mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Group.GeometryOfNumbers

Goal state before:
pp: E : Type u_1
inst✝⁸ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : Nontrivial E
inst✝² : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝¹ : Countable ↥L
inst✝ : DiscreteTopology ↥L
fund : IsAddFundamentalDomain (↥L) F μ
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h_cpt : IsCompact s
h : μ F * 2 ^ finrank ℝ E ≤ μ s
h_mes : μ s ≠ 0
h_nemp : s.Nonempty
u : ℕ → ℝ≥0 := ⋯.choose
K : ConvexBody E := { carrier := s, convex' := h_conv, isCompact' := h_cpt, nonempty' := h_nemp }
S : ℕ → ConvexBody E := fun n ↦ (1 + u n) • K
Z : ℕ → Set E := fun n ↦ ↑(S n) ∩ (↑L \ {0})
h_zero : 0 ∈ K
this : (⋂ n, ↑((1 + u n) • K) ∩ (↑L \ {0})).Nonempty
⊢ ∃ x, x ≠ 0 ∧ ↑x ∈ s

Tactic:
ConvexBody.coe_smul',

Goal state after:
E : Type u_1
inst✝⁸ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : Nontrivial E
inst✝² : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝¹ : Countable ↥L
inst✝ : DiscreteTopology ↥L
fund : IsAddFundamentalDomain (↥L) F μ
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h_cpt : IsCompact s
h : μ F * 2 ^ finrank ℝ E ≤ μ s
h_mes : μ s ≠ 0
h_nemp : s.Nonempty
u : ℕ → ℝ≥0 := ⋯.choose
K : ConvexBody E := { carrier := s, convex' := h_conv, isCompact' := h_cpt, nonempty' := h_nemp }
S : ℕ → ConvexBody E := fun n ↦ (1 + u n) • K
Z : ℕ → Set E := fun n ↦ ↑(S n) ∩ (↑L \ {0})
h_zero : 0 ∈ K
this : (⋂ n, (1 + u n) • ↑K ∩ (↑L \ {0})).Nonempty
⊢ ∃ x, x ≠ 0 ∧ ↑x ∈ s
Module: Mathlib.MeasureTheory.Group.GeometryOfNumbers

Goal state before:
pp: case refine_2
E : Type u_1
inst✝⁸ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : Nontrivial E
inst✝² : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝¹ : Countable ↥L
inst✝ : DiscreteTopology ↥L
fund : IsAddFundamentalDomain (↥L) F μ
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h_cpt : IsCompact s
h : μ F * 2 ^ finrank ℝ E ≤ μ s
h_mes : μ s ≠ 0
h_nemp : s.Nonempty
u : ℕ → ℝ≥0 := ⋯.choose
K : ConvexBody E := { carrier := s, convex' := h_conv, isCompact' := h_cpt, nonempty' := h_nemp }
S : ℕ → ConvexBody E := fun n ↦ (1 + u n) • K
Z : ℕ → Set E := fun n ↦ ↑(S n) ∩ (↑L \ {0})
h_zero : 0 ∈ K
h_clos : IsClosed (↑L \ {0})
n : ℕ
⊢ (Z n).Nonempty

Tactic:
suffices μ F * 2 ^ finrank ℝ E < μ (S n : Set E) by
      have h_symm' : ∀ x ∈ S n, -x ∈ S n := by
        rintro _ ⟨y, hy, rfl⟩
        exact ⟨-y, h_symm _ hy, by simp⟩
      obtain ⟨x, hx_nz, hx_mem⟩ := exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure
        fund h_symm' (S n).convex this
      exact ⟨x, hx_mem, by simp_all⟩

Goal state after:
case refine_2
E : Type u_1
inst✝⁸ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : Nontrivial E
inst✝² : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝¹ : Countable ↥L
inst✝ : DiscreteTopology ↥L
fund : IsAddFundamentalDomain (↥L) F μ
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h_cpt : IsCompact s
h : μ F * 2 ^ finrank ℝ E ≤ μ s
h_mes : μ s ≠ 0
h_nemp : s.Nonempty
u : ℕ → ℝ≥0 := ⋯.choose
K : ConvexBody E := { carrier := s, convex' := h_conv, isCompact' := h_cpt, nonempty' := h_nemp }
S : ℕ → ConvexBody E := fun n ↦ (1 + u n) • K
Z : ℕ → Set E := fun n ↦ ↑(S n) ∩ (↑L \ {0})
h_zero : 0 ∈ K
h_clos : IsClosed (↑L \ {0})
n : ℕ
⊢ μ F * 2 ^ finrank ℝ E < μ ↑(S n)
Module: Mathlib.MeasureTheory.Group.GeometryOfNumbers

Goal state before:
pp: case refine_2
E : Type u_1
inst✝⁸ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : Nontrivial E
inst✝² : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝¹ : Countable ↥L
inst✝ : DiscreteTopology ↥L
fund : IsAddFundamentalDomain (↥L) F μ
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h_cpt : IsCompact s
h : μ F * 2 ^ finrank ℝ E ≤ μ s
h_mes : μ s ≠ 0
h_nemp : s.Nonempty
u : ℕ → ℝ≥0 := ⋯.choose
K : ConvexBody E := { carrier := s, convex' := h_conv, isCompact' := h_cpt, nonempty' := h_nemp }
S : ℕ → ConvexBody E := fun n ↦ (1 + u n) • K
Z : ℕ → Set E := fun n ↦ ↑(S n) ∩ (↑L \ {0})
h_zero : 0 ∈ K
h_clos : IsClosed (↑L \ {0})
n : ℕ
⊢ μ s < μ ↑(S n)

Tactic:
ConvexBody.coe_smul',

Goal state after:
case refine_2
E : Type u_1
inst✝⁸ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : Nontrivial E
inst✝² : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝¹ : Countable ↥L
inst✝ : DiscreteTopology ↥L
fund : IsAddFundamentalDomain (↥L) F μ
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h_cpt : IsCompact s
h : μ F * 2 ^ finrank ℝ E ≤ μ s
h_mes : μ s ≠ 0
h_nemp : s.Nonempty
u : ℕ → ℝ≥0 := ⋯.choose
K : ConvexBody E := { carrier := s, convex' := h_conv, isCompact' := h_cpt, nonempty' := h_nemp }
S : ℕ → ConvexBody E := fun n ↦ (1 + u n) • K
Z : ℕ → Set E := fun n ↦ ↑(S n) ∩ (↑L \ {0})
h_zero : 0 ∈ K
h_clos : IsClosed (↑L \ {0})
n : ℕ
⊢ μ s < μ ((1 + u n) • ↑K)
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝⁵ : TopologicalSpace α
inst✝⁴ : Zero β
inst✝³ : TopologicalSpace β
G : Type u_5
inst✝² : FunLike G α β
inst✝¹ : ContinuousMapClass G α β
inst✝ : CompactSpace α
⊢ ∀ (f : G), HasCompactSupport ⇑f

Tactic:
by
    intro f
    exact HasCompactSupport.of_compactSpace f

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

Goal state before:
pp: case e_a
p q : ℝ
hp : 1 ≤ p
hq : -2 < q
⊢ ∫ (x : ℝ) in Ioi 0, x * |x| ^ q * rexp (-|x| ^ p) = ∫ (x : ℝ) in Ioi 0, x ^ (q + 1) * rexp (-x ^ p)

Tactic:
refine setIntegral_congr_fun measurableSet_Ioi (fun x hx => ?_)

Goal state after:
case e_a
p q : ℝ
hp : 1 ≤ p
hq : -2 < q
x : ℝ
hx : x ∈ Ioi 0
⊢ x * |x| ^ q * rexp (-|x| ^ p) = x ^ (q + 1) * rexp (-x ^ p)
Module: Mathlib.MeasureTheory.Integral.Gamma

Goal state before:
pp: case e_a
p q b : ℝ
hp : 1 ≤ p
hq : -2 < q
hb : 0 < b
⊢ ∫ (x : ℝ) in Ioi 0, x * |x| ^ q * rexp (-b * |x| ^ p) = ∫ (x : ℝ) in Ioi 0, x ^ (q + 1) * rexp (-b * x ^ p)

Tactic:
refine setIntegral_congr_fun measurableSet_Ioi (fun x hx => ?_)

Goal state after:
case e_a
p q b : ℝ
hp : 1 ≤ p
hq : -2 < q
hb : 0 < b
x : ℝ
hx : x ∈ Ioi 0
⊢ x * |x| ^ q * rexp (-b * |x| ^ p) = x ^ (q + 1) * rexp (-b * x ^ p)
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
a b : E
ω : E → E →L[𝕜] F
inst✝ : NormedSpace ℝ E
⊢ EqOn (curveIntegralFun ω (Path.segment a b)) (fun t ↦ (ω ((lineMap a b) t)) (b - a)) (Ioc 0 1)

Tactic:
exact .mono Ioc_subset_Icc_self fun _t ↦ curveIntegralFun_segment ω a b

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

Goal state before:
pp: E✝ : Type u_1
inst✝³ : NormedAddCommGroup E✝
inst✝² : NormedSpace ℝ E✝
f✝ : ℝ → E✝
a b : ℝ
B : ℝ → ℝ
hab : a ≤ b
hBi : IntervalIntegrable B volume a b
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
hfc : ContinuousOn f (Icc a b)
hfd : DifferentiableOn ℝ f (Ioo a b)
hfB : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ B t
hE : CompleteSpace E
⊢ (fun x ↦ ‖deriv f x‖) ≤ᵐ[volume.restrict (Ioo a b)] B

Tactic:
EventuallyLE,

Goal state after:
E✝ : Type u_1
inst✝³ : NormedAddCommGroup E✝
inst✝² : NormedSpace ℝ E✝
f✝ : ℝ → E✝
a b : ℝ
B : ℝ → ℝ
hab : a ≤ b
hBi : IntervalIntegrable B volume a b
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
hfc : ContinuousOn f (Icc a b)
hfd : DifferentiableOn ℝ f (Ioo a b)
hfB : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ B t
hE : CompleteSpace E
⊢ ∀ᵐ (x : ℝ) ∂volume.restrict (Ioo a b), ‖deriv f x‖ ≤ B x
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.MeanValue

Goal state before:
pp: a✝ b✝ : ℝ
f g : ℝ → ℝ
μ : Measure ℝ
a b : ℝ
hf : ContinuousOn f [[a, b]]
hg : IntervalIntegrable g μ a b
hg0 : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), 0 ≤ g x
h : ¬a = b
hab : a < b
⊢ ∃ c ∈ [[a, b]], ∫ (x : ℝ) in a..b, f x * g x ∂μ = f c * ∫ (x : ℝ) in a..b, g x ∂μ

Tactic:
let s := Ι a b

Goal state after:
a✝ b✝ : ℝ
f g : ℝ → ℝ
μ : Measure ℝ
a b : ℝ
hf : ContinuousOn f [[a, b]]
hg : IntervalIntegrable g μ a b
hg0 : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), 0 ≤ g x
h : ¬a = b
hab : a < b
s : Set ℝ := Ι a b
⊢ ∃ c ∈ [[a, b]], ∫ (x : ℝ) in a..b, f x * g x ∂μ = f c * ∫ (x : ℝ) in a..b, g x ∂μ
Module: Mathlib.MeasureTheory.Integral.LebesgueNormedSpace

Goal state before:
pp: case mp.ht
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : SecondCountableTopology E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
f : α → ℝ≥0
hf : Measurable f
g g' : α → E
g'meas : Measurable g'
hg' : g =ᵐ[μ.withDensity fun x ↦ ↑(f x)] g'
A : MeasurableSet {x | f x ≠ 0}
⊢ ∀ᵐ (x : α) ∂μ.restrict {x | f x ≠ 0}, (fun x ↦ ↑(f x) • g x) x = (fun x ↦ ↑(f x) • g' x) x

Tactic:
rw [EventuallyEq, ae_withDensity_iff hf.coe_nnreal_ennreal] at hg'

Goal state after:
case mp.ht
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : SecondCountableTopology E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
f : α → ℝ≥0
hf : Measurable f
g g' : α → E
g'meas : Measurable g'
hg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x
A : MeasurableSet {x | f x ≠ 0}
⊢ ∀ᵐ (x : α) ∂μ.restrict {x | f x ≠ 0}, (fun x ↦ ↑(f x) • g x) x = (fun x ↦ ↑(f x) • g' x) x
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: 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
z : ℝ
⊢ z ∈ ⋃ i ∈ Finset.range N, Ioc (a + ↑i * ε) (a + ↑(i + 1) * ε) → z ∈ ⋃ n, Ioc (y n - ε) (y n)

Tactic:
simp only [Finset.mem_range, mem_iUnion, mem_Ioc, forall_exists_index, and_imp, y]

Goal state after:
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
z : ℝ
⊢ ∀ x < N, a + ↑x * ε < z → z ≤ a + ↑(x + 1) * ε → ∃ i, a + ε * (↑↑i + 1) - ε < z ∧ z ≤ a + ε * (↑↑i + 1)
Module: Mathlib.MeasureTheory.MeasurableSpace.Card

Goal state before:
pp: 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
⊢ ∀ j < i, #↑(generateMeasurableRec s j) ≤ max (#↑s) 2 ^ ℵ₀

pp: 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
⊢ Cardinal.lift.{u, v} i.card ≤ Cardinal.lift.{v, u} (max (#↑s) 2 ^ ℵ₀)

Tactic:
· intro j hj
      exact IH j hj (hj.trans_le hi).le

Goal state after:
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
⊢ Cardinal.lift.{u, v} i.card ≤ Cardinal.lift.{v, u} (max (#↑s) 2 ^ ℵ₀)
Module: Mathlib.Topology.Separation.CompletelyRegular

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
⊢ CompletelyRegularSpace X = ?m.25

Tactic:
completelyRegularSpace_iff,

Goal state after:
X : Type u
inst✝ : TopologicalSpace X
⊢ (∀ (x : X) (K : Set X), IsClosed[inst✝] K → x ∉ K → ∃ f, Continuous[inst✝, _] f ∧ f x = 0 ∧ EqOn f 1 K) = ?m.25
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✝ : OpensMeasurableSpace E
L : StrongDual ℝ E
⊢ charFunDual μ L = charFun (Measure.map (⇑L) μ) 1

Tactic:
by
  rw [charFunDual_apply]
  have : ∫ x, cexp (L x * I) ∂μ = ∫ x, cexp (x * I) ∂(μ.map L) := by
    rw [integral_map]
    · fun_prop
    · exact Measurable.aestronglyMeasurable <| by fun_prop
  rw [this, charFun_apply]
  simp

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

Goal state before:
pp: case a
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
x : X
U : Set X
hxU : x ∈ U
hU : IsOpen[inst✝¹] U
f : X → ↑I
hf : Continuous[inst✝¹, _] f
efx : f x = 0
hfU : stoneCechUnit ⁻¹' stoneCechExtend hf ⁻¹' {1}ᶜ ⊆ U
⊢ stoneCechExtend hf (stoneCechUnit x) ∈ {1}ᶜ

Tactic:
stoneCechExtend_stoneCechUnit,

Goal state after:
case a
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
x : X
U : Set X
hxU : x ∈ U
hU : IsOpen[inst✝¹] U
f : X → ↑I
hf : Continuous[inst✝¹, _] f
efx : f x = 0
hfU : stoneCechUnit ⁻¹' stoneCechExtend hf ⁻¹' {1}ᶜ ⊆ U
⊢ f x ∈ {1}ᶜ
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace Ω
μ ν : Measure Ω
δ : ℝ≥0∞
h :
  ∀ (ε : ℝ≥0∞) (B : Set Ω),
    0 < ε →
      ε < ∞ →
        MeasurableSet B → μ B ≤ ν (thickening (δ + ε).toReal B) + δ + ε ∧ ν B ≤ μ (thickening (δ + ε).toReal B) + δ + ε
⊢ levyProkhorovEDist μ ν ≤ δ

Tactic:
apply ENNReal.le_of_forall_pos_le_add

Goal state after:
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace Ω
μ ν : Measure Ω
δ : ℝ≥0∞
h :
  ∀ (ε : ℝ≥0∞) (B : Set Ω),
    0 < ε →
      ε < ∞ →
        MeasurableSet B → μ B ≤ ν (thickening (δ + ε).toReal B) + δ + ε ∧ ν B ≤ μ (thickening (δ + ε).toReal B) + δ + ε
⊢ ∀ (ε : ℝ≥0), 0 < ε → δ < ∞ → levyProkhorovEDist μ ν ≤ δ + ↑ε
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) ≠ ∞
V : Fin N → Opens X
hV : ∀ (n : Fin N), E n ⊆ ↑(V n) ∧ (∀ x ∈ V n, f x < y n + ε') ∧ μ ↑(V n) ≤ μ (E n) + ENNReal.ofReal (ε' / ↑N)
g : Fin N → X →C_c ℝ
hg :
  (∀ (n : Fin N), tsupport ⇑(g n) ⊆ (V n).carrier) ∧
    EqOn (∑ n, ⇑(g n)) 1 (tsupport f.toFun) ∧
      (∀ (n : Fin N) (x : X), (g n) x ∈ Icc 0 1) ∧ ∀ (n : Fin N), HasCompactSupport ⇑(g n)
h : ∀ (n : Fin N), (y n - ε') * μ.real (E n) ≤ ∫ (x : X) in E n, f x ∂μ
⊢ ∑ n, ∫ (x : X) in E n, f x ∂μ = ∫ (x : X) in ⋃ n, E n, f x ∂μ

Tactic:
by
        refine Eq.symm <| integral_iUnion_fintype hE.2.2.2 (fun _ _ ↦ hE.2.1 trivial trivial) ?_
        dsimp [μ, rieszMeasure]
        exact fun _ ↦
          Integrable.integrableOn <| Continuous.integrable_of_hasCompactSupport f.1.2 f.2

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.MulEquivHaarChar

Goal state before:
pp: G : Type u_1
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : IsTopologicalGroup G
inst✝² : LocallyCompactSpace G
μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ.Regular
φ : G ≃ₜ* G
e : G ≃ᵐ G := φ.toHomeomorph.toMeasurableEquiv
⊢ mulEquivHaarChar φ • μ = comap (⇑φ) μ

Tactic:
show ⇑φ = ⇑e from rfl,

Goal state after:
G : Type u_1
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : IsTopologicalGroup G
inst✝² : LocallyCompactSpace G
μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ.Regular
φ : G ≃ₜ* G
e : G ≃ᵐ G := φ.toHomeomorph.toMeasurableEquiv
⊢ mulEquivHaarChar φ • μ = comap (⇑e) μ
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: 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

Tactic:
· subsingleton

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

Goal state before:
pp: case inr
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : ∀ (x : ι → ℝ), 0 ≤ ∑ i, |x i| ^ p
h₂ : ∀ (x : ι → ℝ), 0 ≤ (∑ i, |x i| ^ p) ^ (1 / p)
hr : 0 < r
⊢ volume {x | (∑ i, |x i| ^ p) ^ (1 / p) < r} =
    ENNReal.ofReal r ^ card ι * ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (↑(card ι) / p + 1))

Tactic:
· rw [← volume_sum_rpow_lt_one _ hp, ← ofReal_pow (le_of_lt hr), ← finrank_pi ℝ]
    convert! addHaar_smul_of_nonneg volume (le_of_lt hr) {x : ι → ℝ | ∑ i, |x i| ^ p < 1} using 2
    simp_rw [← Set.preimage_smul_inv₀ (ne_of_gt hr), Set.preimage_ofPred_eq, Pi.smul_apply,
      smul_eq_mul, abs_mul, mul_rpow (abs_nonneg _) (abs_nonneg _), abs_inv,
      inv_rpow (abs_nonneg _), ← Finset.mul_sum, abs_eq_self.mpr (le_of_lt hr),
      inv_mul_lt_iff₀ (rpow_pos_of_pos hr _), mul_one, ← rpow_lt_rpow_iff
      (rpow_nonneg (h₁ _) _) (le_of_lt hr) (by linarith : 0 < p), ← rpow_mul
      (h₁ _), div_mul_cancel₀ _ (ne_of_gt (by linarith) : p ≠ 0), Real.rpow_one]

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

Goal state before:
pp: E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
B : (Measure.sum fun n ↦ ↑(ν n)) univ ≤ ↑C
μ : FiniteMeasure E := ⟨Measure.sum fun n ↦ ↑(ν n), ⋯⟩
g : E →ᵇ ℝ
ε : ℝ
εpos : ε > 0
⊢ Tendsto (fun n ↦ ∫ (x : E), g x ∂∑ i ∈ Finset.range n, ↑(ν i)) atTop (𝓝 (∫ (x : E), g x ∂Measure.sum fun i ↦ ↑(ν i)))

Tactic:
rw [integral_sum_measure (g.integrable (μ := μ))]

Goal state after:
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
B : (Measure.sum fun n ↦ ↑(ν n)) univ ≤ ↑C
μ : FiniteMeasure E := ⟨Measure.sum fun n ↦ ↑(ν n), ⋯⟩
g : E →ᵇ ℝ
ε : ℝ
εpos : ε > 0
⊢ Tendsto (fun n ↦ ∫ (x : E), g x ∂∑ i ∈ Finset.range n, ↑(ν i)) atTop (𝓝 (∑' (i : ℕ), ∫ (x : E), g x ∂↑(ν i)))
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: 𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst✝² : PseudoMetricSpace 𝓧
inst✝¹ : OpensMeasurableSpace 𝓧
inst✝ : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
U : ℕ → Set 𝓧
O : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
Cov : ⋃ i, U i = univ
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
ε : ℝ
hε' : 0 ≤ ε
hε : 0 < ε
hεbound : ε ≤ 1
μ : ℕ → ProbabilityMeasure 𝓧
hμInS : ∀ (k : ℕ), μ k ∈ S
hcontradiction : ∀ (k : ℕ), ↑((μ k) (⋃ i, ⋃ (_ : i ≤ k), U i)) ≤ 1 - ↑(NNReal.mk ε hε')
μlim : ProbabilityMeasure 𝓧
left✝ : μlim ∈ closure[ProbabilityMeasure.instTopologicalSpace] S
sub : ℕ → ℕ
hsubmono : StrictMono sub
hμconverges : Tendsto (μ ∘ sub) atTop (𝓝 μlim)
n : ℕ
hopen : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ i, ⋃ (_ : i ≤ n), U i)
this : ↑μlim (⋃ i, ⋃ (_ : i ≤ n), U i) ≤ liminf (fun i ↦ ↑((μ ∘ sub) i) (⋃ i, ⋃ (_ : i ≤ n), U i)) atTop
⊢ ↑(μlim (⋃ i, ⋃ (_ : i ≤ n), U i)) ≤ liminf (fun k ↦ ↑((μ (sub k)) (⋃ i, ⋃ (_ : i ≤ n), U i))) atTop

Tactic:
simp_rw [Function.comp_apply, ← ennreal_coeFn_eq_coeFn_toMeasure] at this

Goal state after:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst✝² : PseudoMetricSpace 𝓧
inst✝¹ : OpensMeasurableSpace 𝓧
inst✝ : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
U : ℕ → Set 𝓧
O : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
Cov : ⋃ i, U i = univ
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
ε : ℝ
hε' : 0 ≤ ε
hε : 0 < ε
hεbound : ε ≤ 1
μ : ℕ → ProbabilityMeasure 𝓧
hμInS : ∀ (k : ℕ), μ k ∈ S
hcontradiction : ∀ (k : ℕ), ↑((μ k) (⋃ i, ⋃ (_ : i ≤ k), U i)) ≤ 1 - ↑(NNReal.mk ε hε')
μlim : ProbabilityMeasure 𝓧
left✝ : μlim ∈ closure[ProbabilityMeasure.instTopologicalSpace] S
sub : ℕ → ℕ
hsubmono : StrictMono sub
hμconverges : Tendsto (μ ∘ sub) atTop (𝓝 μlim)
n : ℕ
hopen : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⋃ i, ⋃ (_ : i ≤ n), U i)
this : ↑(μlim (⋃ i, ⋃ (_ : i ≤ n), U i)) ≤ liminf (fun i ↦ ↑((μ (sub i)) (⋃ i, ⋃ (_ : i ≤ n), U i))) atTop
⊢ ↑(μlim (⋃ i, ⋃ (_ : i ≤ n), U i)) ≤ liminf (fun k ↦ ↑((μ (sub k)) (⋃ i, ⋃ (_ : i ≤ n), U i))) atTop
Module: Mathlib.MeasureTheory.Measure.SeparableMeasure

Goal state before:
pp: case refine_2.base
X : Type u_1
m : MeasurableSpace X
μ : Measure X
inst✝¹ : CountablyGenerated X
inst✝ : SigmaFinite μ
h : (countableGeneratingSet X).Countable
hgen : MeasurableSpace.generateFrom (countableGeneratingSet X) = m
𝒜 : Set (Set X) := countableGeneratingSet X ∪ {x | ∃ n, μ.toFiniteSpanningSetsIn.set n = x}
count_𝒜 : 𝒜.Countable
s t : Set X
t_mem : t ∈ 𝒜
⊢ MeasurableSet t

Tactic:
rcases t_mem with t_mem | ⟨n, rfl⟩
        · exact hgen ▸ measurableSet_generateFrom t_mem
        · exact μ.toFiniteSpanningSetsIn.set_mem n

Goal state after: