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

Goal state before:
pp: case refine_2.inr.inr.inr
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
β : ι → Type u_3
inst✝⁵ : (i : ι) → TopologicalSpace (β i)
inst✝⁴ : ∀ (i : ι), MetrizableSpace (β i)
mβ : (i : ι) → MeasurableSpace (β i)
inst✝³ : ∀ (i : ι), BorelSpace (β i)
inst✝² : Preorder ι
inst✝¹ : (i : ι) → MulZeroOneClass (β i)
inst✝ : ∀ (i : ι), Nontrivial (β i)
s : ι → Set Ω
hsm : ∀ (i : ι), MeasurableSet (s i)
i : ι
t✝ : Set Ω
n : ι
ht : MeasurableSet t✝
t : Set Ω
hn : n ≤ i
u : Set (β n)
left✝ : MeasurableSet u
hu' : ((s n).indicator fun x ↦ 1) ⁻¹' u = t
heq : ((s n).indicator fun x ↦ 1) ⁻¹' u ∈ {∅}
⊢ MeasurableSet t

Tactic:
rw [Set.mem_singleton_iff] at heq

Goal state after:
case refine_2.inr.inr.inr
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
β : ι → Type u_3
inst✝⁵ : (i : ι) → TopologicalSpace (β i)
inst✝⁴ : ∀ (i : ι), MetrizableSpace (β i)
mβ : (i : ι) → MeasurableSpace (β i)
inst✝³ : ∀ (i : ι), BorelSpace (β i)
inst✝² : Preorder ι
inst✝¹ : (i : ι) → MulZeroOneClass (β i)
inst✝ : ∀ (i : ι), Nontrivial (β i)
s : ι → Set Ω
hsm : ∀ (i : ι), MeasurableSet (s i)
i : ι
t✝ : Set Ω
n : ι
ht : MeasurableSet t✝
t : Set Ω
hn : n ≤ i
u : Set (β n)
left✝ : MeasurableSet u
hu' : ((s n).indicator fun x ↦ 1) ⁻¹' u = t
heq : ((s n).indicator fun x ↦ 1) ⁻¹' u = ∅
⊢ MeasurableSet t
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
⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)

Tactic:
have hμ : μ ≠ 0 := by rintro rfl; exact hr₀ <| le_bot_iff.1 <| hr 0

Goal state after:
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
⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)
Module: Mathlib.Probability.Process.Filtration

Goal state before:
pp: Ω : Type u_1
ι✝ : Type u_2
m : MeasurableSpace Ω
β : ι✝ → Type u_3
inst✝⁴ : (i : ι✝) → TopologicalSpace (β i)
inst✝³ : ∀ (i : ι✝), MetrizableSpace (β i)
mβ : (i : ι✝) → MeasurableSpace (β i)
inst✝² : ∀ (i : ι✝), BorelSpace (β i)
inst✝¹ : Preorder ι✝
ι : Type u_4
X : ι → Type u_5
inst✝ : (i : ι) → MeasurableSpace (X i)
s t : Finset ι
hst : s ⊆ t
⊢ MeasurableSpace.comap s.restrict MeasurableSpace.pi ≤ MeasurableSpace.comap t.restrict MeasurableSpace.pi

Tactic:
rw [← restrict₂_comp_restrict hst, ← comap_comp]

Goal state after:
Ω : Type u_1
ι✝ : Type u_2
m : MeasurableSpace Ω
β : ι✝ → Type u_3
inst✝⁴ : (i : ι✝) → TopologicalSpace (β i)
inst✝³ : ∀ (i : ι✝), MetrizableSpace (β i)
mβ : (i : ι✝) → MeasurableSpace (β i)
inst✝² : ∀ (i : ι✝), BorelSpace (β i)
inst✝¹ : Preorder ι✝
ι : Type u_4
X : ι → Type u_5
inst✝ : (i : ι) → MeasurableSpace (X i)
s t : Finset ι
hst : s ⊆ t
⊢ MeasurableSpace.comap t.restrict (MeasurableSpace.comap (restrict₂ hst) MeasurableSpace.pi) ≤
    MeasurableSpace.comap t.restrict MeasurableSpace.pi
Module: Mathlib.MeasureTheory.Function.SpecialFunctions.Sinc

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
f : α → ℝ
μ : Measure α
hf : AEStronglyMeasurable f μ
⊢ AEStronglyMeasurable (fun x ↦ sinc (f x)) μ

Tactic:
rw [aestronglyMeasurable_iff_aemeasurable] at hf ⊢
  exact hf.sinc

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

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝⁴ : MeasurableSpace X
μ : Measure X
inst✝³ : Group G
inst✝² : MulAction G X
ι : Type u_3
l : Filter ι
F : ι → Set X
inst✝¹ : SMulInvariantMeasure G X μ
inst✝ : l.NeBot
hfoel : IsFoelner G μ l F
⊢ ∃ m,
    m univ = 1 ∧
      (∀ (s t : Set X), MeasurableSet t → Disjoint s t → m (s ∪ t) = m s + m t) ∧ ∀ (g : G) (s : Set X), m (g • s) = m s

Tactic:
use mean μ (Ultrafilter.of l) F
  refine ⟨?_, ?_, ?_⟩
  · exact (hfoel.mono <| Ultrafilter.of_le l).mean_univ_eq_one
  · exact (hfoel.mono <| Ultrafilter.of_le l).mean_union_eq_add_of_disjoint
  · exact (hfoel.mono <| Ultrafilter.of_le l).mean_smul_eq_mean

Goal state after:
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
U : Set E
hU : IsOpen U ∧ U ∩ ↑L = {0}
⊢ IsClosed (↑L \ {0})

Tactic:
sdiff_eq_sdiff_iff_inf_eq_inf (z := U).mpr (by simp [Set.inter_comm .. ▸ hU.2, zero_mem])

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
U : Set E
hU : IsOpen U ∧ U ∩ ↑L = {0}
⊢ IsClosed (↑L \ U)
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : TopologicalSpace α
Λ : (α →C_c ℝ) →ₚ[ℝ] ℝ
f g : α →C_c ℝ≥0
⊢ NNReal.mk (Λ (toRealLinearMap (f + g))) ⋯ = NNReal.mk (Λ (toRealLinearMap f)) ⋯ + NNReal.mk (Λ (toRealLinearMap g)) ⋯

Tactic:
simp; rfl

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
hx : 0 < x
⊢ x * |x| ^ q * rexp (-|x| ^ p) = x ^ (q + 1) * rexp (-x ^ p)

Tactic:
rw [abs_eq_self.mpr hx.le, rpow_add hx, rpow_one]

Goal state after:
case e_a
p q : ℝ
hp : 1 ≤ p
hq : -2 < q
x : ℝ
hx : 0 < x
⊢ x * x ^ q * rexp (-x ^ p) = x ^ q * x * 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
hx : 0 < x
⊢ x * |x| ^ q * rexp (-b * |x| ^ p) = x ^ (q + 1) * rexp (-b * x ^ p)

Tactic:
rw [abs_eq_self.mpr hx.le, rpow_add hx, rpow_one]

Goal state after:
case e_a
p q b : ℝ
hp : 1 ≤ p
hq : -2 < q
hb : 0 < b
x : ℝ
hx : 0 < x
⊢ x * x ^ q * rexp (-b * x ^ p) = x ^ q * x * 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✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
a b c : E
ω : E → E →L[𝕜] F
γab : Path a b
γbc : Path b c
h₁ : CurveIntegrable ω γab
h₂ : CurveIntegrable ω γbc
instF : NormedSpace ℝ F := NormedSpace.restrictScalars ℝ 𝕜 F
⊢ ∫ (t : ℝ) in 0..1, curveIntegralFun (fun x ↦ ω x) (γab.trans γbc) t = ∫ᶜ (x : E) in γab, ω x + ∫ᶜ (x : E) in γbc, ω x

Tactic:
← intervalIntegral.integral_add_adjacent_intervals
    (h₁.intervalIntegrable_curveIntegralFun_trans_left γbc)
    (h₂.intervalIntegrable_curveIntegralFun_trans_right γab),

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
a b c : E
ω : E → E →L[𝕜] F
γab : Path a b
γbc : Path b c
h₁ : CurveIntegrable ω γab
h₂ : CurveIntegrable ω γbc
instF : NormedSpace ℝ F := NormedSpace.restrictScalars ℝ 𝕜 F
⊢ (∫ (x : ℝ) in 0..1 / 2, curveIntegralFun ω (γab.trans γbc) x) +
      ∫ (x : ℝ) in 1 / 2..1, curveIntegralFun ω (γab.trans γbc) x =
    ∫ᶜ (x : E) in γab, ω x + ∫ᶜ (x : E) in γbc, ω x
Module: Mathlib.MeasureTheory.Integral.MeanValue

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
s : Set α
f g : α → ℝ
μ : Measure α
hs_conn : IsConnected s
hs_meas : MeasurableSet s
hf : ContinuousOn f s
hg : IntegrableOn g s μ
hfg : IntegrableOn (fun x ↦ f x * g x) s μ
hg0 : ∀ᵐ (x : α) ∂μ.restrict s, 0 ≤ g x
ρ : α → ENNReal := fun x ↦ ENNReal.ofReal (g x)
ν : Measure α := μ.withDensity ρ
hρ_ae : AEMeasurable ρ (μ.restrict s)
hρ_top : ∀ᵐ (x : α) ∂μ.restrict s, ρ x < ⊤
h_toReal_ae : (fun x ↦ (ρ x).toReal) =ᵐ[μ.restrict s] g
heq : ∫ (x : α) in s, f x * g x ∂μ = ∫ (x : α) in s, f x ∂ν
h : (∫ (x : α) in s, 1 ∂μ.withDensity fun x ↦ ρ x) = ∫ (x : α) in s, (ρ x).toReal • 1 ∂μ
⊢ ∫ (x : α) in s, g x ∂μ = ∫ (x : α) in s, (ρ x).toReal ∂μ

Tactic:
by rw [integral_congr_ae h_toReal_ae]

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✝ : NormedSpace ℝ F
ω : E → E →L[𝕜] F
a b : E
⊢ ∫ (t : ℝ) in 0..1, curveIntegralFun (fun x ↦ ω x) (Path.segment a b) t =
    ∫ (t : ℝ) in 0..1, (ω ((lineMap a b) t)) (b - a)

Tactic:
refine intervalIntegral.integral_congr fun t ht ↦ ?_

Goal state after:
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
ω : E → E →L[𝕜] F
a b : E
t : ℝ
ht : t ∈ [[0, 1]]
⊢ curveIntegralFun (fun x ↦ ω x) (Path.segment a b) t = (ω ((lineMap a b) t)) (b - a)
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 ℝ E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
a : E
s : Set E
ω : E → E →L[𝕜] F
hs : Convex ℝ s
hω : ∀ᶠ (x : E) in 𝓝[s] a, ContinuousWithinAt ω s x
ha : a ∈ s
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₀ : δ > 0
hδ : ball a δ ∩ s ⊆ {z | ContinuousWithinAt ω s z ∧ dist (ω z) (ω a) ≤ ε}
⊢ ∀ᶠ (x : E) in 𝓝 a, x ∈ s → ‖∫ᶜ (x : E) in Path.segment a x, ω x - (ω a) (x - a)‖ ≤ ε * ‖x - a‖

Tactic:
filter_upwards [Metric.ball_mem_nhds _ hδ₀] with b hb hbs

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
a : E
s : Set E
ω : E → E →L[𝕜] F
hs : Convex ℝ s
hω : ∀ᶠ (x : E) in 𝓝[s] a, ContinuousWithinAt ω s x
ha : a ∈ s
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₀ : δ > 0
hδ : ball a δ ∩ s ⊆ {z | ContinuousWithinAt ω s z ∧ dist (ω z) (ω a) ≤ ε}
b : E
hb : b ∈ ball a δ
hbs : b ∈ s
⊢ ‖∫ᶜ (x : E) in Path.segment a b, ω x - (ω a) (b - a)‖ ≤ ε * ‖b - a‖
Module: Mathlib.MeasureTheory.Integral.MeanValue

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
s : Set α
f g : α → ℝ
μ : Measure α
hs_conn : IsConnected s
hs_meas : MeasurableSet s
hf : ContinuousOn f s
hg : IntegrableOn g s μ
hfg : IntegrableOn (fun x ↦ f x * g x) s μ
hg0 : ∀ᵐ (x : α) ∂μ.restrict s, 0 ≤ g x
ρ : α → ENNReal := fun x ↦ ENNReal.ofReal (g x)
ν : Measure α := μ.withDensity ρ
hρ_ae : AEMeasurable ρ (μ.restrict s)
hρ_top : ∀ᵐ (x : α) ∂μ.restrict s, ρ x < ⊤
h_toReal_ae : (fun x ↦ (ρ x).toReal) =ᵐ[μ.restrict s] g
heq : ∫ (x : α) in s, f x * g x ∂μ = ∫ (x : α) in s, f x ∂ν
hg1 : ∫ (x : α) in s, g x ∂μ = ∫ (x : α) in s, 1 ∂ν
hν0 : ¬∫ (x : α) in s, 1 ∂ν = 0
hν0' : (ν s).toReal ≠ 0
hνfin : ν s ≠ ⊤
hν0'' : ν s ≠ 0
hmul_ae : (fun x ↦ (ρ x).toReal * f x) =ᵐ[μ.restrict s] fun x ↦ f x * g x
h_IntOn : IntegrableOn (fun x ↦ (ρ x).toReal * f x) s μ
⊢ Integrable f ((μ.restrict s).withDensity ρ)

Tactic:
integrable_withDensity_iff_integrable_smul₀' hρ_ae hρ_top,

Goal state after:
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
s : Set α
f g : α → ℝ
μ : Measure α
hs_conn : IsConnected s
hs_meas : MeasurableSet s
hf : ContinuousOn f s
hg : IntegrableOn g s μ
hfg : IntegrableOn (fun x ↦ f x * g x) s μ
hg0 : ∀ᵐ (x : α) ∂μ.restrict s, 0 ≤ g x
ρ : α → ENNReal := fun x ↦ ENNReal.ofReal (g x)
ν : Measure α := μ.withDensity ρ
hρ_ae : AEMeasurable ρ (μ.restrict s)
hρ_top : ∀ᵐ (x : α) ∂μ.restrict s, ρ x < ⊤
h_toReal_ae : (fun x ↦ (ρ x).toReal) =ᵐ[μ.restrict s] g
heq : ∫ (x : α) in s, f x * g x ∂μ = ∫ (x : α) in s, f x ∂ν
hg1 : ∫ (x : α) in s, g x ∂μ = ∫ (x : α) in s, 1 ∂ν
hν0 : ¬∫ (x : α) in s, 1 ∂ν = 0
hν0' : (ν s).toReal ≠ 0
hνfin : ν s ≠ ⊤
hν0'' : ν s ≠ 0
hmul_ae : (fun x ↦ (ρ x).toReal * f x) =ᵐ[μ.restrict s] fun x ↦ f x * g x
h_IntOn : IntegrableOn (fun x ↦ (ρ x).toReal * f x) s μ
⊢ Integrable (fun x ↦ (ρ x).toReal • f x) (μ.restrict s)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule

Goal state before:
pp: f : ℝ → ℝ
N : ℕ
a h : ℝ
N_nonzero : 0 < N
h_f_int : IntervalIntegrable f volume a (a + ↑N * h)
k : ℕ
hk : k < N
⊢ ∀ {k : ℕ}, k ≤ N → a + ↑k * h ∈ [[a, a + ↑N * h]]

Tactic:
rcases le_total h 0 with h_neg | h_pos <;> intro k hk

Goal state after:
case inl
f : ℝ → ℝ
N : ℕ
a h : ℝ
N_nonzero : 0 < N
h_f_int : IntervalIntegrable f volume a (a + ↑N * h)
k✝ : ℕ
hk✝ : k✝ < N
h_neg : h ≤ 0
k : ℕ
hk : k ≤ N
⊢ a + ↑k * h ∈ [[a, a + ↑N * h]]

case inr
f : ℝ → ℝ
N : ℕ
a h : ℝ
N_nonzero : 0 < N
h_f_int : IntervalIntegrable f volume a (a + ↑N * h)
k✝ : ℕ
hk✝ : k✝ < N
h_pos : 0 ≤ h
k : ℕ
hk : k ≤ N
⊢ a + ↑k * h ∈ [[a, a + ↑N * h]]
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}) + (↑ε).toNNReal
⊢ ↑b.toNNReal ≤ ↑(sInf (⇑Λ '' {f | ∀ x ∈ K, 1 ≤ f x})) + ↑ε

Tactic:
Real.toNNReal_of_nonneg 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
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 ⋯
⊢ ↑b.toNNReal ≤ ↑(sInf (⇑Λ '' {f | ∀ x ∈ K, 1 ≤ f x})) + ↑ε
Module: Mathlib.MeasureTheory.MeasurableSpace.Card

Goal state before:
pp: α : Type u
s : Set (Set α)
i : Ordinal.{u_1}
t✝ : Set α
p : Set α → Prop
hs : ∀ t ∈ s, p t
h0 : p ∅
hc : ∀ (u : Set α), p u → (∃ j < i, u ∈ generateMeasurableRec s j) → p uᶜ
hn : ∀ (f : ℕ → Set α), (∀ (n : ℕ), p (f n) ∧ ∃ j < i, f n ∈ generateMeasurableRec s j) → p (⋃ n, f n)
k✝ k : Ordinal.{u_1}
hk : k ≤ i
t : Set α
IH : ∀ j < k, ∀ t ∈ generateMeasurableRec s j, p t
⊢ (t ∈
      have S := ⋃ j, ⋃ (_ : j < k), generateMeasurableRec s j;
      s ∪ {∅} ∪ compl '' S ∪ range fun f ↦ ⋃ n, ↑(f n)) →
    p t

Tactic:
rintro (((ht | rfl) | ht) | ⟨f, rfl⟩)

Goal state after:
case inl.inl.inl
α : Type u
s : Set (Set α)
i : Ordinal.{u_1}
t✝ : Set α
p : Set α → Prop
hs : ∀ t ∈ s, p t
h0 : p ∅
hc : ∀ (u : Set α), p u → (∃ j < i, u ∈ generateMeasurableRec s j) → p uᶜ
hn : ∀ (f : ℕ → Set α), (∀ (n : ℕ), p (f n) ∧ ∃ j < i, f n ∈ generateMeasurableRec s j) → p (⋃ n, f n)
k✝ k : Ordinal.{u_1}
hk : k ≤ i
t : Set α
IH : ∀ j < k, ∀ t ∈ generateMeasurableRec s j, p t
ht : t ∈ s
⊢ p t

case inl.inl.inr
α : Type u
s : Set (Set α)
i : Ordinal.{u_1}
t : Set α
p : Set α → Prop
hs : ∀ t ∈ s, p t
h0 : p ∅
hc : ∀ (u : Set α), p u → (∃ j < i, u ∈ generateMeasurableRec s j) → p uᶜ
hn : ∀ (f : ℕ → Set α), (∀ (n : ℕ), p (f n) ∧ ∃ j < i, f n ∈ generateMeasurableRec s j) → p (⋃ n, f n)
k✝ k : Ordinal.{u_1}
hk : k ≤ i
IH : ∀ j < k, ∀ t ∈ generateMeasurableRec s j, p t
⊢ p ∅

case inl.inr
α : Type u
s : Set (Set α)
i : Ordinal.{u_1}
t✝ : Set α
p : Set α → Prop
hs : ∀ t ∈ s, p t
h0 : p ∅
hc : ∀ (u : Set α), p u → (∃ j < i, u ∈ generateMeasurableRec s j) → p uᶜ
hn : ∀ (f : ℕ → Set α), (∀ (n : ℕ), p (f n) ∧ ∃ j < i, f n ∈ generateMeasurableRec s j) → p (⋃ n, f n)
k✝ k : Ordinal.{u_1}
hk : k ≤ i
t : Set α
IH : ∀ j < k, ∀ t ∈ generateMeasurableRec s j, p t
ht : t ∈ compl '' ⋃ j, ⋃ (_ : j < k), generateMeasurableRec s j
⊢ p t

case inr
α : Type u
s : Set (Set α)
i : Ordinal.{u_1}
t : Set α
p : Set α → Prop
hs : ∀ t ∈ s, p t
h0 : p ∅
hc : ∀ (u : Set α), p u → (∃ j < i, u ∈ generateMeasurableRec s j) → p uᶜ
hn : ∀ (f : ℕ → Set α), (∀ (n : ℕ), p (f n) ∧ ∃ j < i, f n ∈ generateMeasurableRec s j) → p (⋃ n, f n)
k✝ k : Ordinal.{u_1}
hk : k ≤ i
IH : ∀ j < k, ∀ t ∈ generateMeasurableRec s j, p t
f : ℕ → ↑(⋃ j, ⋃ (_ : j < k), generateMeasurableRec s j)
⊢ p ((fun f ↦ ⋃ n, ↑(f n)) 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
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)
⊢ ⋃ j, E j ⊆ ⋃ n, (V n).carrier

Tactic:
gcongr with n

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
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)
n : Fin N
⊢ E n ⊆ (V n).carrier
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)
n : Fin N
hn : n ∈ Finset.univ
⊢ μ ↑(V n) < ∞

Tactic:
apply lt_of_le_of_lt (hV n).2.2

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 := ⋯
K : Set X := ⋯
ε : ℝ
hε : 0 < ε
a b : ℝ
hab : a < b ∧ range ⇑f ⊆ Ioo a b
N : ℕ
hN : 0 < N
ε' : ℝ := ⋯
hε' : 0 < ε' ∧ ε' * (2 * μ.real K + |a| + b + ε') ≤ ε
E : Fin N → Set X
hE :
  tsupport ⇑f = ⋃ j, E j ∧
    univ.PairwiseDisjoint E ∧
      (∀ (n : Fin N), ∀ x ∈ E n, a + ε' * ↑↑n < f x ∧ f x ≤ a + ε' * (↑↑n + 1)) ∧ ∀ (n : Fin N), MeasurableSet (E n)
y : Fin N → ℝ := ⋯
hE' : ∀ (n : Fin N), μ (E n) ≠ ∞
V : Fin N → Opens X
hV : ∀ (n : Fin N), E n ⊆ ↑(V n) ∧ (∀ x ∈ V n, f x < y n + ε') ∧ μ ↑(V n) ≤ μ (E n) + ENNReal.ofReal (ε' / ↑N)
g : Fin N → X →C_c ℝ
hg :
  (∀ (n : Fin N), tsupport ⇑(g n) ⊆ (V n).carrier) ∧
    EqOn (∑ n, ⇑(g n)) 1 (tsupport f.toFun) ∧
      (∀ (n : Fin N) (x : X), (g n) x ∈ Icc 0 1) ∧ ∀ (n : Fin N), HasCompactSupport ⇑(g n)
n : Fin N
hn : n ∈ Finset.univ
⊢ μ (E n) + ENNReal.ofReal (ε' / ↑N) < ∞
Module: Mathlib.Topology.Separation.CompletelyRegular

Goal state before:
pp: case a
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
x : X
⊢ ∀ (s : Set X), x ∈ s → IsOpen[inst✝¹] s → s ∈ comap stoneCechUnit (𝓝 (stoneCechUnit x))

Tactic:
((nhds_basis_opens _).comap _).mem_iff,

Goal state after:
case a
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
x : X
⊢ ∀ (s : Set X), x ∈ s → IsOpen[inst✝¹] s → ∃ i, (stoneCechUnit x ∈ i ∧ IsOpen i) ∧ stoneCechUnit ⁻¹' i ⊆ s