awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.PeakFunction (hypothesis)
Goal:
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst : TopologicalSpace α
inst_1 : BorelSpace α
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s t : Set α
φ : ι → α → ℝ
a : E
hs : MeasurableSet s
h'st : t ∈ 𝓝[s] x₀
hlφ : ∀ (u : Set α), IsOpen[inst] u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (β := ℝ) (fun (e : ι) ↦ ‖HSub.hSub (α := ℝ) (∫ (x : α) in t, φ e x ∂μ) 1‖) l (𝓝 0)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 a)
u : Set α
u_open : IsOpen[inst] u
x₀u : x₀ ∈ u
ut : s ∩ u ⊆ t
hu : ∀ x ∈ u ∩ s, g x ∈ ball a 1
i : ι
hi : ∀ x ∈ s \ u, dist (0 x) (φ i x) < 1
h'i : ‖HSub.hSub (α := ℝ) (∫ (x : α) in t, φ i x ∂μ) 1‖ < 1
h''i : AEStronglyMeasurable (φ i) (μ.restrict s)
⊢ IntegrableOn.{u_1, u_2} (ε := E) (fun (x : α) ↦ φ i x • g x) s μ
have I : IntegrableOn (φ i) t μ
Tactic: have I : IntegrableOn (φ i) t μ := .of_integral_ne_zero (fun h ↦ by simp [h] at h'i)
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.PeakFunction (goal)
Goal:
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst : TopologicalSpace α
inst_1 : BorelSpace α
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s t : Set α
φ : ι → α → ℝ
a : E
hs : MeasurableSet s
h'st : t ∈ 𝓝[s] x₀
hlφ : ∀ (u : Set α), IsOpen[inst] u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (β := ℝ) (fun (e : ι) ↦ ‖HSub.hSub (α := ℝ) (∫ (x : α) in t, φ e x ∂μ) 1‖) l (𝓝 0)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 a)
u : Set α
u_open : IsOpen[inst] u
x₀u : x₀ ∈ u
ut : s ∩ u ⊆ t
hu : ∀ x ∈ u ∩ s, g x ∈ ball a 1
i : ι
hi : ∀ x ∈ s \ u, dist (0 x) (φ i x) < 1
h'i : ‖HSub.hSub (α := ℝ) (∫ (x : α) in t, φ i x ∂μ) 1‖ < 1
h''i : AEStronglyMeasurable (φ i) (μ.restrict s)
I : IntegrableOn (φ i) t μ
⊢ IntegrableOn.{u_1, u_2} (ε := E) (fun (x : α) ↦ φ i x • g x) (s \ u) μ
have refine : MemLp (φ i) ∞ (μ.restrict (s \ u))
Tactic: refine Integrable.smul_of_top_right (hmg.mono diff_subset le_rfl) ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.PeakFunction (goal)
Goal:
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst : TopologicalSpace α
inst_1 : BorelSpace α
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s t : Set α
φ : ι → α → ℝ
a : E
hs : MeasurableSet s
h'st : t ∈ 𝓝[s] x₀
hlφ : ∀ (u : Set α), IsOpen[inst] u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (β := ℝ) (fun (e : ι) ↦ ‖HSub.hSub (α := ℝ) (∫ (x : α) in t, φ e x ∂μ) 1‖) l (𝓝 0)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 a)
u : Set α
u_open : IsOpen[inst] u
x₀u : x₀ ∈ u
ut : s ∩ u ⊆ t
hu : ∀ x ∈ u ∩ s, g x ∈ ball a 1
i : ι
hi : ∀ x ∈ s \ u, dist (0 x) (φ i x) < 1
h'i : ‖HSub.hSub (α := ℝ) (∫ (x : α) in t, φ i x ∂μ) 1‖ < 1
h''i : AEStronglyMeasurable (φ i) (μ.restrict s)
I : IntegrableOn (φ i) t μ
⊢ MemLp (φ i) ∞ (μ.restrict (s \ u))
have apply : ∀ᵐ (x : α) ∂μ.restrict (s \ u), ‖φ i x‖ ≤ 1
Tactic: apply memLp_top_of_bound (h''i.mono_set diff_subset) 1
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.PeakFunction (goal)
Goal:
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst : TopologicalSpace α
inst_1 : BorelSpace α
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s t : Set α
φ : ι → α → ℝ
a : E
hs : MeasurableSet s
h'st : t ∈ 𝓝[s] x₀
hlφ : ∀ (u : Set α), IsOpen[inst] u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (β := ℝ) (fun (e : ι) ↦ ‖HSub.hSub (α := ℝ) (∫ (x : α) in t, φ e x ∂μ) 1‖) l (𝓝 0)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 a)
u : Set α
u_open : IsOpen[inst] u
x₀u : x₀ ∈ u
ut : s ∩ u ⊆ t
hu : ∀ x ∈ u ∩ s, g x ∈ ball a 1
i : ι
hi : ∀ x ∈ s \ u, dist (0 x) (φ i x) < 1
h'i : ‖HSub.hSub (α := ℝ) (∫ (x : α) in t, φ i x ∂μ) 1‖ < 1
h''i : AEStronglyMeasurable (φ i) (μ.restrict s)
I : IntegrableOn (φ i) t μ
⊢ ∀ᵐ (x : α) ∂μ.restrict (s \ u), ‖φ i x‖ ≤ 1
have h : ∀ (x : α) (hx : x ∈ s \ u), ‖φ i x‖ ≤ 1
Tactic: filter_upwards [self_mem_ae_restrict (hs.diff u_open.measurableSet)] with x hx
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.PeakFunction (hypothesis)
Goal:
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst : TopologicalSpace α
inst_1 : BorelSpace α
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s t : Set α
φ : ι → α → ℝ
a : E
hs : MeasurableSet s
h'st : t ∈ 𝓝[s] x₀
hlφ : ∀ (u : Set α), IsOpen[inst] u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (β := ℝ) (fun (e : ι) ↦ ‖HSub.hSub (α := ℝ) (∫ (x : α) in t, φ e x ∂μ) 1‖) l (𝓝 0)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 a)
u : Set α
u_open : IsOpen[inst] u
x₀u : x₀ ∈ u
ut : s ∩ u ⊆ t
hu : ∀ x ∈ u ∩ s, g x ∈ ball a 1
i : ι
hi : ∀ x ∈ s \ u, dist (0 x) (φ i x) < 1
h'i : ‖HSub.hSub (α := ℝ) (∫ (x : α) in t, φ i x ∂μ) 1‖ < 1
h''i : AEStronglyMeasurable (φ i) (μ.restrict s)
I : IntegrableOn (φ i) t μ
⊢ IntegrableOn.{u_1, u_2} (ε := E) (fun (x : α) ↦ φ i x • g x) s μ
have A : IntegrableOn.{u_1, u_2} (ε := E) (fun (x : α) ↦ φ i x • g x) (s \ u) μ
Tactic: have A : IntegrableOn (fun x => φ i x • g x) (s \ u) μ := by
    refine Integrable.smul_of_top_right (hmg.mono diff_subset le_rfl) ?_
    apply memLp_top_of_bound (h''i.mono_set diff_subset) 1
    filter_upwards [self_mem_ae_restrict (hs.diff u_open.measurableSet)] with x hx
    simpa only [Pi.zero_apply, dist_zero_left] using (hi x hx).le
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.PeakFunction (goal)
Goal:
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst : TopologicalSpace α
inst_1 : BorelSpace α
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s t : Set α
φ : ι → α → ℝ
a : E
hs : MeasurableSet s
h'st : t ∈ 𝓝[s] x₀
hlφ : ∀ (u : Set α), IsOpen[inst] u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (β := ℝ) (fun (e : ι) ↦ ‖HSub.hSub (α := ℝ) (∫ (x : α) in t, φ e x ∂μ) 1‖) l (𝓝 0)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 a)
u : Set α
u_open : IsOpen[inst] u
x₀u : x₀ ∈ u
ut : s ∩ u ⊆ t
hu : ∀ x ∈ u ∩ s, g x ∈ ball a 1
i : ι
hi : ∀ x ∈ s \ u, dist (0 x) (φ i x) < 1
h'i : ‖HSub.hSub (α := ℝ) (∫ (x : α) in t, φ i x ∂μ) 1‖ < 1
h''i : AEStronglyMeasurable (φ i) (μ.restrict s)
I : IntegrableOn (φ i) t μ
A : IntegrableOn.{u_1, u_2} (ε := E) (fun (x : α) ↦ φ i x • g x) (s \ u) μ
⊢ IntegrableOn.{u_1, u_2} (ε := E) (fun (x : α) ↦ φ i x • g x) (s ∩ u) μ
have hφ : Integrable (φ i) (μ.restrict (s ∩ u))
Tactic: apply Integrable.smul_of_top_left
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.PeakFunction (goal)
Goal:
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst : TopologicalSpace α
inst_1 : BorelSpace α
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s t : Set α
φ : ι → α → ℝ
a : E
hs : MeasurableSet s
h'st : t ∈ 𝓝[s] x₀
hlφ : ∀ (u : Set α), IsOpen[inst] u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (β := ℝ) (fun (e : ι) ↦ ‖HSub.hSub (α := ℝ) (∫ (x : α) in t, φ e x ∂μ) 1‖) l (𝓝 0)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 a)
u : Set α
u_open : IsOpen[inst] u
x₀u : x₀ ∈ u
ut : s ∩ u ⊆ t
hu : ∀ x ∈ u ∩ s, g x ∈ ball a 1
i : ι
hi : ∀ x ∈ s \ u, dist (0 x) (φ i x) < 1
h'i : ‖HSub.hSub (α := ℝ) (∫ (x : α) in t, φ i x ∂μ) 1‖ < 1
h''i : AEStronglyMeasurable (φ i) (μ.restrict s)
I : IntegrableOn (φ i) t μ
A : IntegrableOn.{u_1, u_2} (ε := E) (fun (x : α) ↦ φ i x • g x) (s \ u) μ
hφ : Integrable (φ i) (μ.restrict (s ∩ u))
⊢ IntegrableOn.{u_1, u_2} (ε := E) (fun (x : α) ↦ φ i x • g x) (s ∩ u) μ
have hf : MemLp g ∞ (μ.restrict (s ∩ u))
Tactic: apply Integrable.smul_of_top_left
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.PeakFunction (goal)
Goal:
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst : TopologicalSpace α
inst_1 : BorelSpace α
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s t : Set α
φ : ι → α → ℝ
a : E
hs : MeasurableSet s
h'st : t ∈ 𝓝[s] x₀
hlφ : ∀ (u : Set α), IsOpen[inst] u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (β := ℝ) (fun (e : ι) ↦ ‖HSub.hSub (α := ℝ) (∫ (x : α) in t, φ e x ∂μ) 1‖) l (𝓝 0)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 a)
u : Set α
u_open : IsOpen[inst] u
x₀u : x₀ ∈ u
ut : s ∩ u ⊆ t
hu : ∀ x ∈ u ∩ s, g x ∈ ball a 1
i : ι
hi : ∀ x ∈ s \ u, dist (0 x) (φ i x) < 1
h'i : ‖HSub.hSub (α := ℝ) (∫ (x : α) in t, φ i x ∂μ) 1‖ < 1
h''i : AEStronglyMeasurable (φ i) (μ.restrict s)
I : IntegrableOn (φ i) t μ
A : IntegrableOn.{u_1, u_2} (ε := E) (fun (x : α) ↦ φ i x • g x) (s \ u) μ
⊢ MemLp g ∞ (μ.restrict (s ∩ u))
have hf : ∀ᵐ (x : α) ∂μ.restrict (s ∩ u), ‖g x‖ ≤ ‖a‖ + 1
Tactic: apply
        memLp_top_of_bound (hmg.mono_set inter_subset_left).aestronglyMeasurable (‖a‖ + 1)
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.PeakFunction (goal)
Goal:
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst : TopologicalSpace α
inst_1 : BorelSpace α
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s t : Set α
φ : ι → α → ℝ
a : E
hs : MeasurableSet s
h'st : t ∈ 𝓝[s] x₀
hlφ : ∀ (u : Set α), IsOpen[inst] u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (β := ℝ) (fun (e : ι) ↦ ‖HSub.hSub (α := ℝ) (∫ (x : α) in t, φ e x ∂μ) 1‖) l (𝓝 0)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 a)
u : Set α
u_open : IsOpen[inst] u
x₀u : x₀ ∈ u
ut : s ∩ u ⊆ t
hu : ∀ x ∈ u ∩ s, g x ∈ ball a 1
i : ι
hi : ∀ x ∈ s \ u, dist (0 x) (φ i x) < 1
h'i : ‖HSub.hSub (α := ℝ) (∫ (x : α) in t, φ i x ∂μ) 1‖ < 1
h''i : AEStronglyMeasurable (φ i) (μ.restrict s)
I : IntegrableOn (φ i) t μ
A : IntegrableOn.{u_1, u_2} (ε := E) (fun (x : α) ↦ φ i x • g x) (s \ u) μ
⊢ ∀ᵐ (x : α) ∂μ.restrict (s ∩ u), ‖g x‖ ≤ ‖a‖ + 1
have h : ∀ (x : α) (hx : x ∈ s ∩ u), ‖g x‖ ≤ ‖a‖ + 1
Tactic: filter_upwards [self_mem_ae_restrict (hs.inter u_open.measurableSet)] with x hx
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.PeakFunction (hypothesis)
Goal:
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst : TopologicalSpace α
inst_1 : BorelSpace α
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s t : Set α
φ : ι → α → ℝ
a : E
hs : MeasurableSet s
h'st : t ∈ 𝓝[s] x₀
hlφ : ∀ (u : Set α), IsOpen[inst] u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (β := ℝ) (fun (e : ι) ↦ ‖HSub.hSub (α := ℝ) (∫ (x : α) in t, φ e x ∂μ) 1‖) l (𝓝 0)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 a)
u : Set α
u_open : IsOpen[inst] u
x₀u : x₀ ∈ u
ut : s ∩ u ⊆ t
hu : ∀ x ∈ u ∩ s, g x ∈ ball a 1
i : ι
hi : ∀ x ∈ s \ u, dist (0 x) (φ i x) < 1
h'i : ‖HSub.hSub (α := ℝ) (∫ (x : α) in t, φ i x ∂μ) 1‖ < 1
h''i : AEStronglyMeasurable (φ i) (μ.restrict s)
I : IntegrableOn (φ i) t μ
A : IntegrableOn.{u_1, u_2} (ε := E) (fun (x : α) ↦ φ i x • g x) (s \ u) μ
x : α
hx : x ∈ s ∩ u
⊢ ‖g x‖ ≤ ‖a‖ + 1
have hx₁ : x ∈ u ∩ s
Tactic: rw [inter_comm] at hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.PeakFunction (hypothesis)
Goal:
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst : TopologicalSpace α
inst_1 : BorelSpace α
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s t : Set α
φ : ι → α → ℝ
a : E
hs : MeasurableSet s
h'st : t ∈ 𝓝[s] x₀
hlφ : ∀ (u : Set α), IsOpen[inst] u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (β := ℝ) (fun (e : ι) ↦ ‖HSub.hSub (α := ℝ) (∫ (x : α) in t, φ e x ∂μ) 1‖) l (𝓝 0)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 a)
u : Set α
u_open : IsOpen[inst] u
x₀u : x₀ ∈ u
ut : s ∩ u ⊆ t
hu : ∀ x ∈ u ∩ s, g x ∈ ball a 1
i : ι
hi : ∀ x ∈ s \ u, dist (0 x) (φ i x) < 1
h'i : ‖HSub.hSub (α := ℝ) (∫ (x : α) in t, φ i x ∂μ) 1‖ < 1
h''i : AEStronglyMeasurable (φ i) (μ.restrict s)
I : IntegrableOn (φ i) t μ
A : IntegrableOn.{u_1, u_2} (ε := E) (fun (x : α) ↦ φ i x • g x) (s \ u) μ
⊢ IntegrableOn.{u_1, u_2} (ε := E) (fun (x : α) ↦ φ i x • g x) s μ
have B : IntegrableOn.{u_1, u_2} (ε := E) (fun (x : α) ↦ φ i x • g x) (s ∩ u) μ
Tactic: have B : IntegrableOn (fun x => φ i x • g x) (s ∩ u) μ := by
    apply Integrable.smul_of_top_left
    · exact IntegrableOn.mono_set I ut
    · apply
        memLp_top_of_bound (hmg.mono_set inter_subset_left).aestronglyMeasurable (‖a‖ + 1)
      filter_upwards [self_mem_ae_restrict (hs.inter u_open.measurableSet)] with x hx
      rw [inter_comm] at hx
      exact (norm_lt_of_mem_ball (hu x hx)).le
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.PeakFunction (goal)
Goal:
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst : TopologicalSpace α
inst_1 : BorelSpace α
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s t : Set α
φ : ι → α → ℝ
a : E
hs : MeasurableSet s
h'st : t ∈ 𝓝[s] x₀
hlφ : ∀ (u : Set α), IsOpen[inst] u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (β := ℝ) (fun (e : ι) ↦ ‖HSub.hSub (α := ℝ) (∫ (x : α) in t, φ e x ∂μ) 1‖) l (𝓝 0)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 a)
u : Set α
u_open : IsOpen[inst] u
x₀u : x₀ ∈ u
ut : s ∩ u ⊆ t
hu : ∀ x ∈ u ∩ s, g x ∈ ball a 1
i : ι
hi : ∀ x ∈ s \ u, dist (0 x) (φ i x) < 1
h'i : ‖HSub.hSub (α := ℝ) (∫ (x : α) in t, φ i x ∂μ) 1‖ < 1
h''i : AEStronglyMeasurable (φ i) (μ.restrict s)
I : IntegrableOn (φ i) t μ
A : IntegrableOn.{u_1, u_2} (ε := E) (fun (x : α) ↦ φ i x • g x) (s \ u) μ
B : IntegrableOn.{u_1, u_2} (ε := E) (fun (x : α) ↦ φ i x • g x) (s ∩ u) μ
⊢ IntegrableOn.{u_1, u_2} (ε := E) (fun (x : α) ↦ φ i x • g x) s μ
have h : s = s \ u ∪ s ∩ u
Tactic: convert! A.union B
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.PeakFunction (goal)
Goal:
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst : TopologicalSpace α
inst_1 : BorelSpace α
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s t : Set α
φ : ι → α → ℝ
hs : MeasurableSet s
ht : MeasurableSet t
hts : t ⊆ s
h'ts : t ∈ 𝓝[s] x₀
hnφ : ∀ᶠ (i : ι) in l, ∀ x ∈ s, 0 ≤ φ i x
hlφ : ∀ (u : Set α), IsOpen[inst] u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (β := ℝ) (fun (i : ι) ↦ ∫ (x : α) in t, φ i x ∂μ) l (𝓝 1)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 0)
⊢ Tendsto.{u_3, u_2} (β := E) (fun (i : ι) ↦ ∫ (x : α) in s, φ i x • g x ∂μ) l (𝓝 0)
have refine : ∀ (ε : ℝ) (εpos : ε > 0), ∀ᶠ (x : ι) in l, dist.{u_2} (α := E) (∫ (x_1 : α) in s, φ x x_1 • g x_1 ∂μ) 0 < ε
Tactic: refine Metric.tendsto_nhds.2 fun ε εpos => ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.PeakFunction (goal)
Goal:
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst : TopologicalSpace α
inst_1 : BorelSpace α
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s t : Set α
φ : ι → α → ℝ
hs : MeasurableSet s
ht : MeasurableSet t
hts : t ⊆ s
h'ts : t ∈ 𝓝[s] x₀
hnφ : ∀ᶠ (i : ι) in l, ∀ x ∈ s, 0 ≤ φ i x
hlφ : ∀ (u : Set α), IsOpen[inst] u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (β := ℝ) (fun (i : ι) ↦ ∫ (x : α) in t, φ i x ∂μ) l (𝓝 1)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 0)
ε : ℝ
εpos : ε > 0
⊢ Tendsto (fun (δ : ℝ) ↦ δ * ∫ (x : α) in s, ‖g x‖ ∂μ + 2 * δ) (𝓝[>] 0)
    (𝓝 (HMul.hMul (α := ℝ) 0 (∫ (x : α) in s, ‖g x‖ ∂μ) + HMul.hMul (α := ℝ) 2 0))
have apply : Tendsto (fun (δ : ℝ) ↦ δ * ∫ (x : α) in s, ‖g x‖ ∂μ + 2 * δ) (𝓝 0)
  (𝓝 (HMul.hMul (α := ℝ) 0 (∫ (x : α) in s, ‖g x‖ ∂μ) + HMul.hMul (α := ℝ) 2 0))
Tactic: apply Tendsto.mono_left _ nhdsWithin_le_nhds
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.PeakFunction (hypothesis)
Goal:
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst : TopologicalSpace α
inst_1 : BorelSpace α
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s t : Set α
φ : ι → α → ℝ
hs : MeasurableSet s
ht : MeasurableSet t
hts : t ⊆ s
h'ts : t ∈ 𝓝[s] x₀
hnφ : ∀ᶠ (i : ι) in l, ∀ x ∈ s, 0 ≤ φ i x
hlφ : ∀ (u : Set α), IsOpen[inst] u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (β := ℝ) (fun (i : ι) ↦ ∫ (x : α) in t, φ i x ∂μ) l (𝓝 1)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 0)
ε : ℝ
εpos : ε > 0
⊢ ∃ (δ : ℝ), δ * ∫ (x : α) in s, ‖g x‖ ∂μ + 2 * δ < ε ∧ 0 < δ ∧ δ < 1
have A : Tendsto (fun (δ : ℝ) ↦ δ * ∫ (x : α) in s, ‖g x‖ ∂μ + 2 * δ) (𝓝[>] 0)
  (𝓝 (HMul.hMul (α := ℝ) 0 (∫ (x : α) in s, ‖g x‖ ∂μ) + HMul.hMul (α := ℝ) 2 0))
Tactic: have A :
      Tendsto (fun δ => (δ * ∫ x in s, ‖g x‖ ∂μ) + 2 * δ) (𝓝[>] 0)
        (𝓝 ((0 * ∫ x in s, ‖g x‖ ∂μ) + 2 * 0)) := by
      apply Tendsto.mono_left _ nhdsWithin_le_nhds
      exact (tendsto_id.mul tendsto_const_nhds).add (tendsto_id.const_mul _)
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.PeakFunction (hypothesis)
Goal:
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst : TopologicalSpace α
inst_1 : BorelSpace α
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s t : Set α
φ : ι → α → ℝ
hs : MeasurableSet s
ht : MeasurableSet t
hts : t ⊆ s
h'ts : t ∈ 𝓝[s] x₀
hnφ : ∀ᶠ (i : ι) in l, ∀ x ∈ s, 0 ≤ φ i x
hlφ : ∀ (u : Set α), IsOpen[inst] u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (β := ℝ) (fun (i : ι) ↦ ∫ (x : α) in t, φ i x ∂μ) l (𝓝 1)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 0)
ε : ℝ
εpos : ε > 0
A :
  Tendsto (fun (δ : ℝ) ↦ δ * ∫ (x : α) in s, ‖g x‖ ∂μ + 2 * δ) (𝓝[>] 0)
    (𝓝 (HMul.hMul (α := ℝ) 0 (∫ (x : α) in s, ‖g x‖ ∂μ) + HMul.hMul (α := ℝ) 2 0))
⊢ ∃ (δ : ℝ), δ * ∫ (x : α) in s, ‖g x‖ ∂μ + 2 * δ < ε ∧ 0 < δ ∧ δ < 1
have A₁ : Tendsto (β := ℝ) (fun (δ : ℝ) ↦ δ * ∫ (x : α) in s, ‖g x‖ ∂μ + 2 * δ) (𝓝[>] 0) (𝓝 0)
Tactic: rw [zero_mul, zero_add, mul_zero] at A
Cleared hypotheses: A
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.PeakFunction (hypothesis)
Goal:
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst : TopologicalSpace α
inst_1 : BorelSpace α
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s t : Set α
φ : ι → α → ℝ
hs : MeasurableSet s
ht : MeasurableSet t
hts : t ⊆ s
h'ts : t ∈ 𝓝[s] x₀
hnφ : ∀ᶠ (i : ι) in l, ∀ x ∈ s, 0 ≤ φ i x
hlφ : ∀ (u : Set α), IsOpen[inst] u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (β := ℝ) (fun (i : ι) ↦ ∫ (x : α) in t, φ i x ∂μ) l (𝓝 1)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 0)
ε : ℝ
εpos : ε > 0
A : Tendsto (β := ℝ) (fun (δ : ℝ) ↦ δ * ∫ (x : α) in s, ‖g x‖ ∂μ + 2 * δ) (𝓝[>] 0) (𝓝 0)
⊢ ∃ (δ : ℝ), δ * ∫ (x : α) in s, ‖g x‖ ∂μ + 2 * δ < ε ∧ 0 < δ ∧ δ < 1
have this : Membership.mem (γ := Filter ℝ) (𝓝[>] 0) (Ioo 0 1)
Tactic: have : Ioo (0 : ℝ) 1 ∈ 𝓝[>] 0 := Ioo_mem_nhdsGT zero_lt_one
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.PeakFunction (goal)
Goal:
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst : TopologicalSpace α
inst_1 : BorelSpace α
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s t : Set α
φ : ι → α → ℝ
hs : MeasurableSet s
ht : MeasurableSet t
hts : t ⊆ s
h'ts : t ∈ 𝓝[s] x₀
hnφ : ∀ᶠ (i : ι) in l, ∀ x ∈ s, 0 ≤ φ i x
hlφ : ∀ (u : Set α), IsOpen[inst] u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (β := ℝ) (fun (i : ι) ↦ ∫ (x : α) in t, φ i x ∂μ) l (𝓝 1)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 0)
ε : ℝ
εpos : ε > 0
A : Tendsto (β := ℝ) (fun (δ : ℝ) ↦ δ * ∫ (x : α) in s, ‖g x‖ ∂μ + 2 * δ) (𝓝[>] 0) (𝓝 0)
this : Membership.mem (γ := Filter ℝ) (𝓝[>] 0) (Ioo 0 1)
⊢ ∃ (δ : ℝ), δ * ∫ (x : α) in s, ‖g x‖ ∂μ + 2 * δ < ε ∧ 0 < δ ∧ δ < 1
have rcases : ∀ (δ : ℝ) (hδ : δ * ∫ (x : α) in s, ‖g x‖ ∂μ + 2 * δ < ε) (h'δ : 0 < δ ∧ δ < 1),
  ∃ (δ : ℝ), δ * ∫ (x : α) in s, ‖g x‖ ∂μ + 2 * δ < ε ∧ 0 < δ ∧ δ < 1
Tactic: rcases (((tendsto_order.1 A).2 ε εpos).and this).exists with ⟨δ, hδ, h'δ⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.PeakFunction (goal)
Goal:
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst : TopologicalSpace α
inst_1 : BorelSpace α
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s t : Set α
φ : ι → α → ℝ
hs : MeasurableSet s
ht : MeasurableSet t
hts : t ⊆ s
h'ts : t ∈ 𝓝[s] x₀
hnφ : ∀ᶠ (i : ι) in l, ∀ x ∈ s, 0 ≤ φ i x
hlφ : ∀ (u : Set α), IsOpen[inst] u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (β := ℝ) (fun (i : ι) ↦ ∫ (x : α) in t, φ i x ∂μ) l (𝓝 1)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 0)
ε : ℝ
εpos : ε > 0
⊢ ∀ᶠ (x : ι) in l, dist.{u_2} (α := E) (∫ (x_1 : α) in s, φ x x_1 • g x_1 ∂μ) 0 < ε
have obtain : ∀ (δ : ℝ) (hδ : δ * ∫ (x : α) in s, ‖g x‖ ∂μ + 2 * δ < ε) (δpos : 0 < δ) (δone : δ < 1),
  ∀ᶠ (x : ι) in l, dist.{u_2} (α := E) (∫ (x_1 : α) in s, φ x x_1 • g x_1 ∂μ) 0 < ε
Tactic: obtain ⟨δ, hδ, δpos, δone⟩ : ∃ δ, (δ * ∫ x in s, ‖g x‖ ∂μ) + 2 * δ < ε ∧ 0 < δ ∧ δ < 1 := by
    have A :
      Tendsto (fun δ => (δ * ∫ x in s, ‖g x‖ ∂μ) + 2 * δ) (𝓝[>] 0)
        (𝓝 ((0 * ∫ x in s, ‖g x‖ ∂μ) + 2 * 0)) := by
      apply Tendsto.mono_left _ nhdsWithin_le_nhds
      exact (tendsto_id.mul tendsto_const_nhds).add (tendsto_id.const_mul _)
    rw [zero_mul, zero_add, mul_zero] at A
    have : Ioo (0 : ℝ) 1 ∈ 𝓝[>] 0 := Ioo_mem_nhdsGT zero_lt_one
    rcases (((tendsto_order.1 A).2 ε εpos).and this).exists with ⟨δ, hδ, h'δ⟩
    exact ⟨δ, hδ, h'δ.1, h'δ.2⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Integral.PeakFunction (goal)
Goal:
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst : TopologicalSpace α
inst_1 : BorelSpace α
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s t : Set α
φ : ι → α → ℝ
hs : MeasurableSet s
ht : MeasurableSet t
hts : t ⊆ s
h'ts : t ∈ 𝓝[s] x₀
hnφ : ∀ᶠ (i : ι) in l, ∀ x ∈ s, 0 ≤ φ i x
hlφ : ∀ (u : Set α), IsOpen[inst] u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (β := ℝ) (fun (i : ι) ↦ ∫ (x : α) in t, φ i x ∂μ) l (𝓝 1)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 0)
ε : ℝ
εpos : ε > 0
δ : ℝ
hδ : δ * ∫ (x : α) in s, ‖g x‖ ∂μ + 2 * δ < ε
δpos : 0 < δ
δone : δ < 1
this : ∀ᶠ (i : ι) in l, norm.{u_2} (E := E) (∫ (x : α) in s, φ i x • g x ∂μ) ≤ δ * ∫ (x : α) in s, ‖g x‖ ∂μ + 2 * δ
⊢ ∀ᶠ (x : ι) in l, dist.{u_2} (α := E) (∫ (x_1 : α) in s, φ x x_1 • g x_1 ∂μ) 0 < ε
have h : ∀ (i : ι) (hi : norm.{u_2} (E := E) (∫ (x : α) in s, φ i x • g x ∂μ) ≤ δ * ∫ (x : α) in s, ‖g x‖ ∂μ + 2 * δ),
  dist.{u_2} (α := E) (∫ (x : α) in s, φ i x • g x ∂μ) 0 < ε
Tactic: filter_upwards [this] with i hi