mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝² : NormedAddCommGroup γ
inst✝¹ : NormedSpace ℝ γ
inst✝ : BorelSpace γ
h : IdentDistrib f g μ ν
hf : AEStronglyMeasurable f μ
t : Set γ
t_sep : IsSeparable t
ht : ∀ᵐ (x : α) ∂μ, f x ∈ t
⊢ ∀ᵐ (x : α) ∂μ, id (f x) ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t

Tactic:
filter_upwards [ht] with x hx using subset_closure hx

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

Goal state before:
pp: case inl
R : Type u_2
inst✝⁷ : Semiring R
inst✝⁶ : LinearOrder R
inst✝⁵ : FloorSemiring R
inst✝⁴ : TopologicalSpace R
inst✝³ : OrderTopology R
inst✝² : MeasurableSpace R
inst✝¹ : OpensMeasurableSpace R
inst✝ : IsStrictOrderedRing R
n : R
h : ⌊n⌋₊ = 0
⊢ MeasurableSet (floor ⁻¹' {⌊n⌋₊})

Tactic:
simp [h, Nat.preimage_floor_of_ne_zero, -floor_eq_zero]

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

Goal state before:
pp: case inr
R : Type u_2
inst✝⁷ : Semiring R
inst✝⁶ : LinearOrder R
inst✝⁵ : FloorSemiring R
inst✝⁴ : TopologicalSpace R
inst✝³ : OrderTopology R
inst✝² : MeasurableSpace R
inst✝¹ : OpensMeasurableSpace R
inst✝ : IsStrictOrderedRing R
n : R
h : ⌊n⌋₊ ≠ 0
⊢ MeasurableSet (floor ⁻¹' {⌊n⌋₊})

Tactic:
simp [h, Nat.preimage_floor_of_ne_zero, -floor_eq_zero]

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

Goal state before:
pp: case neg.inl
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝ : PartialOrder ι
𝓕 : Filtration ι m
this : TopologicalSpace ι := Preorder.topology ι
j : ι
hj : ¬(𝓝[>] j).NeBot
hij : j ≤ j
hi : (𝓝[>] j).NeBot
⊢ ⨅ j_1, ⨅ (_ : j < j_1), ↑𝓕 j_1 ≤ ↑𝓕 j

pp: case neg.inr
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝ : PartialOrder ι
𝓕 : Filtration ι m
this : TopologicalSpace ι := Preorder.topology ι
i j : ι
hij : i ≤ j
hi : (𝓝[>] i).NeBot
hj✝ : ¬(𝓝[>] j).NeBot
hj : j ≠ i
⊢ ⨅ j, ⨅ (_ : i < j), ↑𝓕 j ≤ ↑𝓕 j

Tactic:
· contradiction

Goal state after:
case neg.inr
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝ : PartialOrder ι
𝓕 : Filtration ι m
this : TopologicalSpace ι := Preorder.topology ι
i j : ι
hij : i ≤ j
hi : (𝓝[>] i).NeBot
hj✝ : ¬(𝓝[>] j).NeBot
hj : j ≠ i
⊢ ⨅ j, ⨅ (_ : i < j), ↑𝓕 j ≤ ↑𝓕 j
Module: Mathlib.MeasureTheory.Function.Intersectivity

Goal state before:
pp: 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
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
hrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ
hμ : μ ≠ 0
this : ∫⁻ (x : α), limsup (fun x_1 ↦ f x_1 x) atTop ∂μ ≤ μ univ
⊢ μ univ ≠ ∞

Tactic:
finiteness

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

Goal state before:
pp: case pos
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 = ∞
⊢ eLpNormEssSup f (dirac i) = ‖f i‖ₑ

Tactic:
simp [eLpNormEssSup, essSup, limsup, limsSup, Set.Ici_def]

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

Goal state before:
pp: 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
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
hrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ
hμ : μ ≠ 0
this : ∫⁻ (x : α), limsup (fun x_1 ↦ f x_1 x) atTop ∂μ ≤ μ univ
x : α
hxN : x ∉ N
hx : r / μ univ ≤ limsup (fun x_1 ↦ f x_1 x) atTop
hxs : {n | x ∈ s n}.Finite
⊢ μ univ ≠ ∞

Tactic:
finiteness

Goal state after:
Module: Mathlib.MeasureTheory.Group.FoelnerFilter

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝³ : MeasurableSpace X
μ : Measure X
inst✝² : Group G
inst✝¹ : MulAction G X
ι : Type u_3
u : Ultrafilter ι
F : ι → Set X
inst✝ : SMulInvariantMeasure G X μ
hfoel : IsFoelner G μ (↑u) F
g h : G
s : Set X
⊢ ∀ (g h : G), mean μ u F (g • s) ≤ mean μ u F (h • s)

Tactic:
intro g h

Goal state after:
G : Type u_1
X : Type u_2
inst✝³ : MeasurableSpace X
μ : Measure X
inst✝² : Group G
inst✝¹ : MulAction G X
ι : Type u_3
u : Ultrafilter ι
F : ι → Set X
inst✝ : SMulInvariantMeasure G X μ
hfoel : IsFoelner G μ (↑u) F
g✝ h✝ : G
s : Set X
g h : G
⊢ mean μ u F (g • s) ≤ mean μ u F (h • s)
Module: Mathlib.MeasureTheory.Function.UnifTight

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp_top : p ≠ ∞
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), MemLp (f i) p μ) → UnifTight f p μ
f : Fin (n + 1) → α → β
hfLp : ∀ (i : Fin (n + 1)), MemLp (f i) p μ
ε : ℝ≥0
hε : 0 < ε
hε_top : ¬↑ε = ∞
g : Fin n → α → β := fun k ↦ f k.castSucc
hgLp : ∀ (i : Fin n), MemLp (g i) p μ
S : Set α
hμS : μ S ≠ ∞
hFε : ∀ (i : Fin n), eLpNorm (Sᶜ.indicator (g i)) p μ ≤ ↑ε
s : Set α
left✝ : MeasurableSet s
hμs : μ s < ∞
hfε : eLpNorm (sᶜ.indicator (f (Fin.last n))) p μ < ↑ε
⊢ μ (s ∪ S) ≠ ∞

Tactic:
finiteness

Goal state after:
Module: Mathlib.MeasureTheory.Group.GeometryOfNumbers

Goal state before:
pp: E : Type u_1
inst✝⁶ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝ : Countable ↥L
fund : IsAddFundamentalDomain (↥L) F μ
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h : μ F * 2 ^ finrank ℝ E < μ s
⊢ 2 ^ finrank ℝ E ≠ ∞

Tactic:
finiteness

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

Goal state before:
pp: case e_f
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
hfinε : ε ≠ ∞
hμ : μ ≠ 0
hε' : 0 < ε / 3
Eg : Set α
hmEg : MeasurableSet Eg
hμEg : μ Eg < ∞
hgε : eLpNorm (Egᶜ.indicator g) p μ < ε / 3
Ef : Set α
hmEf : MeasurableSet Ef
hμEf : μ Ef < ∞
hfε : ∀ (i : ℕ), eLpNorm (Efᶜ.indicator (f i)) p μ ≤ ε / 3
E : Set α := Ef ∪ Eg
hmE : MeasurableSet E
hfmE : μ E < ∞
hgE' : MemLp g p (μ.restrict E)
huiE : UnifIntegrable f p (μ.restrict E)
hfgE : ∀ᵐ (x : α) ∂μ.restrict E, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ffmE : Fact (μ E < ∞)
hInner : ∀ ε > 0, ∃ N, ∀ n ≥ N, eLpNorm (f n - g) p (μ.restrict E) ≤ ε
N : ℕ
hfngε : ∀ n ≥ N, eLpNorm (f n - g) p (μ.restrict E) ≤ ε / 3
n : ℕ
hn : n ≥ N
hmfngE : AEStronglyMeasurable (E.indicator (f n - g)) μ
hfngEε : eLpNorm (E.indicator (f n - g)) p μ ≤ ε / 3
hmgEc : AEStronglyMeasurable (Eᶜ.indicator g) μ
hgEcε : eLpNorm (Eᶜ.indicator g) p μ ≤ ε / 3
hmfnEc : AEStronglyMeasurable (Eᶜ.indicator (f n)) μ
hfnEcε : eLpNorm (Eᶜ.indicator (f n)) p μ ≤ ε / 3
hmfngEc : AEStronglyMeasurable (Eᶜ.indicator (f n - g)) μ
hfngEcε : eLpNorm (Eᶜ.indicator (f n - g)) p μ ≤ ε / 3 + ε / 3
⊢ f n - g = Eᶜ.indicator (f n - g) + E.indicator (f n - g)

Tactic:
exact (E.indicator_compl_add_self _).symm

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

Goal state before:
pp: α : Type u_1
inst✝² : MeasurableSpace α
A : Set α
ι : Type u_2
L : Filter ι
inst✝¹ : L.IsCountablyGenerated
As : ι → Set α
μ : Measure α
inst✝ : IsFiniteMeasure μ
A_mble : MeasurableSet A
As_mble : ∀ (i : ι), MeasurableSet (As i)
h_lim : ∀ᵐ (x : α) ∂μ, ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A
⊢ μ univ ≠ ∞

Tactic:
finiteness

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

Goal state before:
pp: 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✝ : CompleteSpace F
ω : E →L[𝕜] F
a b : E
⊢ ∫ᶜ (x : E) in Path.segment a b, ω = ω (b - a)

Tactic:
let : NormedSpace ℝ F := .restrictScalars ℝ 𝕜 F
  simp [curveIntegral_segment]

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

Goal state before:
pp: case right
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 := ⋯
hso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ
soopen : ∀ (j : Fin 2), IsOpen (so j)
x : X
hx : x ∈ t
h : x ∈ s₀ᶜ ∨ x ∈ s₁ᶜ
⊢ x ∈ s₁ᶜ → ∃ i, x ∈ if i = 0 then s₀ᶜ else s₁ᶜ

Tactic:
· intro h1
      use 1
      simp only [Fin.isValue, one_ne_zero, ↓reduceIte]
      exact h1

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

Goal state before:
pp: case e_6.refine_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
a✝ b c d : E
γ₁ : Path a✝ b
γ₂ : Path c d
s : Set (↑I × ↑I)
t : Set E
ω : E → E →L[ℝ] F
dω : E → E →L[ℝ] E →L[ℝ] F
φ : (↑γ₁).Homotopy ↑γ₂
hs : s.Countable
hφt : ∀ a ∈ Ioo 0 1, ∀ b_1 ∈ Ioo 0 1, φ (a, b_1) ∈ t
hω : ∀ a ∈ Ioo 0 1, ∀ b_1 ∈ Ioo 0 1, (a, b_1) ∉ s → HasFDerivWithinAt ω (dω (φ (a, b_1))) t (φ (a, b_1))
hωc : ContinuousOn ω (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t)
hdω_symm :
  ∀ a ∈ Ioo 0 1,
    ∀ b_1 ∈ Ioo 0 1,
      (a, b_1) ∉ s →
        ∀ u ∈ tangentConeAt ℝ t (φ (a, b_1)),
          ∀ v ∈ tangentConeAt ℝ t (φ (a, b_1)), ((dω (φ (a, b_1))) u) v = ((dω (φ (a, b_1))) v) u
U : Set (ℝ × ℝ) := Ioo 0 1 ×ˢ Ioo 0 1
hU : U = Ioo 0 1 ×ˢ Ioo 0 1
hinterior : interior (Icc 0 1) = U
hunique : UniqueDiffOn ℝ (Icc 0 1)
hUopen : IsOpen U
hU_subset : U ⊆ Icc 0 1
hclosure : closure U = Icc 0 1
ψ : ℝ × ℝ → E := fun xy ↦ Set.IccExtend ⋯ (⇑(φ.extend xy.1)) xy.2
hcontdiff : ContDiffOn ℝ 2 ψ (Icc 0 1)
hψ : ψ = fun xy ↦ Set.IccExtend ⋯ (⇑(φ.extend xy.1)) xy.2
hψφ : ∀ (a b_1 : ↑I), ψ (↑a, ↑b_1) = φ (a, b_1)
hψ_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ψ
hψUt : MapsTo ψ U t
dψ : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fderivWithin ℝ ψ (Icc 0 1)
s' : Set (ℝ × ℝ) := Prod.map Subtype.val Subtype.val '' s
hs' : s' = Prod.map Subtype.val Subtype.val '' s
hmem_s' : ∀ (x y : ↑I), (↑x, ↑y) ∈ s' ↔ (x, y) ∈ s
hs'c : s'.Countable
hdψ : ∀ a ∈ U, HasFDerivAt ψ (dψ a) a
d2ψ : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ →L[ℝ] E := fderivWithin ℝ dψ (Icc 0 1)
hd2ψ : ∀ a ∈ U, HasFDerivAt dψ (d2ψ a) a
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)
a : ℝ × ℝ
ha : a ∈ Icc 0 1
this : HasDerivWithinAt (fun x ↦ (a.1, x)) (0, 1) I a.2
⊢ DifferentiableWithinAt ℝ ψ (Icc 0 1) (a.1, a.2)

Tactic:
exact hcontdiff.differentiableOn (by decide) _ ha

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

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K : Compacts X
b : ℝ≥0∞
hb : ∀ (i : Compacts X), b ≤ ⨅ (_ : ↑K ⊆ interior ↑i), ↑(rieszContentAux Λ i)
this : b < ∞
ε : ℝ
hε : 0 < ε
⊢ ↑b.toNNReal ≤ ↑(sInf (⇑Λ '' {f | ∀ x ∈ K, 1 ≤ f x})) + ε

Tactic:
lift ε to ℝ≥0 using hε.le

Goal state after:
X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K : Compacts X
b : ℝ≥0∞
hb : ∀ (i : Compacts X), b ≤ ⨅ (_ : ↑K ⊆ interior ↑i), ↑(rieszContentAux Λ i)
this : b < ∞
ε : ℝ≥0
hε : 0 < ↑ε
⊢ ↑b.toNNReal ≤ ↑(sInf (⇑Λ '' {f | ∀ x ∈ K, 1 ≤ f x})) + ↑ε
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

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

Tactic:
← map_smul

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

Tactic:
Real.toNNReal_coe

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' := ⋯ }
ε✝ : NNReal
hε : 0 < ε✝
g : X →C_c NNReal
hg :
  (∀ x ∈ { carrier := tsupport ⇑f, isCompact' := ⋯ }, 1 ≤ g x) ∧
    (toNNRealLinear Λ) g < rieszContentAux (toNNRealLinear Λ) { carrier := tsupport ⇑f, isCompact' := ⋯ } + ε✝
⊢ NNReal.mk (Λ f) ⋯ ≤ rieszContentAux (toNNRealLinear Λ) { carrier := tsupport ⇑f, isCompact' := ⋯ } + ε✝
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: 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
⊢ Λ f ≤ rieszContentAux Λ { carrier := K, isCompact' := hK } + ε

Tactic:
apply le_trans _ hg.2.le

Goal state after:
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
⊢ Λ f ≤ Λ g
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare

Goal state before:
pp: case e_6.refine_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
a✝ b c d : E
γ₁ : Path a✝ b
γ₂ : Path c d
s : Set (↑I × ↑I)
t : Set E
ω : E → E →L[ℝ] F
dω : E → E →L[ℝ] E →L[ℝ] F
φ : (↑γ₁).Homotopy ↑γ₂
hs : s.Countable
hφt : ∀ a ∈ Ioo 0 1, ∀ b_1 ∈ Ioo 0 1, φ (a, b_1) ∈ t
hω : ∀ a ∈ Ioo 0 1, ∀ b_1 ∈ Ioo 0 1, (a, b_1) ∉ s → HasFDerivWithinAt ω (dω (φ (a, b_1))) t (φ (a, b_1))
hωc : ContinuousOn ω (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t)
hdω_symm :
  ∀ a ∈ Ioo 0 1,
    ∀ b_1 ∈ Ioo 0 1,
      (a, b_1) ∉ s →
        ∀ u ∈ tangentConeAt ℝ t (φ (a, b_1)),
          ∀ v ∈ tangentConeAt ℝ t (φ (a, b_1)), ((dω (φ (a, b_1))) u) v = ((dω (φ (a, b_1))) v) u
U : Set (ℝ × ℝ) := Ioo 0 1 ×ˢ Ioo 0 1
hU : U = Ioo 0 1 ×ˢ Ioo 0 1
hinterior : interior (Icc 0 1) = U
hunique : UniqueDiffOn ℝ (Icc 0 1)
hUopen : IsOpen U
hU_subset : U ⊆ Icc 0 1
hclosure : closure U = Icc 0 1
ψ : ℝ × ℝ → E := fun xy ↦ Set.IccExtend ⋯ (⇑(φ.extend xy.1)) xy.2
hcontdiff : ContDiffOn ℝ 2 ψ (Icc 0 1)
hψ : ψ = fun xy ↦ Set.IccExtend ⋯ (⇑(φ.extend xy.1)) xy.2
hψφ : ∀ (a b_1 : ↑I), ψ (↑a, ↑b_1) = φ (a, b_1)
hψ_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ψ
hψUt : MapsTo ψ U t
dψ : ℝ × ℝ → ℝ × ℝ →L[ℝ] E := fderivWithin ℝ ψ (Icc 0 1)
s' : Set (ℝ × ℝ) := Prod.map Subtype.val Subtype.val '' s
hs' : s' = Prod.map Subtype.val Subtype.val '' s
hmem_s' : ∀ (x y : ↑I), (↑x, ↑y) ∈ s' ↔ (x, y) ∈ s
hs'c : s'.Countable
hdψ : ∀ a ∈ U, HasFDerivAt ψ (dψ a) a
d2ψ : ℝ × ℝ → ℝ × ℝ →L[ℝ] ℝ × ℝ →L[ℝ] E := fderivWithin ℝ dψ (Icc 0 1)
hd2ψ : ∀ a ∈ U, HasFDerivAt dψ (d2ψ a) a
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)
a : ℝ × ℝ
ha : a ∈ Icc 0 1
this : HasDerivWithinAt (fun x ↦ (x, a.2)) (1, 0) I a.1
⊢ DifferentiableWithinAt ℝ ψ (Icc 0 1) (a.1, a.2)

Tactic:
exact hcontdiff.differentiableOn (by decide) _ ha

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

Goal state before:
pp: case h.refine_2.inl
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
a✝¹ : m ∈ univ
n : Fin N
a✝ : n ∈ univ
hmn : m ≠ n
x : X
hx : y m - ε < f x ∧ f x ≤ y m ∧ x ∈ tsupport ⇑f
hc : m < n
⊢ f x ≤ y n - ε ∨ y n < f x ∨ x ∉ tsupport ⇑f

Tactic:
left

Goal state after:
case h.refine_2.inl
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 := ⋯
y : Fin N → ℝ := ⋯
hy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m
E : Fin N → Set X := ⋯
m : Fin N
a✝¹ : m ∈ univ
n : Fin N
a✝ : n ∈ univ
hmn : m ≠ n
x : X
hx : y m - ε < f x ∧ f x ≤ y m ∧ x ∈ tsupport ⇑f
hc : m < n
⊢ f x ≤ y n - ε
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case h.refine_2.inr
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 := ⋯
y : Fin N → ℝ := ⋯
hy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m
E : Fin N → Set X := ⋯
m : Fin N
a✝¹ : m ∈ univ
n : Fin N
a✝ : n ∈ univ
hmn : m ≠ n
x : X
hx : y m - ε < f x ∧ f x ≤ y m ∧ x ∈ tsupport ⇑f
hc : n < m
⊢ y n < f x ∨ x ∉ tsupport ⇑f

Tactic:
left

Goal state after:
case h.refine_2.inr
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 := ⋯
y : Fin N → ℝ := ⋯
hy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m
E : Fin N → Set X := ⋯
m : Fin N
a✝¹ : m ∈ univ
n : Fin N
a✝ : n ∈ univ
hmn : m ≠ n
x : X
hx : y m - ε < f x ∧ f x ≤ y m ∧ x ∈ tsupport ⇑f
hc : n < m
⊢ y n < f x
Module: Mathlib.MeasureTheory.Measure.FiniteMeasureExt

Goal state before:
pp: E : Type u_1
𝕜 : Type u_2
inst✝⁷ : RCLike 𝕜
inst✝⁶ : MeasurableSpace E
inst✝⁵ : PseudoEMetricSpace E
inst✝⁴ : BorelSpace E
inst✝³ : CompleteSpace E
inst✝² : SecondCountableTopology E
P P' : Measure E
inst✝¹ : IsFiniteMeasure P
inst✝ : IsFiniteMeasure P'
A : StarSubalgebra 𝕜 (E →ᵇ 𝕜)
hA : (StarSubalgebra.map (toContinuousMapStarₐ 𝕜) A).SeparatesPoints
heq : ∀ g ∈ A, ∫ (x : E), g x ∂P = ∫ (x : E), g x ∂P'
A_toReal : Subalgebra ℝ (E →ᵇ ℝ) :=
  Subalgebra.comap (AlgHom.compLeftContinuousBounded ℝ ofRealAm ⋯) (Subalgebra.restrictScalars ℝ A.toSubalgebra)
⊢ (Subalgebra.comap (AlgHom.compLeftContinuous ℝ ofRealAm ⋯)
      (Subalgebra.restrictScalars ℝ (StarSubalgebra.map (toContinuousMapStarₐ 𝕜) A).toSubalgebra)).SeparatesPoints

Tactic:
exact Subalgebra.SeparatesPoints.rclike_to_real hA

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

Goal state before:
pp: μ : Measure ℝ
inst✝ : IsFiniteMeasure μ
n : ℕ
hint : MemLp id (↑n) μ
⊢ iteratedDeriv n (charFun μ) 0 = I ^ n * ↑(∫ (x : ℝ), x ^ n ∂μ)

Tactic:
simp [iteratedDeriv_charFun hint]
  norm_cast

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 := ⋯
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
⊢ ENNReal.ofReal (ε' / ↑N) ≠ ∞

Tactic:
finiteness

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

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
h : TopologicalSpace.IsTopologicalBasis {s | IsClopen s}
x : X
K : Set X
hK : IsClosed[inst✝] K
hx : x ∉ K
⊢ ∃ f, Continuous[inst✝, _] f ∧ f x = 0 ∧ EqOn f 1 K

Tactic:
obtain ⟨s, hs, hx, hsK⟩ := h.exists_subset_of_mem_open hx hK.isOpen_compl
    refine ⟨sᶜ.indicator 1, ?_, by simpa, fun x hx ↦ indicator_of_mem ?_ _⟩
    · exact hs.compl.continuous_indicator continuous_const
    · exact (mem_compl_iff s x).mpr fun hs ↦ hsK hs hx

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

Goal state before:
pp: case ha
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoEMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
LPμν_finite : ¬levyProkhorovEDist μ ν = ∞
LPνκ_finite : ¬levyProkhorovEDist ν κ = ∞
ε : ℝ≥0∞
B : Set Ω
ε_pos : 0 < ε
ε_lt_top : ε < ∞
B_mble : MeasurableSet B
half_ε_pos : 0 < ε / 2
r : ℝ≥0∞ := levyProkhorovEDist μ ν + ε / 2
s : ℝ≥0∞ := levyProkhorovEDist ν κ + ε / 2
lt_r : levyProkhorovEDist μ ν < r
lt_s : levyProkhorovEDist ν κ < s
hs_add_r : s + r = levyProkhorovEDist μ ν + levyProkhorovEDist ν κ + ε
⊢ s ≠ ∞

Tactic:
finiteness

Goal state after: