝ : CompactSpace E
C : ℝ≥0
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C} ∈ f
g : E →C_c ℝ
μ : FiniteMeasure E
hμ : μ.mass ≤ C
⊢ (↑μ).real univ ≤ ↑C
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: E : Type u_1
inst✝⁴ : MeasurableSpace E
inst✝³ : TopologicalSpace E
inst✝² : T2Space E
inst✝¹ : BorelSpace E
inst✝ : CompactSpace E
C : ℝ≥0
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C} ∈ f
g : E →C_c ℝ
μ : FiniteMeasure E
hμ : μ.mass ≤ C
⊢ ∫ (x : E), -‖g.toBoundedContinuousFunction‖ ∂↑μ ≤ ∫ (x : E), g x ∂↑μ

Tactic:
gcongr

Goal state after:
case hf
E : Type u_1
inst✝⁴ : MeasurableSpace E
inst✝³ : TopologicalSpace E
inst✝² : T2Space E
inst✝¹ : BorelSpace E
inst✝ : CompactSpace E
C : ℝ≥0
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C} ∈ f
g : E →C_c ℝ
μ : FiniteMeasure E
hμ : μ.mass ≤ C
⊢ Integrable (fun x ↦ -‖g.toBoundedContinuousFunction‖) ↑μ

case hg
E : Type u_1
inst✝⁴ : MeasurableSpace E
inst✝³ : TopologicalSpace E
inst✝² : T2Space E
inst✝¹ : BorelSpace E
inst✝ : CompactSpace E
C : ℝ≥0
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C} ∈ f
g : E →C_c ℝ
μ : FiniteMeasure E
hμ : μ.mass ≤ C
⊢ Integrable (fun x ↦ g x) ↑μ

case h
E : Type u_1
inst✝⁴ : MeasurableSpace E
inst✝³ : TopologicalSpace E
inst✝² : T2Space E
inst✝¹ : BorelSpace E
inst✝ : CompactSpace E
C : ℝ≥0
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C} ∈ f
g : E →C_c ℝ
μ : FiniteMeasure E
hμ : μ.mass ≤ C
⊢ (fun x ↦ -‖g.toBoundedContinuousFunction‖) ≤ fun x ↦ g x