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leandojo-lean4-formal-informal-strings-split

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Split (3)

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formal stringlengths 41 427k	informal stringclasses 1 value
essSup_map_measure_of_measurable α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α inst✝⁵ : CompleteLattice β γ : Type u_3 mγ : MeasurableSpace γ f : α → γ g : γ → β inst✝⁴ : MeasurableSpace β inst✝³ : TopologicalSpace β inst✝² : SecondCountableTopology β inst✝¹ : OrderClosedTopology β inst✝ : OpensMeasurableSpace β hg : Measurable g hf : AEMeasurable f ⊢ essSup g (Measure.map f μ) = essSup (g ∘ f) μ refine' le_antisymm _ (essSup_comp_le_essSup_map_measure hf) α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α inst✝⁵ : CompleteLattice β γ : Type u_3 mγ : MeasurableSpace γ f : α → γ g : γ → β inst✝⁴ : MeasurableSpace β inst✝³ : TopologicalSpace β inst✝² : SecondCountableTopology β inst✝¹ : OrderClosedTopology β inst✝ : OpensMeasurableSpace β hg : Measurable g hf : AEMeasurable f ⊢ essSup g (Measure.map f μ) ≤ essSup (g ∘ f) μ refine' limsSup_le_limsSup (by isBoundedDefault) (by isBoundedDefault) (fun c h_le => _) α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α inst✝⁵ : CompleteLattice β γ : Type u_3 mγ : MeasurableSpace γ f : α → γ g : γ → β inst✝⁴ : MeasurableSpace β inst✝³ : TopologicalSpace β inst✝² : SecondCountableTopology β inst✝¹ : OrderClosedTopology β inst✝ : OpensMeasurableSpace β hg : Measurable g hf : AEMeasurable f c : β h_le : ∀ᶠ (n : β) in map (g ∘ f) (Measure.ae μ), n ≤ c ⊢ ∀ᶠ (n : β) in map g (Measure.ae (Measure.map f μ)), n ≤ c rw [eventually_map] at h_le ⊢ α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α inst✝⁵ : CompleteLattice β γ : Type u_3 mγ : MeasurableSpace γ f : α → γ g : γ → β inst✝⁴ : MeasurableSpace β inst✝³ : TopologicalSpace β inst✝² : SecondCountableTopology β inst✝¹ : OrderClosedTopology β inst✝ : OpensMeasurableSpace β hg : Measurable g hf : AEMeasurable f c : β h_le : ∀ᵐ (a : α) ∂μ, (g ∘ f) a ≤ c ⊢ ∀ᵐ (a : γ) ∂Measure.map f μ, g a ≤ c rw [ae_map_iff hf (measurableSet_le hg measurable_const)] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α inst✝⁵ : CompleteLattice β γ : Type u_3 mγ : MeasurableSpace γ f : α → γ g : γ → β inst✝⁴ : MeasurableSpace β inst✝³ : TopologicalSpace β inst✝² : SecondCountableTopology β inst✝¹ : OrderClosedTopology β inst✝ : OpensMeasurableSpace β hg : Measurable g hf : AEMeasurable f c : β h_le : ∀ᵐ (a : α) ∂μ, (g ∘ f) a ≤ c ⊢ ∀ᵐ (x : α) ∂μ, g (f x) ≤ c exact h_le α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α inst✝⁵ : CompleteLattice β γ : Type u_3 mγ : MeasurableSpace γ f : α → γ g : γ → β inst✝⁴ : MeasurableSpace β inst✝³ : TopologicalSpace β inst✝² : SecondCountableTopology β inst✝¹ : OrderClosedTopology β inst✝ : OpensMeasurableSpace β hg : Measurable g hf : AEMeasurable f ⊢ IsCobounded (fun x x_1 => x ≤ x_1) (map g (Measure.ae (Measure.map f μ))) isBoundedDefault α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α inst✝⁵ : CompleteLattice β γ : Type u_3 mγ : MeasurableSpace γ f : α → γ g : γ → β inst✝⁴ : MeasurableSpace β inst✝³ : TopologicalSpace β inst✝² : SecondCountableTopology β inst✝¹ : OrderClosedTopology β inst✝ : OpensMeasurableSpace β hg : Measurable g hf : AEMeasurable f ⊢ IsBounded (fun x x_1 => x ≤ x_1) (map (g ∘ f) (Measure.ae μ)) isBoundedDefault ** Qed
VitaliFamily.eventually_measure_lt_top α : Type u_1 inst✝² : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝¹ : NormedAddCommGroup E inst✝ : IsLocallyFiniteMeasure μ x : α ⊢ ∀ᶠ (a : Set α) in filterAt v x, ↑↑μ a < ⊤ obtain ⟨ε, εpos, με⟩ : ∃ (ε : ℝ), 0 < ε ∧ μ (closedBall x ε) < ∞ := (μ.finiteAt_nhds x).exists_mem_basis nhds_basis_closedBall case intro.intro α : Type u_1 inst✝² : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝¹ : NormedAddCommGroup E inst✝ : IsLocallyFiniteMeasure μ x : α ε : ℝ εpos : 0 < ε με : ↑↑μ (closedBall x ε) < ⊤ ⊢ ∀ᶠ (a : Set α) in filterAt v x, ↑↑μ a < ⊤ exact v.eventually_filterAt_iff.2 ⟨ε, εpos, fun a _ haε => (measure_mono haε).trans_lt με⟩ ** Qed
VitaliFamily.measure_le_of_frequently_le α : Type u_1 inst✝⁴ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝³ : NormedAddCommGroup E inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α ρ ν : Measure α inst✝ : IsLocallyFiniteMeasure ν hρ : ρ ≪ μ s : Set α hs : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑ν a ⊢ ↑↑ρ s ≤ ↑↑ν s apply ENNReal.le_of_forall_pos_le_add fun ε εpos _ => ?_ α : Type u_1 inst✝⁴ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝³ : NormedAddCommGroup E inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α ρ ν : Measure α inst✝ : IsLocallyFiniteMeasure ν hρ : ρ ≪ μ s : Set α hs : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑ν a ε : ℝ≥0 εpos : 0 < ε x✝ : ↑↑ν s < ⊤ ⊢ ↑↑ρ s ≤ ↑↑ν s + ↑ε obtain ⟨U, sU, U_open, νU⟩ : ∃ (U : Set α), s ⊆ U ∧ IsOpen U ∧ ν U ≤ ν s + ε := exists_isOpen_le_add s ν (ENNReal.coe_pos.2 εpos).ne' case intro.intro.intro α : Type u_1 inst✝⁴ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝³ : NormedAddCommGroup E inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α ρ ν : Measure α inst✝ : IsLocallyFiniteMeasure ν hρ : ρ ≪ μ s : Set α hs : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑ν a ε : ℝ≥0 εpos : 0 < ε x✝ : ↑↑ν s < ⊤ U : Set α sU : s ⊆ U U_open : IsOpen U νU : ↑↑ν U ≤ ↑↑ν s + ↑ε ⊢ ↑↑ρ s ≤ ↑↑ν s + ↑ε let f : α → Set (Set α) := fun _ => {a \| ρ a ≤ ν a ∧ a ⊆ U} case intro.intro.intro α : Type u_1 inst✝⁴ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝³ : NormedAddCommGroup E inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α ρ ν : Measure α inst✝ : IsLocallyFiniteMeasure ν hρ : ρ ≪ μ s : Set α hs : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑ν a ε : ℝ≥0 εpos : 0 < ε x✝ : ↑↑ν s < ⊤ U : Set α sU : s ⊆ U U_open : IsOpen U νU : ↑↑ν U ≤ ↑↑ν s + ↑ε f : α → Set (Set α) := fun x => {a \| ↑↑ρ a ≤ ↑↑ν a ∧ a ⊆ U} ⊢ ↑↑ρ s ≤ ↑↑ν s + ↑ε have h : v.FineSubfamilyOn f s := by apply v.fineSubfamilyOn_of_frequently f s fun x hx => ?_ have := (hs x hx).and_eventually ((v.eventually_filterAt_mem_sets x).and (v.eventually_filterAt_subset_of_nhds (U_open.mem_nhds (sU hx)))) apply Frequently.mono this rintro a ⟨ρa, _, aU⟩ exact ⟨ρa, aU⟩ case intro.intro.intro α : Type u_1 inst✝⁴ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝³ : NormedAddCommGroup E inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α ρ ν : Measure α inst✝ : IsLocallyFiniteMeasure ν hρ : ρ ≪ μ s : Set α hs : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑ν a ε : ℝ≥0 εpos : 0 < ε x✝ : ↑↑ν s < ⊤ U : Set α sU : s ⊆ U U_open : IsOpen U νU : ↑↑ν U ≤ ↑↑ν s + ↑ε f : α → Set (Set α) := fun x => {a \| ↑↑ρ a ≤ ↑↑ν a ∧ a ⊆ U} h : FineSubfamilyOn v f s ⊢ ↑↑ρ s ≤ ↑↑ν s + ↑ε haveI : Encodable h.index := h.index_countable.toEncodable case intro.intro.intro α : Type u_1 inst✝⁴ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝³ : NormedAddCommGroup E inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α ρ ν : Measure α inst✝ : IsLocallyFiniteMeasure ν hρ : ρ ≪ μ s : Set α hs : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑ν a ε : ℝ≥0 εpos : 0 < ε x✝ : ↑↑ν s < ⊤ U : Set α sU : s ⊆ U U_open : IsOpen U νU : ↑↑ν U ≤ ↑↑ν s + ↑ε f : α → Set (Set α) := fun x => {a \| ↑↑ρ a ≤ ↑↑ν a ∧ a ⊆ U} h : FineSubfamilyOn v f s this : Encodable ↑(FineSubfamilyOn.index h) ⊢ ↑↑ρ s ≤ ↑↑ν s + ↑ε calc ρ s ≤ ∑' x : h.index, ρ (h.covering x) := h.measure_le_tsum_of_absolutelyContinuous hρ _ ≤ ∑' x : h.index, ν (h.covering x) := (ENNReal.tsum_le_tsum fun x => (h.covering_mem x.2).1) _ = ν (⋃ x : h.index, h.covering x) := by rw [measure_iUnion h.covering_disjoint_subtype fun i => h.measurableSet_u i.2] _ ≤ ν U := (measure_mono (iUnion_subset fun i => (h.covering_mem i.2).2)) _ ≤ ν s + ε := νU α : Type u_1 inst✝⁴ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝³ : NormedAddCommGroup E inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α ρ ν : Measure α inst✝ : IsLocallyFiniteMeasure ν hρ : ρ ≪ μ s : Set α hs : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑ν a ε : ℝ≥0 εpos : 0 < ε x✝ : ↑↑ν s < ⊤ U : Set α sU : s ⊆ U U_open : IsOpen U νU : ↑↑ν U ≤ ↑↑ν s + ↑ε f : α → Set (Set α) := fun x => {a \| ↑↑ρ a ≤ ↑↑ν a ∧ a ⊆ U} ⊢ FineSubfamilyOn v f s apply v.fineSubfamilyOn_of_frequently f s fun x hx => ?_ α : Type u_1 inst✝⁴ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝³ : NormedAddCommGroup E inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α ρ ν : Measure α inst✝ : IsLocallyFiniteMeasure ν hρ : ρ ≪ μ s : Set α hs : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑ν a ε : ℝ≥0 εpos : 0 < ε x✝ : ↑↑ν s < ⊤ U : Set α sU : s ⊆ U U_open : IsOpen U νU : ↑↑ν U ≤ ↑↑ν s + ↑ε f : α → Set (Set α) := fun x => {a \| ↑↑ρ a ≤ ↑↑ν a ∧ a ⊆ U} x : α hx : x ∈ s ⊢ ∃ᶠ (a : Set α) in filterAt v x, a ∈ f x have := (hs x hx).and_eventually ((v.eventually_filterAt_mem_sets x).and (v.eventually_filterAt_subset_of_nhds (U_open.mem_nhds (sU hx)))) α : Type u_1 inst✝⁴ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝³ : NormedAddCommGroup E inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α ρ ν : Measure α inst✝ : IsLocallyFiniteMeasure ν hρ : ρ ≪ μ s : Set α hs : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑ν a ε : ℝ≥0 εpos : 0 < ε x✝ : ↑↑ν s < ⊤ U : Set α sU : s ⊆ U U_open : IsOpen U νU : ↑↑ν U ≤ ↑↑ν s + ↑ε f : α → Set (Set α) := fun x => {a \| ↑↑ρ a ≤ ↑↑ν a ∧ a ⊆ U} x : α hx : x ∈ s this : ∃ᶠ (x_1 : Set α) in filterAt v x, ↑↑ρ x_1 ≤ ↑↑ν x_1 ∧ x_1 ∈ setsAt v x ∧ x_1 ⊆ U ⊢ ∃ᶠ (a : Set α) in filterAt v x, a ∈ f x apply Frequently.mono this α : Type u_1 inst✝⁴ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝³ : NormedAddCommGroup E inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α ρ ν : Measure α inst✝ : IsLocallyFiniteMeasure ν hρ : ρ ≪ μ s : Set α hs : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑ν a ε : ℝ≥0 εpos : 0 < ε x✝ : ↑↑ν s < ⊤ U : Set α sU : s ⊆ U U_open : IsOpen U νU : ↑↑ν U ≤ ↑↑ν s + ↑ε f : α → Set (Set α) := fun x => {a \| ↑↑ρ a ≤ ↑↑ν a ∧ a ⊆ U} x : α hx : x ∈ s this : ∃ᶠ (x_1 : Set α) in filterAt v x, ↑↑ρ x_1 ≤ ↑↑ν x_1 ∧ x_1 ∈ setsAt v x ∧ x_1 ⊆ U ⊢ ∀ (x_1 : Set α), ↑↑ρ x_1 ≤ ↑↑ν x_1 ∧ x_1 ∈ setsAt v x ∧ x_1 ⊆ U → x_1 ∈ f x rintro a ⟨ρa, _, aU⟩ case intro.intro α : Type u_1 inst✝⁴ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝³ : NormedAddCommGroup E inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α ρ ν : Measure α inst✝ : IsLocallyFiniteMeasure ν hρ : ρ ≪ μ s : Set α hs : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑ν a ε : ℝ≥0 εpos : 0 < ε x✝ : ↑↑ν s < ⊤ U : Set α sU : s ⊆ U U_open : IsOpen U νU : ↑↑ν U ≤ ↑↑ν s + ↑ε f : α → Set (Set α) := fun x => {a \| ↑↑ρ a ≤ ↑↑ν a ∧ a ⊆ U} x : α hx : x ∈ s this : ∃ᶠ (x_1 : Set α) in filterAt v x, ↑↑ρ x_1 ≤ ↑↑ν x_1 ∧ x_1 ∈ setsAt v x ∧ x_1 ⊆ U a : Set α ρa : ↑↑ρ a ≤ ↑↑ν a left✝ : a ∈ setsAt v x aU : a ⊆ U ⊢ a ∈ f x exact ⟨ρa, aU⟩ α : Type u_1 inst✝⁴ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝³ : NormedAddCommGroup E inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α ρ ν : Measure α inst✝ : IsLocallyFiniteMeasure ν hρ : ρ ≪ μ s : Set α hs : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑ν a ε : ℝ≥0 εpos : 0 < ε x✝ : ↑↑ν s < ⊤ U : Set α sU : s ⊆ U U_open : IsOpen U νU : ↑↑ν U ≤ ↑↑ν s + ↑ε f : α → Set (Set α) := fun x => {a \| ↑↑ρ a ≤ ↑↑ν a ∧ a ⊆ U} h : FineSubfamilyOn v f s this : Encodable ↑(FineSubfamilyOn.index h) ⊢ ∑' (x : ↑(FineSubfamilyOn.index h)), ↑↑ν (FineSubfamilyOn.covering h ↑x) = ↑↑ν (⋃ x, FineSubfamilyOn.covering h ↑x) rw [measure_iUnion h.covering_disjoint_subtype fun i => h.measurableSet_u i.2] ** Qed
VitaliFamily.ae_eventually_measure_zero_of_singular α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 0) have A : ∀ ε > (0 : ℝ≥0), ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, ρ a < ε * μ a := by intro ε εpos set s := {x \| ¬∀ᶠ a in v.filterAt x, ρ a < ε * μ a} with hs change μ s = 0 obtain ⟨o, _, ρo, μo⟩ : ∃ o : Set α, MeasurableSet o ∧ ρ o = 0 ∧ μ oᶜ = 0 := hρ apply le_antisymm _ bot_le calc μ s ≤ μ (s ∩ o ∪ oᶜ) := by conv_lhs => rw [← inter_union_compl s o] exact measure_mono (union_subset_union_right _ (inter_subset_right _ _)) _ ≤ μ (s ∩ o) + μ oᶜ := (measure_union_le _ _) _ = μ (s ∩ o) := by rw [μo, add_zero] _ = (ε : ℝ≥0∞)⁻¹ * (ε • μ) (s ∩ o) := by simp only [coe_nnreal_smul_apply, ← mul_assoc, mul_comm _ (ε : ℝ≥0∞)] rw [ENNReal.mul_inv_cancel (ENNReal.coe_pos.2 εpos).ne' ENNReal.coe_ne_top, one_mul] _ ≤ (ε : ℝ≥0∞)⁻¹ * ρ (s ∩ o) := by refine' mul_le_mul_left' _ _ refine' v.measure_le_of_frequently_le ρ ((Measure.AbsolutelyContinuous.refl μ).smul ε) _ _ intro x hx rw [hs] at hx simp only [mem_inter_iff, not_lt, not_eventually, mem_setOf_eq] at hx exact hx.1 _ ≤ (ε : ℝ≥0∞)⁻¹ * ρ o := (mul_le_mul_left' (measure_mono (inter_subset_right _ _)) _) _ = 0 := by rw [ρo, mul_zero] ** α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ A : ∀ (ε : ℝ≥0), ε > 0 → ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 0) obtain ⟨u, _, u_pos, u_lim⟩ : ∃ u : ℕ → ℝ≥0, StrictAnti u ∧ (∀ n : ℕ, 0 < u n) ∧ Tendsto u atTop (𝓝 0) := exists_seq_strictAnti_tendsto (0 : ℝ≥0) case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ A : ∀ (ε : ℝ≥0), ε > 0 → ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a u : ℕ → ℝ≥0 left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 0) ** have B : ∀ᵐ x ∂μ, ∀ n, ∀ᶠ a in v.filterAt x, ρ a < u n * μ a := ae_all_iff.2 fun n => A (u n) (u_pos n) ** case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ A : ∀ (ε : ℝ≥0), ε > 0 → ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a u : ℕ → ℝ≥0 left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) B : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 0) filter_upwards [B, v.ae_eventually_measure_pos] case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ A : ∀ (ε : ℝ≥0), ε > 0 → ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a u : ℕ → ℝ≥0 left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) B : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a ⊢ ∀ (a : α), (∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v a, ↑↑ρ a < ↑(u n) * ↑↑μ a) → (∀ᶠ (a : Set α) in filterAt v a, 0 < ↑↑μ a) → Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v a) (𝓝 0) intro x hx h'x case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ A : ∀ (ε : ℝ≥0), ε > 0 → ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a u : ℕ → ℝ≥0 left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) B : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a x : α hx : ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a ⊢ Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 0) refine' tendsto_order.2 ⟨fun z hz => (ENNReal.not_lt_zero hz).elim, fun z hz => _⟩ case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ A : ∀ (ε : ℝ≥0), ε > 0 → ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a u : ℕ → ℝ≥0 left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) B : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a x : α hx : ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a z : ℝ≥0∞ hz : z > 0 ⊢ ∀ᶠ (b : Set α) in filterAt v x, ↑↑ρ b / ↑↑μ b < z obtain ⟨w, w_pos, w_lt⟩ : ∃ w : ℝ≥0, (0 : ℝ≥0∞) < w ∧ (w : ℝ≥0∞) < z := ENNReal.lt_iff_exists_nnreal_btwn.1 hz case h.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ A : ∀ (ε : ℝ≥0), ε > 0 → ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a u : ℕ → ℝ≥0 left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) B : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a x : α hx : ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a z : ℝ≥0∞ hz : z > 0 w : ℝ≥0 w_pos : 0 < ↑w w_lt : ↑w < z ⊢ ∀ᶠ (b : Set α) in filterAt v x, ↑↑ρ b / ↑↑μ b < z obtain ⟨n, hn⟩ : ∃ n, u n < w := ((tendsto_order.1 u_lim).2 w (ENNReal.coe_pos.1 w_pos)).exists case h.intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ A : ∀ (ε : ℝ≥0), ε > 0 → ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a u : ℕ → ℝ≥0 left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) B : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a x : α hx : ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a z : ℝ≥0∞ hz : z > 0 w : ℝ≥0 w_pos : 0 < ↑w w_lt : ↑w < z n : ℕ hn : u n < w ⊢ ∀ᶠ (b : Set α) in filterAt v x, ↑↑ρ b / ↑↑μ b < z filter_upwards [hx n, h'x, v.eventually_measure_lt_top x] case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ A : ∀ (ε : ℝ≥0), ε > 0 → ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a u : ℕ → ℝ≥0 left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) B : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a x : α hx : ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a z : ℝ≥0∞ hz : z > 0 w : ℝ≥0 w_pos : 0 < ↑w w_lt : ↑w < z n : ℕ hn : u n < w ⊢ ∀ (a : Set α), ↑↑ρ a < ↑(u n) * ↑↑μ a → 0 < ↑↑μ a → ↑↑μ a < ⊤ → ↑↑ρ a / ↑↑μ a < z intro a ha μa_pos μa_lt_top case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ A : ∀ (ε : ℝ≥0), ε > 0 → ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a u : ℕ → ℝ≥0 left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) B : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a x : α hx : ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a z : ℝ≥0∞ hz : z > 0 w : ℝ≥0 w_pos : 0 < ↑w w_lt : ↑w < z n : ℕ hn : u n < w a : Set α ha : ↑↑ρ a < ↑(u n) * ↑↑μ a μa_pos : 0 < ↑↑μ a μa_lt_top : ↑↑μ a < ⊤ ⊢ ↑↑ρ a / ↑↑μ a < z rw [ENNReal.div_lt_iff (Or.inl μa_pos.ne') (Or.inl μa_lt_top.ne)] case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ A : ∀ (ε : ℝ≥0), ε > 0 → ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a u : ℕ → ℝ≥0 left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) B : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a x : α hx : ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a z : ℝ≥0∞ hz : z > 0 w : ℝ≥0 w_pos : 0 < ↑w w_lt : ↑w < z n : ℕ hn : u n < w a : Set α ha : ↑↑ρ a < ↑(u n) * ↑↑μ a μa_pos : 0 < ↑↑μ a μa_lt_top : ↑↑μ a < ⊤ ⊢ ↑↑ρ a < z * ↑↑μ a exact ha.trans_le (mul_le_mul_right' ((ENNReal.coe_le_coe.2 hn.le).trans w_lt.le) _) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ ⊢ ∀ (ε : ℝ≥0), ε > 0 → ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a intro ε εpos α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ ε : ℝ≥0 εpos : ε > 0 ⊢ ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a ** set s := {x \| ¬∀ᶠ a in v.filterAt x, ρ a < ε * μ a} with hs ** α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} ⊢ ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a change μ s = 0 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} ⊢ ↑↑μ s = 0 obtain ⟨o, _, ρo, μo⟩ : ∃ o : Set α, MeasurableSet o ∧ ρ o = 0 ∧ μ oᶜ = 0 := hρ case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} o : Set α left✝ : MeasurableSet o ρo : ↑↑ρ o = 0 μo : ↑↑μ oᶜ = 0 ⊢ ↑↑μ s = 0 apply le_antisymm _ bot_le α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} o : Set α left✝ : MeasurableSet o ρo : ↑↑ρ o = 0 μo : ↑↑μ oᶜ = 0 ⊢ ↑↑μ s ≤ ⊥ ** calc μ s ≤ μ (s ∩ o ∪ oᶜ) := by conv_lhs => rw [← inter_union_compl s o] exact measure_mono (union_subset_union_right _ (inter_subset_right _ _)) _ ≤ μ (s ∩ o) + μ oᶜ := (measure_union_le _ _) _ = μ (s ∩ o) := by rw [μo, add_zero] _ = (ε : ℝ≥0∞)⁻¹ * (ε • μ) (s ∩ o) := by simp only [coe_nnreal_smul_apply, ← mul_assoc, mul_comm _ (ε : ℝ≥0∞)] rw [ENNReal.mul_inv_cancel (ENNReal.coe_pos.2 εpos).ne' ENNReal.coe_ne_top, one_mul] _ ≤ (ε : ℝ≥0∞)⁻¹ * ρ (s ∩ o) := by refine' mul_le_mul_left' _ _ refine' v.measure_le_of_frequently_le ρ ((Measure.AbsolutelyContinuous.refl μ).smul ε) _ _ intro x hx rw [hs] at hx simp only [mem_inter_iff, not_lt, not_eventually, mem_setOf_eq] at hx exact hx.1 _ ≤ (ε : ℝ≥0∞)⁻¹ * ρ o := (mul_le_mul_left' (measure_mono (inter_subset_right _ _)) _) _ = 0 := by rw [ρo, mul_zero] ** α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} o : Set α left✝ : MeasurableSet o ρo : ↑↑ρ o = 0 μo : ↑↑μ oᶜ = 0 ⊢ ↑↑μ s ≤ ↑↑μ (s ∩ o ∪ oᶜ) conv_lhs => rw [← inter_union_compl s o] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} o : Set α left✝ : MeasurableSet o ρo : ↑↑ρ o = 0 μo : ↑↑μ oᶜ = 0 ⊢ ↑↑μ (s ∩ o ∪ s ∩ oᶜ) ≤ ↑↑μ (s ∩ o ∪ oᶜ) exact measure_mono (union_subset_union_right _ (inter_subset_right _ _)) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} o : Set α left✝ : MeasurableSet o ρo : ↑↑ρ o = 0 μo : ↑↑μ oᶜ = 0 ⊢ ↑↑μ (s ∩ o) + ↑↑μ oᶜ = ↑↑μ (s ∩ o) rw [μo, add_zero] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} o : Set α left✝ : MeasurableSet o ρo : ↑↑ρ o = 0 μo : ↑↑μ oᶜ = 0 ⊢ ↑↑μ (s ∩ o) = (↑ε)⁻¹ * ↑↑(ε • μ) (s ∩ o) simp only [coe_nnreal_smul_apply, ← mul_assoc, mul_comm _ (ε : ℝ≥0∞)] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} o : Set α left✝ : MeasurableSet o ρo : ↑↑ρ o = 0 μo : ↑↑μ oᶜ = 0 ⊢ ↑↑μ ({x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} ∩ o) = ↑ε * (↑ε)⁻¹ * ↑↑μ ({x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} ∩ o) rw [ENNReal.mul_inv_cancel (ENNReal.coe_pos.2 εpos).ne' ENNReal.coe_ne_top, one_mul] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} o : Set α left✝ : MeasurableSet o ρo : ↑↑ρ o = 0 μo : ↑↑μ oᶜ = 0 ⊢ (↑ε)⁻¹ * ↑↑(ε • μ) (s ∩ o) ≤ (↑ε)⁻¹ * ↑↑ρ (s ∩ o) refine' mul_le_mul_left' _ _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} o : Set α left✝ : MeasurableSet o ρo : ↑↑ρ o = 0 μo : ↑↑μ oᶜ = 0 ⊢ ↑↑(ε • μ) (s ∩ o) ≤ ↑↑ρ (s ∩ o) refine' v.measure_le_of_frequently_le ρ ((Measure.AbsolutelyContinuous.refl μ).smul ε) _ _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} o : Set α left✝ : MeasurableSet o ρo : ↑↑ρ o = 0 μo : ↑↑μ oᶜ = 0 ⊢ ∀ (x : α), x ∈ s ∩ o → ∃ᶠ (a : Set α) in filterAt v x, ↑↑(ε • μ) a ≤ ↑↑ρ a intro x hx α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} o : Set α left✝ : MeasurableSet o ρo : ↑↑ρ o = 0 μo : ↑↑μ oᶜ = 0 x : α hx : x ∈ s ∩ o ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑(ε • μ) a ≤ ↑↑ρ a rw [hs] at hx α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} o : Set α left✝ : MeasurableSet o ρo : ↑↑ρ o = 0 μo : ↑↑μ oᶜ = 0 x : α hx : x ∈ {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} ∩ o ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑(ε • μ) a ≤ ↑↑ρ a simp only [mem_inter_iff, not_lt, not_eventually, mem_setOf_eq] at hx α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} o : Set α left✝ : MeasurableSet o ρo : ↑↑ρ o = 0 μo : ↑↑μ oᶜ = 0 x : α hx : (∃ᶠ (x : Set α) in filterAt v x, ↑ε * ↑↑μ x ≤ ↑↑ρ x) ∧ x ∈ o ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑(ε • μ) a ≤ ↑↑ρ a exact hx.1 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} o : Set α left✝ : MeasurableSet o ρo : ↑↑ρ o = 0 μo : ↑↑μ oᶜ = 0 ⊢ (↑ε)⁻¹ * ↑↑ρ o = 0 rw [ρo, mul_zero] Qed
VitaliFamily.null_of_frequently_le_of_frequently_ge ** α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ c d : ℝ≥0 hcd : c < d s : Set α hc : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑c * ↑↑μ a hd : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑d * ↑↑μ a ≤ ↑↑ρ a ⊢ ↑↑μ s = 0 apply null_of_locally_null s fun x _ => ?_ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ c d : ℝ≥0 hcd : c < d s : Set α hc : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑c * ↑↑μ a hd : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑d * ↑↑μ a ≤ ↑↑ρ a x : α x✝ : x ∈ s ⊢ ∃ u, u ∈ 𝓝[s] x ∧ ↑↑μ u = 0 obtain ⟨o, xo, o_open, μo⟩ : ∃ o : Set α, x ∈ o ∧ IsOpen o ∧ μ o < ∞ := Measure.exists_isOpen_measure_lt_top μ x case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ c d : ℝ≥0 hcd : c < d s : Set α hc : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑c * ↑↑μ a hd : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑d * ↑↑μ a ≤ ↑↑ρ a x : α x✝ : x ∈ s o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ ⊢ ∃ u, u ∈ 𝓝[s] x ∧ ↑↑μ u = 0 refine' ⟨s ∩ o, inter_mem_nhdsWithin _ (o_open.mem_nhds xo), _⟩ case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ c d : ℝ≥0 hcd : c < d s : Set α hc : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑c * ↑↑μ a hd : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑d * ↑↑μ a ≤ ↑↑ρ a x : α x✝ : x ∈ s o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ ⊢ ↑↑μ (s ∩ o) = 0 let s' := s ∩ o case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ c d : ℝ≥0 hcd : c < d s : Set α hc : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑c * ↑↑μ a hd : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑d * ↑↑μ a ≤ ↑↑ρ a x : α x✝ : x ∈ s o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s' : Set α := s ∩ o ⊢ ↑↑μ (s ∩ o) = 0 by_contra h case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ c d : ℝ≥0 hcd : c < d s : Set α hc : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑c * ↑↑μ a hd : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑d * ↑↑μ a ≤ ↑↑ρ a x : α x✝ : x ∈ s o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s' : Set α := s ∩ o h : ¬↑↑μ (s ∩ o) = 0 ⊢ False apply lt_irrefl (ρ s') case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ c d : ℝ≥0 hcd : c < d s : Set α hc : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑c * ↑↑μ a hd : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑d * ↑↑μ a ≤ ↑↑ρ a x : α x✝ : x ∈ s o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s' : Set α := s ∩ o h : ¬↑↑μ (s ∩ o) = 0 ⊢ ↑↑ρ s' < ↑↑ρ s' ** calc ρ s' ≤ c * μ s' := v.measure_le_of_frequently_le (c • μ) hρ s' fun x hx => hc x hx.1 _ < d * μ s' := by apply (ENNReal.mul_lt_mul_right h _).2 (ENNReal.coe_lt_coe.2 hcd) exact (lt_of_le_of_lt (measure_mono (inter_subset_right _ _)) μo).ne _ ≤ ρ s' := v.measure_le_of_frequently_le ρ ((Measure.AbsolutelyContinuous.refl μ).smul d) s' fun x hx => hd x hx.1 ** α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ c d : ℝ≥0 hcd : c < d s : Set α hc : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑c * ↑↑μ a hd : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑d * ↑↑μ a ≤ ↑↑ρ a x : α x✝ : x ∈ s o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s' : Set α := s ∩ o h : ¬↑↑μ (s ∩ o) = 0 ⊢ ↑c * ↑↑μ s' < ↑d * ↑↑μ s' apply (ENNReal.mul_lt_mul_right h _).2 (ENNReal.coe_lt_coe.2 hcd) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ c d : ℝ≥0 hcd : c < d s : Set α hc : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑c * ↑↑μ a hd : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑d * ↑↑μ a ≤ ↑↑ρ a x : α x✝ : x ∈ s o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s' : Set α := s ∩ o h : ¬↑↑μ (s ∩ o) = 0 ⊢ ↑↑μ (s ∩ o) ≠ ⊤ exact (lt_of_le_of_lt (measure_mono (inter_subset_right _ _)) μo).ne Qed
VitaliFamily.ae_tendsto_div α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ ⊢ ∀ᵐ (x : α) ∂μ, ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c) obtain ⟨w, w_count, w_dense, _, w_top⟩ : ∃ w : Set ℝ≥0∞, w.Countable ∧ Dense w ∧ 0 ∉ w ∧ ∞ ∉ w := ENNReal.exists_countable_dense_no_zero_top case intro.intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w ⊢ ∀ᵐ (x : α) ∂μ, ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c) have I : ∀ x ∈ w, x ≠ ∞ := fun x xs hx => w_top (hx ▸ xs) case intro.intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ A : ∀ (c : ℝ≥0∞), c ∈ w → ∀ (d : ℝ≥0∞), d ∈ w → c < d → ∀ᵐ (x : α) ∂μ, ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a) ⊢ ∀ᵐ (x : α) ∂μ, ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c) have B : ∀ᵐ x ∂μ, ∀ c ∈ w, ∀ d ∈ w, c < d → ¬((∃ᶠ a in v.filterAt x, ρ a / μ a < c) ∧ ∃ᶠ a in v.filterAt x, d < ρ a / μ a) := by simpa only [ae_ball_iff w_count, ae_all_iff] case intro.intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ A : ∀ (c : ℝ≥0∞), c ∈ w → ∀ (d : ℝ≥0∞), d ∈ w → c < d → ∀ᵐ (x : α) ∂μ, ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a) B : ∀ᵐ (x : α) ∂μ, ∀ (c : ℝ≥0∞), c ∈ w → ∀ (d : ℝ≥0∞), d ∈ w → c < d → ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a) ⊢ ∀ᵐ (x : α) ∂μ, ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c) filter_upwards [B] case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ A : ∀ (c : ℝ≥0∞), c ∈ w → ∀ (d : ℝ≥0∞), d ∈ w → c < d → ∀ᵐ (x : α) ∂μ, ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a) B : ∀ᵐ (x : α) ∂μ, ∀ (c : ℝ≥0∞), c ∈ w → ∀ (d : ℝ≥0∞), d ∈ w → c < d → ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a) ⊢ ∀ (a : α), (∀ (c : ℝ≥0∞), c ∈ w → ∀ (d : ℝ≥0∞), d ∈ w → c < d → ¬((∃ᶠ (a : Set α) in filterAt v a, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v a, d < ↑↑ρ a / ↑↑μ a)) → ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v a) (𝓝 c) intro x hx case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ A : ∀ (c : ℝ≥0∞), c ∈ w → ∀ (d : ℝ≥0∞), d ∈ w → c < d → ∀ᵐ (x : α) ∂μ, ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a) B : ∀ᵐ (x : α) ∂μ, ∀ (c : ℝ≥0∞), c ∈ w → ∀ (d : ℝ≥0∞), d ∈ w → c < d → ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a) x : α hx : ∀ (c : ℝ≥0∞), c ∈ w → ∀ (d : ℝ≥0∞), d ∈ w → c < d → ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a) ⊢ ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c) exact tendsto_of_no_upcrossings w_dense hx α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ ⊢ ∀ (c : ℝ≥0∞), c ∈ w → ∀ (d : ℝ≥0∞), d ∈ w → c < d → ∀ᵐ (x : α) ∂μ, ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a) intro c hc d hd hcd α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ c : ℝ≥0∞ hc : c ∈ w d : ℝ≥0∞ hd : d ∈ w hcd : c < d ⊢ ∀ᵐ (x : α) ∂μ, ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a) lift c to ℝ≥0 using I c hc case intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ d : ℝ≥0∞ hd : d ∈ w c : ℝ≥0 hc : ↑c ∈ w hcd : ↑c < d ⊢ ∀ᵐ (x : α) ∂μ, ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a) lift d to ℝ≥0 using I d hd case intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ c : ℝ≥0 hc : ↑c ∈ w d : ℝ≥0 hd : ↑d ∈ w hcd : ↑c < ↑d ⊢ ∀ᵐ (x : α) ∂μ, ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c) ∧ ∃ᶠ (a : Set α) in filterAt v x, ↑d < ↑↑ρ a / ↑↑μ a) apply v.null_of_frequently_le_of_frequently_ge hρ (ENNReal.coe_lt_coe.1 hcd) ** case intro.intro.hc α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ c : ℝ≥0 hc : ↑c ∈ w d : ℝ≥0 hd : ↑d ∈ w hcd : ↑c < ↑d ⊢ ∀ (x : α), x ∈ {x \| (fun x => ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c) ∧ ∃ᶠ (a : Set α) in filterAt v x, ↑d < ↑↑ρ a / ↑↑μ a)) x}ᶜ → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑c * ↑↑μ a simp only [and_imp, exists_prop, not_frequently, not_and, not_lt, not_le, not_eventually, mem_setOf_eq, mem_compl_iff, not_forall] case intro.intro.hc α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ c : ℝ≥0 hc : ↑c ∈ w d : ℝ≥0 hd : ↑d ∈ w hcd : ↑c < ↑d ⊢ ∀ (x : α), (∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c) → (∃ᶠ (x : Set α) in filterAt v x, ↑d < ↑↑ρ x / ↑↑μ x) → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑c * ↑↑μ a intro x h1x _ case intro.intro.hc α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ c : ℝ≥0 hc : ↑c ∈ w d : ℝ≥0 hd : ↑d ∈ w hcd : ↑c < ↑d x : α h1x : ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c a✝ : ∃ᶠ (x : Set α) in filterAt v x, ↑d < ↑↑ρ x / ↑↑μ x ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑c * ↑↑μ a apply h1x.mono fun a ha => ?_ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ c : ℝ≥0 hc : ↑c ∈ w d : ℝ≥0 hd : ↑d ∈ w hcd : ↑c < ↑d x : α h1x : ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c a✝ : ∃ᶠ (x : Set α) in filterAt v x, ↑d < ↑↑ρ x / ↑↑μ x a : Set α ha : ↑↑ρ a / ↑↑μ a < ↑c ⊢ ↑↑ρ a ≤ ↑c * ↑↑μ a refine' (ENNReal.div_le_iff_le_mul _ (Or.inr (bot_le.trans_lt ha).ne')).1 ha.le α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ c : ℝ≥0 hc : ↑c ∈ w d : ℝ≥0 hd : ↑d ∈ w hcd : ↑c < ↑d x : α h1x : ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c a✝ : ∃ᶠ (x : Set α) in filterAt v x, ↑d < ↑↑ρ x / ↑↑μ x a : Set α ha : ↑↑ρ a / ↑↑μ a < ↑c ⊢ ↑↑μ a ≠ 0 ∨ ↑c ≠ ⊤ simp only [ENNReal.coe_ne_top, Ne.def, or_true_iff, not_false_iff] case intro.intro.hd α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ c : ℝ≥0 hc : ↑c ∈ w d : ℝ≥0 hd : ↑d ∈ w hcd : ↑c < ↑d ⊢ ∀ (x : α), x ∈ {x \| (fun x => ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c) ∧ ∃ᶠ (a : Set α) in filterAt v x, ↑d < ↑↑ρ a / ↑↑μ a)) x}ᶜ → ∃ᶠ (a : Set α) in filterAt v x, ↑d * ↑↑μ a ≤ ↑↑ρ a simp only [and_imp, exists_prop, not_frequently, not_and, not_lt, not_le, not_eventually, mem_setOf_eq, mem_compl_iff, not_forall] case intro.intro.hd α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ c : ℝ≥0 hc : ↑c ∈ w d : ℝ≥0 hd : ↑d ∈ w hcd : ↑c < ↑d ⊢ ∀ (x : α), (∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c) → (∃ᶠ (x : Set α) in filterAt v x, ↑d < ↑↑ρ x / ↑↑μ x) → ∃ᶠ (a : Set α) in filterAt v x, ↑d * ↑↑μ a ≤ ↑↑ρ a intro x _ h2x case intro.intro.hd α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ c : ℝ≥0 hc : ↑c ∈ w d : ℝ≥0 hd : ↑d ∈ w hcd : ↑c < ↑d x : α a✝ : ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c h2x : ∃ᶠ (x : Set α) in filterAt v x, ↑d < ↑↑ρ x / ↑↑μ x ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑d * ↑↑μ a ≤ ↑↑ρ a apply h2x.mono fun a ha => ?_ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ c : ℝ≥0 hc : ↑c ∈ w d : ℝ≥0 hd : ↑d ∈ w hcd : ↑c < ↑d x : α a✝ : ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c h2x : ∃ᶠ (x : Set α) in filterAt v x, ↑d < ↑↑ρ x / ↑↑μ x a : Set α ha : ↑d < ↑↑ρ a / ↑↑μ a ⊢ ↑d * ↑↑μ a ≤ ↑↑ρ a exact ENNReal.mul_le_of_le_div ha.le α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ A : ∀ (c : ℝ≥0∞), c ∈ w → ∀ (d : ℝ≥0∞), d ∈ w → c < d → ∀ᵐ (x : α) ∂μ, ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a) ⊢ ∀ᵐ (x : α) ∂μ, ∀ (c : ℝ≥0∞), c ∈ w → ∀ (d : ℝ≥0∞), d ∈ w → c < d → ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a) simpa only [ae_ball_iff w_count, ae_all_iff] Qed
VitaliFamily.ae_tendsto_limRatio α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatio v ρ x)) filter_upwards [v.ae_tendsto_div hρ] case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ ⊢ ∀ (a : α), (∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v a) (𝓝 c)) → Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v a) (𝓝 (limRatio v ρ a)) intro x hx case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α hx : ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c) ⊢ Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatio v ρ x)) exact tendsto_nhds_limUnder hx ** Qed
VitaliFamily.exists_measurable_supersets_limRatio α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q ⊢ ∃ a b, MeasurableSet a ∧ MeasurableSet b ∧ {x \| limRatio v ρ x < ↑p} ⊆ a ∧ {x \| ↑q < limRatio v ρ x} ⊆ b ∧ ↑↑μ (a ∩ b) = 0 let s := {x \| ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)} α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ⊢ ∃ a b, MeasurableSet a ∧ MeasurableSet b ∧ {x \| limRatio v ρ x < ↑p} ⊆ a ∧ {x \| ↑q < limRatio v ρ x} ⊆ b ∧ ↑↑μ (a ∩ b) = 0 let o : ℕ → Set α := spanningSets (ρ + μ) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) ⊢ ∃ a b, MeasurableSet a ∧ MeasurableSet b ∧ {x \| limRatio v ρ x < ↑p} ⊆ a ∧ {x \| ↑q < limRatio v ρ x} ⊆ b ∧ ↑↑μ (a ∩ b) = 0 let u n := s ∩ {x \| v.limRatio ρ x < p} ∩ o n α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n ⊢ ∃ a b, MeasurableSet a ∧ MeasurableSet b ∧ {x \| limRatio v ρ x < ↑p} ⊆ a ∧ {x \| ↑q < limRatio v ρ x} ⊆ b ∧ ↑↑μ (a ∩ b) = 0 let w n := s ∩ {x \| (q : ℝ≥0∞) < v.limRatio ρ x} ∩ o n α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n ⊢ ∃ a b, MeasurableSet a ∧ MeasurableSet b ∧ {x \| limRatio v ρ x < ↑p} ⊆ a ∧ {x \| ↑q < limRatio v ρ x} ⊆ b ∧ ↑↑μ (a ∩ b) = 0 refine' ⟨toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n), toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n), _, _, _, _, _⟩ case refine'_5 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n ⊢ ↑↑μ ((toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n)) ∩ (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n))) = 0 suffices H : ∀ m n : ℕ, μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n ⊢ ∀ (m n : ℕ), ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 intro m n case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ ⊢ ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 have I : (ρ + μ) (u m) ≠ ∞ := by apply (lt_of_le_of_lt (measure_mono _) (measure_spanningSets_lt_top (ρ + μ) m)).ne exact inter_subset_right _ _ case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ ⊢ ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 have J : (ρ + μ) (w n) ≠ ∞ := by apply (lt_of_le_of_lt (measure_mono _) (measure_spanningSets_lt_top (ρ + μ) n)).ne exact inter_subset_right _ _ case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ ⊢ ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 have A : ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ p * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) := calc ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = ρ (u m ∩ toMeasurable (ρ + μ) (w n)) := measure_toMeasurable_add_inter_left (measurableSet_toMeasurable _ _) I _ ≤ (p • μ) (u m ∩ toMeasurable (ρ + μ) (w n)) := by refine' v.measure_le_of_frequently_le (p • μ) hρ _ fun x hx => _ have L : Tendsto (fun a : Set α => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatio ρ x)) := tendsto_nhds_limUnder hx.1.1.1 have I : ∀ᶠ b : Set α in v.filterAt x, ρ b / μ b < p := (tendsto_order.1 L).2 _ hx.1.1.2 apply I.frequently.mono fun a ha => ?_ rw [coe_nnreal_smul_apply] refine' (ENNReal.div_le_iff_le_mul _ (Or.inr (bot_le.trans_lt ha).ne')).1 ha.le simp only [ENNReal.coe_ne_top, Ne.def, or_true_iff, not_false_iff] _ = p * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) := by simp only [coe_nnreal_smul_apply, measure_toMeasurable_add_inter_right (measurableSet_toMeasurable _ _) I] ** case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ⊢ ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 ** have B : (q : ℝ≥0∞) * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) := calc (q : ℝ≥0∞) * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = (q : ℝ≥0∞) * μ (toMeasurable (ρ + μ) (u m) ∩ w n) := by conv_rhs => rw [inter_comm] rw [inter_comm, measure_toMeasurable_add_inter_right (measurableSet_toMeasurable _ _) J] _ ≤ ρ (toMeasurable (ρ + μ) (u m) ∩ w n) := by rw [← coe_nnreal_smul_apply] refine' v.measure_le_of_frequently_le _ (AbsolutelyContinuous.rfl.smul _) _ _ intro x hx have L : Tendsto (fun a : Set α => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatio ρ x)) := tendsto_nhds_limUnder hx.2.1.1 have I : ∀ᶠ b : Set α in v.filterAt x, (q : ℝ≥0∞) < ρ b / μ b := (tendsto_order.1 L).1 _ hx.2.1.2 apply I.frequently.mono fun a ha => ?_ rw [coe_nnreal_smul_apply] exact ENNReal.mul_le_of_le_div ha.le _ = ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) := by conv_rhs => rw [inter_comm] rw [inter_comm] exact (measure_toMeasurable_add_inter_left (measurableSet_toMeasurable _ _) J).symm ** case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) B : ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ⊢ ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 by_contra h case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) B : ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) h : ¬↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 ⊢ False apply lt_irrefl (ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))) case refine'_1 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n ⊢ MeasurableSet (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n)) exact (measurableSet_toMeasurable _ _).union (MeasurableSet.iUnion fun n => measurableSet_toMeasurable _ _) case refine'_2 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n ⊢ MeasurableSet (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n)) exact (measurableSet_toMeasurable _ _).union (MeasurableSet.iUnion fun n => measurableSet_toMeasurable _ _) case refine'_3 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n ⊢ {x \| limRatio v ρ x < ↑p} ⊆ toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n) intro x hx case refine'_3 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n x : α hx : x ∈ {x \| limRatio v ρ x < ↑p} ⊢ x ∈ toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n) by_cases h : x ∈ s case pos α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n x : α hx : x ∈ {x \| limRatio v ρ x < ↑p} h : x ∈ s ⊢ x ∈ toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n) refine' Or.inr (mem_iUnion.2 ⟨spanningSetsIndex (ρ + μ) x, _⟩) case pos α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n x : α hx : x ∈ {x \| limRatio v ρ x < ↑p} h : x ∈ s ⊢ x ∈ toMeasurable (ρ + μ) (u (spanningSetsIndex (ρ + μ) x)) exact subset_toMeasurable _ _ ⟨⟨h, hx⟩, mem_spanningSetsIndex _ _⟩ case neg α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n x : α hx : x ∈ {x \| limRatio v ρ x < ↑p} h : ¬x ∈ s ⊢ x ∈ toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n) exact Or.inl (subset_toMeasurable μ sᶜ h) case refine'_4 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n ⊢ {x \| ↑q < limRatio v ρ x} ⊆ toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n) intro x hx case refine'_4 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n x : α hx : x ∈ {x \| ↑q < limRatio v ρ x} ⊢ x ∈ toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n) by_cases h : x ∈ s case pos α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n x : α hx : x ∈ {x \| ↑q < limRatio v ρ x} h : x ∈ s ⊢ x ∈ toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n) refine' Or.inr (mem_iUnion.2 ⟨spanningSetsIndex (ρ + μ) x, _⟩) case pos α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n x : α hx : x ∈ {x \| ↑q < limRatio v ρ x} h : x ∈ s ⊢ x ∈ toMeasurable (ρ + μ) (w (spanningSetsIndex (ρ + μ) x)) exact subset_toMeasurable _ _ ⟨⟨h, hx⟩, mem_spanningSetsIndex _ _⟩ case neg α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n x : α hx : x ∈ {x \| ↑q < limRatio v ρ x} h : ¬x ∈ s ⊢ x ∈ toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n) exact Or.inl (subset_toMeasurable μ sᶜ h) case refine'_5 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n H : ∀ (m n : ℕ), ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 A : (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n)) ∩ (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n)) ⊆ toMeasurable μ sᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n) ⊢ ↑↑μ ((toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n)) ∩ (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n))) = 0 refine' le_antisymm ((measure_mono A).trans _) bot_le case refine'_5 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n H : ∀ (m n : ℕ), ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 A : (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n)) ∩ (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n)) ⊆ toMeasurable μ sᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n) ⊢ ↑↑μ (toMeasurable μ sᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ 0 calc μ (toMeasurable μ sᶜ ∪ ⋃ (m) (n), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ μ (toMeasurable μ sᶜ) + μ (⋃ (m) (n), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) := measure_union_le _ _ _ = μ (⋃ (m) (n), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) := by have : μ sᶜ = 0 := v.ae_tendsto_div hρ; rw [measure_toMeasurable, this, zero_add] _ ≤ ∑' (m) (n), μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) := ((measure_iUnion_le _).trans (ENNReal.tsum_le_tsum fun m => measure_iUnion_le _)) _ = 0 := by simp only [H, tsum_zero] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n H : ∀ (m n : ℕ), ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 ⊢ (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n)) ∩ (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n)) ⊆ toMeasurable μ sᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n) simp only [inter_distrib_left, inter_distrib_right, true_and_iff, subset_union_left, union_subset_iff, inter_self] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n H : ∀ (m n : ℕ), ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 ⊢ (⋃ n, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| limRatio v ρ x < ↑p} ∩ spanningSets (ρ + μ) n)) ∩ toMeasurable μ {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)}ᶜ ⊆ toMeasurable μ {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)}ᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| limRatio v ρ x < ↑p} ∩ spanningSets (ρ + μ) m) ∩ toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| ↑q < limRatio v ρ x} ∩ spanningSets (ρ + μ) n) ∧ toMeasurable μ {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)}ᶜ ∩ ⋃ n, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| ↑q < limRatio v ρ x} ∩ spanningSets (ρ + μ) n) ⊆ toMeasurable μ {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)}ᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| limRatio v ρ x < ↑p} ∩ spanningSets (ρ + μ) m) ∩ toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| ↑q < limRatio v ρ x} ∩ spanningSets (ρ + μ) n) ∧ (⋃ n, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| limRatio v ρ x < ↑p} ∩ spanningSets (ρ + μ) n)) ∩ ⋃ n, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| ↑q < limRatio v ρ x} ∩ spanningSets (ρ + μ) n) ⊆ toMeasurable μ {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)}ᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| limRatio v ρ x < ↑p} ∩ spanningSets (ρ + μ) m) ∩ toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| ↑q < limRatio v ρ x} ∩ spanningSets (ρ + μ) n) refine' ⟨_, _, _⟩ case refine'_1 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n H : ∀ (m n : ℕ), ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 ⊢ (⋃ n, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| limRatio v ρ x < ↑p} ∩ spanningSets (ρ + μ) n)) ∩ toMeasurable μ {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)}ᶜ ⊆ toMeasurable μ {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)}ᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| limRatio v ρ x < ↑p} ∩ spanningSets (ρ + μ) m) ∩ toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| ↑q < limRatio v ρ x} ∩ spanningSets (ρ + μ) n) exact (inter_subset_right _ _).trans (subset_union_left _ _) case refine'_2 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n H : ∀ (m n : ℕ), ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 ⊢ toMeasurable μ {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)}ᶜ ∩ ⋃ n, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| ↑q < limRatio v ρ x} ∩ spanningSets (ρ + μ) n) ⊆ toMeasurable μ {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)}ᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| limRatio v ρ x < ↑p} ∩ spanningSets (ρ + μ) m) ∩ toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| ↑q < limRatio v ρ x} ∩ spanningSets (ρ + μ) n) exact (inter_subset_left _ _).trans (subset_union_left _ _) case refine'_3 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n H : ∀ (m n : ℕ), ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 ⊢ (⋃ n, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| limRatio v ρ x < ↑p} ∩ spanningSets (ρ + μ) n)) ∩ ⋃ n, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| ↑q < limRatio v ρ x} ∩ spanningSets (ρ + μ) n) ⊆ toMeasurable μ {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)}ᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| limRatio v ρ x < ↑p} ∩ spanningSets (ρ + μ) m) ∩ toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| ↑q < limRatio v ρ x} ∩ spanningSets (ρ + μ) n) simp_rw [iUnion_inter, inter_iUnion] case refine'_3 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n H : ∀ (m n : ℕ), ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 ⊢ ⋃ i, ⋃ i_1, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| limRatio v ρ x < ↑p} ∩ spanningSets (ρ + μ) i) ∩ toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| ↑q < limRatio v ρ x} ∩ spanningSets (ρ + μ) i_1) ⊆ toMeasurable μ {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)}ᶜ ∪ ⋃ i, ⋃ i_1, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| limRatio v ρ x < ↑p} ∩ spanningSets (ρ + μ) i) ∩ toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| ↑q < limRatio v ρ x} ∩ spanningSets (ρ + μ) i_1) exact subset_union_right _ _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n H : ∀ (m n : ℕ), ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 A : (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n)) ∩ (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n)) ⊆ toMeasurable μ sᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n) ⊢ ↑↑μ (toMeasurable μ sᶜ) + ↑↑μ (⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = ↑↑μ (⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) have : μ sᶜ = 0 := v.ae_tendsto_div hρ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n H : ∀ (m n : ℕ), ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 A : (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n)) ∩ (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n)) ⊆ toMeasurable μ sᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n) this : ↑↑μ sᶜ = 0 ⊢ ↑↑μ (toMeasurable μ sᶜ) + ↑↑μ (⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = ↑↑μ (⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) rw [measure_toMeasurable, this, zero_add] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n H : ∀ (m n : ℕ), ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 A : (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n)) ∩ (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n)) ⊆ toMeasurable μ sᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n) ⊢ ∑' (m : ℕ) (n : ℕ), ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 simp only [H, tsum_zero] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ ⊢ ↑↑(ρ + μ) (u m) ≠ ⊤ apply (lt_of_le_of_lt (measure_mono _) (measure_spanningSets_lt_top (ρ + μ) m)).ne α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ ⊢ u m ⊆ spanningSets (ρ + μ) m exact inter_subset_right _ _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ ⊢ ↑↑(ρ + μ) (w n) ≠ ⊤ apply (lt_of_le_of_lt (measure_mono _) (measure_spanningSets_lt_top (ρ + μ) n)).ne α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ ⊢ w n ⊆ spanningSets (ρ + μ) n exact inter_subset_right _ _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ ⊢ ↑↑ρ (u m ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑↑(p • μ) (u m ∩ toMeasurable (ρ + μ) (w n)) refine' v.measure_le_of_frequently_le (p • μ) hρ _ fun x hx => _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ x : α hx : x ∈ u m ∩ toMeasurable (ρ + μ) (w n) ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑(p • μ) a have L : Tendsto (fun a : Set α => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatio ρ x)) := tendsto_nhds_limUnder hx.1.1.1 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ x : α hx : x ∈ u m ∩ toMeasurable (ρ + μ) (w n) L : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatio v ρ x)) ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑(p • μ) a have I : ∀ᶠ b : Set α in v.filterAt x, ρ b / μ b < p := (tendsto_order.1 L).2 _ hx.1.1.2 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I✝ : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ x : α hx : x ∈ u m ∩ toMeasurable (ρ + μ) (w n) L : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatio v ρ x)) I : ∀ᶠ (b : Set α) in filterAt v x, ↑↑ρ b / ↑↑μ b < ↑p ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑(p • μ) a apply I.frequently.mono fun a ha => ?_ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I✝ : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ x : α hx : x ∈ u m ∩ toMeasurable (ρ + μ) (w n) L : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatio v ρ x)) I : ∀ᶠ (b : Set α) in filterAt v x, ↑↑ρ b / ↑↑μ b < ↑p a : Set α ha : ↑↑ρ a / ↑↑μ a < ↑p ⊢ ↑↑ρ a ≤ ↑↑(p • μ) a rw [coe_nnreal_smul_apply] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I✝ : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ x : α hx : x ∈ u m ∩ toMeasurable (ρ + μ) (w n) L : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatio v ρ x)) I : ∀ᶠ (b : Set α) in filterAt v x, ↑↑ρ b / ↑↑μ b < ↑p a : Set α ha : ↑↑ρ a / ↑↑μ a < ↑p ⊢ ↑↑ρ a ≤ ↑p * ↑↑μ a refine' (ENNReal.div_le_iff_le_mul _ (Or.inr (bot_le.trans_lt ha).ne')).1 ha.le α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I✝ : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ x : α hx : x ∈ u m ∩ toMeasurable (ρ + μ) (w n) L : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatio v ρ x)) I : ∀ᶠ (b : Set α) in filterAt v x, ↑↑ρ b / ↑↑μ b < ↑p a : Set α ha : ↑↑ρ a / ↑↑μ a < ↑p ⊢ ↑↑μ a ≠ 0 ∨ ↑p ≠ ⊤ simp only [ENNReal.coe_ne_top, Ne.def, or_true_iff, not_false_iff] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ ⊢ ↑↑(p • μ) (u m ∩ toMeasurable (ρ + μ) (w n)) = ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) simp only [coe_nnreal_smul_apply, measure_toMeasurable_add_inter_right (measurableSet_toMeasurable _ _) I] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ⊢ ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ w n) conv_rhs => rw [inter_comm] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ⊢ ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = ↑q * ↑↑μ (w n ∩ toMeasurable (ρ + μ) (u m)) rw [inter_comm, measure_toMeasurable_add_inter_right (measurableSet_toMeasurable _ _) J] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ⊢ ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ w n) ≤ ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ w n) rw [← coe_nnreal_smul_apply] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ⊢ ↑↑(q • μ) (toMeasurable (ρ + μ) (u m) ∩ w n) ≤ ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ w n) refine' v.measure_le_of_frequently_le _ (AbsolutelyContinuous.rfl.smul _) _ _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ⊢ ∀ (x : α), x ∈ toMeasurable (ρ + μ) (u m) ∩ w n → ∃ᶠ (a : Set α) in filterAt v x, ↑↑(q • μ) a ≤ ↑↑ρ a intro x hx α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) x : α hx : x ∈ toMeasurable (ρ + μ) (u m) ∩ w n ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑(q • μ) a ≤ ↑↑ρ a have L : Tendsto (fun a : Set α => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatio ρ x)) := tendsto_nhds_limUnder hx.2.1.1 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) x : α hx : x ∈ toMeasurable (ρ + μ) (u m) ∩ w n L : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatio v ρ x)) ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑(q • μ) a ≤ ↑↑ρ a have I : ∀ᶠ b : Set α in v.filterAt x, (q : ℝ≥0∞) < ρ b / μ b := (tendsto_order.1 L).1 _ hx.2.1.2 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I✝ : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) x : α hx : x ∈ toMeasurable (ρ + μ) (u m) ∩ w n L : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatio v ρ x)) I : ∀ᶠ (b : Set α) in filterAt v x, ↑q < ↑↑ρ b / ↑↑μ b ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑(q • μ) a ≤ ↑↑ρ a apply I.frequently.mono fun a ha => ?_ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I✝ : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) x : α hx : x ∈ toMeasurable (ρ + μ) (u m) ∩ w n L : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatio v ρ x)) I : ∀ᶠ (b : Set α) in filterAt v x, ↑q < ↑↑ρ b / ↑↑μ b a : Set α ha : ↑q < ↑↑ρ a / ↑↑μ a ⊢ ↑↑(q • μ) a ≤ ↑↑ρ a rw [coe_nnreal_smul_apply] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I✝ : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) x : α hx : x ∈ toMeasurable (ρ + μ) (u m) ∩ w n L : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatio v ρ x)) I : ∀ᶠ (b : Set α) in filterAt v x, ↑q < ↑↑ρ b / ↑↑μ b a : Set α ha : ↑q < ↑↑ρ a / ↑↑μ a ⊢ ↑q * ↑↑μ a ≤ ↑↑ρ a exact ENNReal.mul_le_of_le_div ha.le α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ⊢ ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ w n) = ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) conv_rhs => rw [inter_comm] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ⊢ ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ w n) = ↑↑ρ (toMeasurable (ρ + μ) (w n) ∩ toMeasurable (ρ + μ) (u m)) rw [inter_comm] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ⊢ ↑↑ρ (w n ∩ toMeasurable (ρ + μ) (u m)) = ↑↑ρ (toMeasurable (ρ + μ) (w n) ∩ toMeasurable (ρ + μ) (u m)) exact (measure_toMeasurable_add_inter_left (measurableSet_toMeasurable _ _) J).symm α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) B : ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) h : ¬↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 ⊢ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) < ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) apply (ENNReal.mul_lt_mul_right h _).2 (ENNReal.coe_lt_coe.2 hpq) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) B : ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) h : ¬↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 ⊢ ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≠ ⊤ suffices H : (ρ + μ) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≠ ∞ case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) B : ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) h : ¬↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 ⊢ ↑↑(ρ + μ) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≠ ⊤ apply (lt_of_le_of_lt (measure_mono (inter_subset_left _ _)) _).ne α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) B : ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) h : ¬↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 ⊢ ↑↑(ρ + μ) (toMeasurable (ρ + μ) (u m)) < ⊤ rw [measure_toMeasurable] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) B : ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) h : ¬↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 ⊢ ↑↑(ρ + μ) (u m) < ⊤ apply lt_of_le_of_lt (measure_mono _) (measure_spanningSets_lt_top (ρ + μ) m) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) B : ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) h : ¬↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 ⊢ u m ⊆ spanningSets (ρ + μ) m exact inter_subset_right _ _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) B : ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) h : ¬↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 H : ↑↑(ρ + μ) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≠ ⊤ ⊢ ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≠ ⊤ simp only [not_or, ENNReal.add_eq_top, Pi.add_apply, Ne.def, coe_add] at H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) B : ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) h : ¬↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 H : ¬↑↑ρ (toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| limRatio v ρ x < ↑p} ∩ spanningSets (ρ + μ) m) ∩ toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| ↑q < limRatio v ρ x} ∩ spanningSets (ρ + μ) n)) = ⊤ ∧ ¬↑↑μ (toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| limRatio v ρ x < ↑p} ∩ spanningSets (ρ + μ) m) ∩ toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| ↑q < limRatio v ρ x} ∩ spanningSets (ρ + μ) n)) = ⊤ ⊢ ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≠ ⊤ exact H.2 Qed
VitaliFamily.ae_tendsto_limRatioMeas α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x)) filter_upwards [v.ae_tendsto_limRatio hρ, AEMeasurable.ae_eq_mk (v.aemeasurable_limRatio hρ)] case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ ⊢ ∀ (a : α), Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v a) (𝓝 (limRatio v ρ a)) → limRatio v ρ a = AEMeasurable.mk (limRatio v ρ) (_ : AEMeasurable (limRatio v ρ)) a → Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v a) (𝓝 (limRatioMeas v hρ a)) intro x hx h'x case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α hx : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatio v ρ x)) h'x : limRatio v ρ x = AEMeasurable.mk (limRatio v ρ) (_ : AEMeasurable (limRatio v ρ)) x ⊢ Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x)) rwa [h'x] at hx ** Qed
VitaliFamily.measure_le_mul_of_subset_limRatioMeas_lt ** α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p : ℝ≥0 s : Set α h : s ⊆ {x \| limRatioMeas v hρ x < ↑p} ⊢ ↑↑ρ s ≤ ↑p * ↑↑μ s let t := {x : α \| Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))} α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p : ℝ≥0 s : Set α h : s ⊆ {x \| limRatioMeas v hρ x < ↑p} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} ⊢ ↑↑ρ s ≤ ↑p * ↑↑μ s have A : μ tᶜ = 0 := v.ae_tendsto_limRatioMeas hρ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p : ℝ≥0 s : Set α h : s ⊆ {x \| limRatioMeas v hρ x < ↑p} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 ⊢ ↑↑ρ s ≤ ↑p * ↑↑μ s suffices H : ρ (s ∩ t) ≤ (p • μ) (s ∩ t) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p : ℝ≥0 s : Set α h : s ⊆ {x \| limRatioMeas v hρ x < ↑p} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 H : ↑↑ρ (s ∩ t) ≤ ↑↑(p • μ) (s ∩ t) ⊢ ↑↑ρ s ≤ ↑p * ↑↑μ s case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p : ℝ≥0 s : Set α h : s ⊆ {x \| limRatioMeas v hρ x < ↑p} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 ⊢ ↑↑ρ (s ∩ t) ≤ ↑↑(p • μ) (s ∩ t) ** exact calc ρ s = ρ (s ∩ t ∪ s ∩ tᶜ) := by rw [inter_union_compl] _ ≤ ρ (s ∩ t) + ρ (s ∩ tᶜ) := (measure_union_le _ _) _ ≤ p * μ (s ∩ t) + 0 := (add_le_add H ((measure_mono (inter_subset_right _ _)).trans (hρ A).le)) _ ≤ p * μ s := by rw [add_zero]; exact mul_le_mul_left' (measure_mono (inter_subset_left _ _)) _ case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p : ℝ≥0 s : Set α h : s ⊆ {x \| limRatioMeas v hρ x < ↑p} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 ⊢ ↑↑ρ (s ∩ t) ≤ ↑↑(p • μ) (s ∩ t) refine' v.measure_le_of_frequently_le (p • μ) hρ _ fun x hx => _ case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p : ℝ≥0 s : Set α h : s ⊆ {x \| limRatioMeas v hρ x < ↑p} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 x : α hx : x ∈ s ∩ t ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑(p • μ) a have I : ∀ᶠ b : Set α in v.filterAt x, ρ b / μ b < p := (tendsto_order.1 hx.2).2 _ (h hx.1) case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p : ℝ≥0 s : Set α h : s ⊆ {x \| limRatioMeas v hρ x < ↑p} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 x : α hx : x ∈ s ∩ t I : ∀ᶠ (b : Set α) in filterAt v x, ↑↑ρ b / ↑↑μ b < ↑p ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑(p • μ) a apply I.frequently.mono fun a ha => ?_ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p : ℝ≥0 s : Set α h : s ⊆ {x \| limRatioMeas v hρ x < ↑p} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 x : α hx : x ∈ s ∩ t I : ∀ᶠ (b : Set α) in filterAt v x, ↑↑ρ b / ↑↑μ b < ↑p a : Set α ha : ↑↑ρ a / ↑↑μ a < ↑p ⊢ ↑↑ρ a ≤ ↑↑(p • μ) a rw [coe_nnreal_smul_apply] ** α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p : ℝ≥0 s : Set α h : s ⊆ {x \| limRatioMeas v hρ x < ↑p} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 x : α hx : x ∈ s ∩ t I : ∀ᶠ (b : Set α) in filterAt v x, ↑↑ρ b / ↑↑μ b < ↑p a : Set α ha : ↑↑ρ a / ↑↑μ a < ↑p ⊢ ↑↑ρ a ≤ ↑p * ↑↑μ a refine' (ENNReal.div_le_iff_le_mul _ (Or.inr (bot_le.trans_lt ha).ne')).1 ha.le α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p : ℝ≥0 s : Set α h : s ⊆ {x \| limRatioMeas v hρ x < ↑p} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 x : α hx : x ∈ s ∩ t I : ∀ᶠ (b : Set α) in filterAt v x, ↑↑ρ b / ↑↑μ b < ↑p a : Set α ha : ↑↑ρ a / ↑↑μ a < ↑p ⊢ ↑↑μ a ≠ 0 ∨ ↑p ≠ ⊤ simp only [ENNReal.coe_ne_top, Ne.def, or_true_iff, not_false_iff] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p : ℝ≥0 s : Set α h : s ⊆ {x \| limRatioMeas v hρ x < ↑p} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 H : ↑↑ρ (s ∩ t) ≤ ↑↑(p • μ) (s ∩ t) ⊢ ↑↑ρ s = ↑↑ρ (s ∩ t ∪ s ∩ tᶜ) rw [inter_union_compl] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p : ℝ≥0 s : Set α h : s ⊆ {x \| limRatioMeas v hρ x < ↑p} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 H : ↑↑ρ (s ∩ t) ≤ ↑↑(p • μ) (s ∩ t) ⊢ ↑p * ↑↑μ (s ∩ t) + 0 ≤ ↑p * ↑↑μ s rw [add_zero] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p : ℝ≥0 s : Set α h : s ⊆ {x \| limRatioMeas v hρ x < ↑p} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 H : ↑↑ρ (s ∩ t) ≤ ↑↑(p • μ) (s ∩ t) ⊢ ↑p * ↑↑μ (s ∩ t) ≤ ↑p * ↑↑μ s exact mul_le_mul_left' (measure_mono (inter_subset_left _ _)) _ Qed
VitaliFamily.mul_measure_le_of_subset_lt_limRatioMeas ** α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} ⊢ ↑q * ↑↑μ s ≤ ↑↑ρ s let t := {x : α \| Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))} α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} ⊢ ↑q * ↑↑μ s ≤ ↑↑ρ s have A : μ tᶜ = 0 := v.ae_tendsto_limRatioMeas hρ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 ⊢ ↑q * ↑↑μ s ≤ ↑↑ρ s suffices H : (q • μ) (s ∩ t) ≤ ρ (s ∩ t) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 H : ↑↑(q • μ) (s ∩ t) ≤ ↑↑ρ (s ∩ t) ⊢ ↑q * ↑↑μ s ≤ ↑↑ρ s case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 ⊢ ↑↑(q • μ) (s ∩ t) ≤ ↑↑ρ (s ∩ t) ** exact calc (q • μ) s = (q • μ) (s ∩ t ∪ s ∩ tᶜ) := by rw [inter_union_compl] _ ≤ (q • μ) (s ∩ t) + (q • μ) (s ∩ tᶜ) := (measure_union_le _ _) _ ≤ ρ (s ∩ t) + q * μ tᶜ := by apply add_le_add H rw [coe_nnreal_smul_apply] exact mul_le_mul_left' (measure_mono (inter_subset_right _ _)) _ _ ≤ ρ s := by rw [A, mul_zero, add_zero]; exact measure_mono (inter_subset_left _ _) case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 ⊢ ↑↑(q • μ) (s ∩ t) ≤ ↑↑ρ (s ∩ t) refine' v.measure_le_of_frequently_le _ (AbsolutelyContinuous.rfl.smul _) _ _ case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 ⊢ ∀ (x : α), x ∈ s ∩ t → ∃ᶠ (a : Set α) in filterAt v x, ↑↑(q • μ) a ≤ ↑↑ρ a intro x hx case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 x : α hx : x ∈ s ∩ t ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑(q • μ) a ≤ ↑↑ρ a have I : ∀ᶠ a in v.filterAt x, (q : ℝ≥0∞) < ρ a / μ a := (tendsto_order.1 hx.2).1 _ (h hx.1) case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 x : α hx : x ∈ s ∩ t I : ∀ᶠ (a : Set α) in filterAt v x, ↑q < ↑↑ρ a / ↑↑μ a ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑(q • μ) a ≤ ↑↑ρ a apply I.frequently.mono fun a ha => ?_ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 x : α hx : x ∈ s ∩ t I : ∀ᶠ (a : Set α) in filterAt v x, ↑q < ↑↑ρ a / ↑↑μ a a : Set α ha : ↑q < ↑↑ρ a / ↑↑μ a ⊢ ↑↑(q • μ) a ≤ ↑↑ρ a rw [coe_nnreal_smul_apply] ** α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 x : α hx : x ∈ s ∩ t I : ∀ᶠ (a : Set α) in filterAt v x, ↑q < ↑↑ρ a / ↑↑μ a a : Set α ha : ↑q < ↑↑ρ a / ↑↑μ a ⊢ ↑q * ↑↑μ a ≤ ↑↑ρ a exact ENNReal.mul_le_of_le_div ha.le α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 H : ↑↑(q • μ) (s ∩ t) ≤ ↑↑ρ (s ∩ t) ⊢ ↑↑(q • μ) s = ↑↑(q • μ) (s ∩ t ∪ s ∩ tᶜ) rw [inter_union_compl] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 H : ↑↑(q • μ) (s ∩ t) ≤ ↑↑ρ (s ∩ t) ⊢ ↑↑(q • μ) (s ∩ t) + ↑↑(q • μ) (s ∩ tᶜ) ≤ ↑↑ρ (s ∩ t) + ↑q * ↑↑μ tᶜ apply add_le_add H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 H : ↑↑(q • μ) (s ∩ t) ≤ ↑↑ρ (s ∩ t) ⊢ ↑↑(q • μ) (s ∩ tᶜ) ≤ ↑q * ↑↑μ tᶜ rw [coe_nnreal_smul_apply] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 H : ↑↑(q • μ) (s ∩ t) ≤ ↑↑ρ (s ∩ t) ⊢ ↑q * ↑↑μ (s ∩ tᶜ) ≤ ↑q * ↑↑μ tᶜ exact mul_le_mul_left' (measure_mono (inter_subset_right _ _)) _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 H : ↑↑(q • μ) (s ∩ t) ≤ ↑↑ρ (s ∩ t) ⊢ ↑↑ρ (s ∩ t) + ↑q * ↑↑μ tᶜ ≤ ↑↑ρ s rw [A, mul_zero, add_zero] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 H : ↑↑(q • μ) (s ∩ t) ≤ ↑↑ρ (s ∩ t) ⊢ ↑↑ρ (s ∩ t) ≤ ↑↑ρ s exact measure_mono (inter_subset_left _ _) Qed
VitaliFamily.measure_limRatioMeas_top α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ ⊢ ↑↑μ {x \| limRatioMeas v hρ x = ⊤} = 0 refine' null_of_locally_null _ fun x _ => _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} ⊢ ∃ u, u ∈ 𝓝[{x \| limRatioMeas v hρ x = ⊤}] x ∧ ↑↑μ u = 0 obtain ⟨o, xo, o_open, μo⟩ : ∃ o : Set α, x ∈ o ∧ IsOpen o ∧ ρ o < ∞ := Measure.exists_isOpen_measure_lt_top ρ x case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ ⊢ ∃ u, u ∈ 𝓝[{x \| limRatioMeas v hρ x = ⊤}] x ∧ ↑↑μ u = 0 let s := {x : α \| v.limRatioMeas hρ x = ∞} ∩ o case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ⊢ ∃ u, u ∈ 𝓝[{x \| limRatioMeas v hρ x = ⊤}] x ∧ ↑↑μ u = 0 refine' ⟨s, inter_mem_nhdsWithin _ (o_open.mem_nhds xo), le_antisymm _ bot_le⟩ case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ⊢ ↑↑μ s ≤ 0 have ρs : ρ s ≠ ∞ := ((measure_mono (inter_subset_right _ _)).trans_lt μo).ne ** case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ρs : ↑↑ρ s ≠ ⊤ A : ∀ (q : ℝ≥0), 1 ≤ q → ↑↑μ s ≤ (↑q)⁻¹ * ↑↑ρ s ⊢ ↑↑μ s ≤ 0 ** have B : Tendsto (fun q : ℝ≥0 => (q : ℝ≥0∞)⁻¹ * ρ s) atTop (𝓝 (∞⁻¹ * ρ s)) := by apply ENNReal.Tendsto.mul_const _ (Or.inr ρs) exact ENNReal.tendsto_inv_iff.2 (ENNReal.tendsto_coe_nhds_top.2 tendsto_id) ** case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ρs : ↑↑ρ s ≠ ⊤ A : ∀ (q : ℝ≥0), 1 ≤ q → ↑↑μ s ≤ (↑q)⁻¹ * ↑↑ρ s B : Tendsto (fun q => (↑q)⁻¹ * ↑↑ρ s) atTop (𝓝 (⊤⁻¹ * ↑↑ρ s)) ⊢ ↑↑μ s ≤ 0 simp only [zero_mul, ENNReal.inv_top] at B case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ρs : ↑↑ρ s ≠ ⊤ A : ∀ (q : ℝ≥0), 1 ≤ q → ↑↑μ s ≤ (↑q)⁻¹ * ↑↑ρ s B : Tendsto (fun q => (↑q)⁻¹ * ↑↑ρ ({x \| limRatioMeas v hρ x = ⊤} ∩ o)) atTop (𝓝 0) ⊢ ↑↑μ s ≤ 0 apply ge_of_tendsto B case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ρs : ↑↑ρ s ≠ ⊤ A : ∀ (q : ℝ≥0), 1 ≤ q → ↑↑μ s ≤ (↑q)⁻¹ * ↑↑ρ s B : Tendsto (fun q => (↑q)⁻¹ * ↑↑ρ ({x \| limRatioMeas v hρ x = ⊤} ∩ o)) atTop (𝓝 0) ⊢ ∀ᶠ (c : ℝ≥0) in atTop, ↑↑μ s ≤ (↑c)⁻¹ * ↑↑ρ ({x \| limRatioMeas v hρ x = ⊤} ∩ o) exact eventually_atTop.2 ⟨1, A⟩ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ρs : ↑↑ρ s ≠ ⊤ ⊢ ∀ (q : ℝ≥0), 1 ≤ q → ↑↑μ s ≤ (↑q)⁻¹ * ↑↑ρ s intro q hq α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ρs : ↑↑ρ s ≠ ⊤ q : ℝ≥0 hq : 1 ≤ q ⊢ ↑↑μ s ≤ (↑q)⁻¹ * ↑↑ρ s rw [mul_comm, ← div_eq_mul_inv, ENNReal.le_div_iff_mul_le _ (Or.inr ρs), mul_comm] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ρs : ↑↑ρ s ≠ ⊤ q : ℝ≥0 hq : 1 ≤ q ⊢ ↑q * ↑↑μ s ≤ ↑↑ρ s apply v.mul_measure_le_of_subset_lt_limRatioMeas hρ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ρs : ↑↑ρ s ≠ ⊤ q : ℝ≥0 hq : 1 ≤ q ⊢ s ⊆ {x \| ↑q < limRatioMeas v hρ x} intro y hy α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ρs : ↑↑ρ s ≠ ⊤ q : ℝ≥0 hq : 1 ≤ q y : α hy : y ∈ s ⊢ y ∈ {x \| ↑q < limRatioMeas v hρ x} have : v.limRatioMeas hρ y = ∞ := hy.1 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ρs : ↑↑ρ s ≠ ⊤ q : ℝ≥0 hq : 1 ≤ q y : α hy : y ∈ s this : limRatioMeas v hρ y = ⊤ ⊢ y ∈ {x \| ↑q < limRatioMeas v hρ x} simp only [this, ENNReal.coe_lt_top, mem_setOf_eq] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ρs : ↑↑ρ s ≠ ⊤ q : ℝ≥0 hq : 1 ≤ q ⊢ ↑q ≠ 0 ∨ ↑↑ρ s ≠ 0 simp only [(zero_lt_one.trans_le hq).ne', true_or_iff, ENNReal.coe_eq_zero, Ne.def, not_false_iff] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ρs : ↑↑ρ s ≠ ⊤ A : ∀ (q : ℝ≥0), 1 ≤ q → ↑↑μ s ≤ (↑q)⁻¹ * ↑↑ρ s ⊢ Tendsto (fun q => (↑q)⁻¹ * ↑↑ρ s) atTop (𝓝 (⊤⁻¹ * ↑↑ρ s)) apply ENNReal.Tendsto.mul_const _ (Or.inr ρs) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ρs : ↑↑ρ s ≠ ⊤ A : ∀ (q : ℝ≥0), 1 ≤ q → ↑↑μ s ≤ (↑q)⁻¹ * ↑↑ρ s ⊢ Tendsto (fun x => (↑x)⁻¹) atTop (𝓝 ⊤⁻¹) exact ENNReal.tendsto_inv_iff.2 (ENNReal.tendsto_coe_nhds_top.2 tendsto_id) Qed
VitaliFamily.measure_limRatioMeas_zero α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ ⊢ ↑↑ρ {x \| limRatioMeas v hρ x = 0} = 0 refine' null_of_locally_null _ fun x _ => _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} ⊢ ∃ u, u ∈ 𝓝[{x \| limRatioMeas v hρ x = 0}] x ∧ ↑↑ρ u = 0 obtain ⟨o, xo, o_open, μo⟩ : ∃ o : Set α, x ∈ o ∧ IsOpen o ∧ μ o < ∞ := Measure.exists_isOpen_measure_lt_top μ x case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ ⊢ ∃ u, u ∈ 𝓝[{x \| limRatioMeas v hρ x = 0}] x ∧ ↑↑ρ u = 0 let s := {x : α \| v.limRatioMeas hρ x = 0} ∩ o case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o ⊢ ∃ u, u ∈ 𝓝[{x \| limRatioMeas v hρ x = 0}] x ∧ ↑↑ρ u = 0 refine' ⟨s, inter_mem_nhdsWithin _ (o_open.mem_nhds xo), le_antisymm _ bot_le⟩ case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o ⊢ ↑↑ρ s ≤ 0 have μs : μ s ≠ ∞ := ((measure_mono (inter_subset_right _ _)).trans_lt μo).ne case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o μs : ↑↑μ s ≠ ⊤ ⊢ ↑↑ρ s ≤ 0 have A : ∀ q : ℝ≥0, 0 < q → ρ s ≤ q * μ s := by intro q hq apply v.measure_le_mul_of_subset_limRatioMeas_lt hρ intro y hy have : v.limRatioMeas hρ y = 0 := hy.1 simp only [this, mem_setOf_eq, hq, ENNReal.coe_pos] ** case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o μs : ↑↑μ s ≠ ⊤ A : ∀ (q : ℝ≥0), 0 < q → ↑↑ρ s ≤ ↑q * ↑↑μ s ⊢ ↑↑ρ s ≤ 0 ** have B : Tendsto (fun q : ℝ≥0 => (q : ℝ≥0∞) * μ s) (𝓝[>] (0 : ℝ≥0)) (𝓝 ((0 : ℝ≥0) * μ s)) := by apply ENNReal.Tendsto.mul_const _ (Or.inr μs) rw [ENNReal.tendsto_coe] exact nhdsWithin_le_nhds ** case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o μs : ↑↑μ s ≠ ⊤ A : ∀ (q : ℝ≥0), 0 < q → ↑↑ρ s ≤ ↑q * ↑↑μ s B : Tendsto (fun q => ↑q * ↑↑μ s) (𝓝[Ioi 0] 0) (𝓝 (↑0 * ↑↑μ s)) ⊢ ↑↑ρ s ≤ 0 simp only [zero_mul, ENNReal.coe_zero] at B case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o μs : ↑↑μ s ≠ ⊤ A : ∀ (q : ℝ≥0), 0 < q → ↑↑ρ s ≤ ↑q * ↑↑μ s B : Tendsto (fun q => ↑q * ↑↑μ ({x \| limRatioMeas v hρ x = 0} ∩ o)) (𝓝[Ioi 0] 0) (𝓝 0) ⊢ ↑↑ρ s ≤ 0 apply ge_of_tendsto B case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o μs : ↑↑μ s ≠ ⊤ A : ∀ (q : ℝ≥0), 0 < q → ↑↑ρ s ≤ ↑q * ↑↑μ s B : Tendsto (fun q => ↑q * ↑↑μ ({x \| limRatioMeas v hρ x = 0} ∩ o)) (𝓝[Ioi 0] 0) (𝓝 0) ⊢ ∀ᶠ (c : ℝ≥0) in 𝓝[Ioi 0] 0, ↑↑ρ s ≤ ↑c * ↑↑μ ({x \| limRatioMeas v hρ x = 0} ∩ o) filter_upwards [self_mem_nhdsWithin] using A α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o μs : ↑↑μ s ≠ ⊤ ⊢ ∀ (q : ℝ≥0), 0 < q → ↑↑ρ s ≤ ↑q * ↑↑μ s intro q hq α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o μs : ↑↑μ s ≠ ⊤ q : ℝ≥0 hq : 0 < q ⊢ ↑↑ρ s ≤ ↑q * ↑↑μ s apply v.measure_le_mul_of_subset_limRatioMeas_lt hρ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o μs : ↑↑μ s ≠ ⊤ q : ℝ≥0 hq : 0 < q ⊢ s ⊆ {x \| limRatioMeas v hρ x < ↑q} intro y hy α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o μs : ↑↑μ s ≠ ⊤ q : ℝ≥0 hq : 0 < q y : α hy : y ∈ s ⊢ y ∈ {x \| limRatioMeas v hρ x < ↑q} have : v.limRatioMeas hρ y = 0 := hy.1 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o μs : ↑↑μ s ≠ ⊤ q : ℝ≥0 hq : 0 < q y : α hy : y ∈ s this : limRatioMeas v hρ y = 0 ⊢ y ∈ {x \| limRatioMeas v hρ x < ↑q} simp only [this, mem_setOf_eq, hq, ENNReal.coe_pos] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o μs : ↑↑μ s ≠ ⊤ A : ∀ (q : ℝ≥0), 0 < q → ↑↑ρ s ≤ ↑q * ↑↑μ s ⊢ Tendsto (fun q => ↑q * ↑↑μ s) (𝓝[Ioi 0] 0) (𝓝 (↑0 * ↑↑μ s)) apply ENNReal.Tendsto.mul_const _ (Or.inr μs) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o μs : ↑↑μ s ≠ ⊤ A : ∀ (q : ℝ≥0), 0 < q → ↑↑ρ s ≤ ↑q * ↑↑μ s ⊢ Tendsto (fun x => ↑x) (𝓝[Ioi 0] 0) (𝓝 ↑0) rw [ENNReal.tendsto_coe] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o μs : ↑↑μ s ≠ ⊤ A : ∀ (q : ℝ≥0), 0 < q → ↑↑ρ s ≤ ↑q * ↑↑μ s ⊢ Tendsto (fun x => x) (𝓝[Ioi 0] 0) (𝓝 0) exact nhdsWithin_le_nhds Qed
VitaliFamily.le_mul_withDensity ** α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t ⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s have t_ne_zero' : t ≠ 0 := (zero_lt_one.trans ht).ne' α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 ⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s have t_ne_zero : (t : ℝ≥0∞) ≠ 0 := by simpa only [ENNReal.coe_eq_zero, Ne.def] using t_ne_zero' α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s let ν := μ.withDensity (v.limRatioMeas hρ) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) ⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s let f := v.limRatioMeas hρ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ ⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s have f_meas : Measurable f := v.limRatioMeas_measurable hρ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f ⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s have A : ρ (s ∩ f ⁻¹' {0}) ≤ (t • ν) (s ∩ f ⁻¹' {0}) := by refine' le_trans (measure_mono (inter_subset_right _ _)) (le_trans (le_of_eq _) (zero_le _)) exact v.measure_limRatioMeas_zero hρ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) ⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s have B : ρ (s ∩ f ⁻¹' {∞}) ≤ (t • ν) (s ∩ f ⁻¹' {∞}) := by apply le_trans (le_of_eq _) (zero_le _) apply hρ rw [← nonpos_iff_eq_zero] exact (measure_mono (inter_subset_right _ _)).trans (v.measure_limRatioMeas_top hρ).le α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) ⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s ** have C : ∀ n : ℤ, ρ (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) ≤ (t • ν) (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) := by intro n let I := Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1)) have M : MeasurableSet (s ∩ f ⁻¹' I) := hs.inter (f_meas measurableSet_Ico) simp only [M, withDensity_apply, coe_nnreal_smul_apply] calc ρ (s ∩ f ⁻¹' I) ≤ (t : ℝ≥0∞) ^ (n + 1) * μ (s ∩ f ⁻¹' I) := by rw [← ENNReal.coe_zpow t_ne_zero'] apply v.measure_le_mul_of_subset_limRatioMeas_lt hρ intro x hx apply hx.2.2.trans_le (le_of_eq _) rw [ENNReal.coe_zpow t_ne_zero'] _ = ∫⁻ x in s ∩ f ⁻¹' I, (t : ℝ≥0∞) ^ (n + 1) ∂μ := by simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter] _ ≤ ∫⁻ x in s ∩ f ⁻¹' I, t * f x ∂μ := by apply lintegral_mono_ae ((ae_restrict_iff' M).2 (eventually_of_forall fun x hx => ?_)) rw [add_comm, ENNReal.zpow_add t_ne_zero ENNReal.coe_ne_top, zpow_one] exact mul_le_mul_left' hx.2.1 _ _ = t * ∫⁻ x in s ∩ f ⁻¹' I, f x ∂μ := lintegral_const_mul _ f_meas ** α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) C : ∀ (n : ℤ), ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s calc ρ s = ρ (s ∩ f ⁻¹' {0}) + ρ (s ∩ f ⁻¹' {∞}) + ∑' n : ℤ, ρ (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) := measure_eq_measure_preimage_add_measure_tsum_Ico_zpow ρ f_meas hs ht _ ≤ (t • ν) (s ∩ f ⁻¹' {0}) + (t • ν) (s ∩ f ⁻¹' {∞}) + ∑' n : ℤ, (t • ν) (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) := (add_le_add (add_le_add A B) (ENNReal.tsum_le_tsum C)) _ = (t • ν) s := (measure_eq_measure_preimage_add_measure_tsum_Ico_zpow (t • ν) f_meas hs ht).symm α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 ⊢ ↑t ≠ 0 simpa only [ENNReal.coe_eq_zero, Ne.def] using t_ne_zero' α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f ⊢ ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) refine' le_trans (measure_mono (inter_subset_right _ _)) (le_trans (le_of_eq _) (zero_le _)) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f ⊢ ↑↑ρ (f ⁻¹' {0}) = 0 exact v.measure_limRatioMeas_zero hρ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) ⊢ ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) apply le_trans (le_of_eq _) (zero_le _) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) ⊢ ↑↑ρ (s ∩ f ⁻¹' {⊤}) = 0 apply hρ case a α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) ⊢ ↑↑μ (s ∩ f ⁻¹' {⊤}) = 0 rw [← nonpos_iff_eq_zero] case a α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) ⊢ ↑↑μ (s ∩ f ⁻¹' {⊤}) ≤ 0 exact (measure_mono (inter_subset_right _ _)).trans (v.measure_limRatioMeas_top hρ).le α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) ⊢ ∀ (n : ℤ), ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) intro n α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) n : ℤ ⊢ ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) let I := Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1)) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) n : ℤ I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1)) ⊢ ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) have M : MeasurableSet (s ∩ f ⁻¹' I) := hs.inter (f_meas measurableSet_Ico) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) n : ℤ I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1)) M : MeasurableSet (s ∩ f ⁻¹' I) ⊢ ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) simp only [M, withDensity_apply, coe_nnreal_smul_apply] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) n : ℤ I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1)) M : MeasurableSet (s ∩ f ⁻¹' I) ⊢ ↑↑ρ (s ∩ limRatioMeas v hρ ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑t * ∫⁻ (a : α) in s ∩ limRatioMeas v hρ ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)), limRatioMeas v hρ a ∂μ ** calc ρ (s ∩ f ⁻¹' I) ≤ (t : ℝ≥0∞) ^ (n + 1) * μ (s ∩ f ⁻¹' I) := by rw [← ENNReal.coe_zpow t_ne_zero'] apply v.measure_le_mul_of_subset_limRatioMeas_lt hρ intro x hx apply hx.2.2.trans_le (le_of_eq _) rw [ENNReal.coe_zpow t_ne_zero'] _ = ∫⁻ x in s ∩ f ⁻¹' I, (t : ℝ≥0∞) ^ (n + 1) ∂μ := by simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter] _ ≤ ∫⁻ x in s ∩ f ⁻¹' I, t * f x ∂μ := by apply lintegral_mono_ae ((ae_restrict_iff' M).2 (eventually_of_forall fun x hx => ?_)) rw [add_comm, ENNReal.zpow_add t_ne_zero ENNReal.coe_ne_top, zpow_one] exact mul_le_mul_left' hx.2.1 _ _ = t * ∫⁻ x in s ∩ f ⁻¹' I, f x ∂μ := lintegral_const_mul _ f_meas ** α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) n : ℤ I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1)) M : MeasurableSet (s ∩ f ⁻¹' I) ⊢ ↑↑ρ (s ∩ f ⁻¹' I) ≤ ↑t ^ (n + 1) * ↑↑μ (s ∩ f ⁻¹' I) rw [← ENNReal.coe_zpow t_ne_zero'] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) n : ℤ I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1)) M : MeasurableSet (s ∩ f ⁻¹' I) ⊢ ↑↑ρ (s ∩ f ⁻¹' I) ≤ ↑(t ^ (n + 1)) * ↑↑μ (s ∩ f ⁻¹' I) apply v.measure_le_mul_of_subset_limRatioMeas_lt hρ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) n : ℤ I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1)) M : MeasurableSet (s ∩ f ⁻¹' I) ⊢ s ∩ f ⁻¹' I ⊆ {x \| limRatioMeas v hρ x < ↑(t ^ (n + 1))} intro x hx α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) n : ℤ I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1)) M : MeasurableSet (s ∩ f ⁻¹' I) x : α hx : x ∈ s ∩ f ⁻¹' I ⊢ x ∈ {x \| limRatioMeas v hρ x < ↑(t ^ (n + 1))} apply hx.2.2.trans_le (le_of_eq _) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) n : ℤ I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1)) M : MeasurableSet (s ∩ f ⁻¹' I) x : α hx : x ∈ s ∩ f ⁻¹' I ⊢ ↑t ^ (n + 1) = ↑(t ^ (n + 1)) rw [ENNReal.coe_zpow t_ne_zero'] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) n : ℤ I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1)) M : MeasurableSet (s ∩ f ⁻¹' I) ⊢ ↑t ^ (n + 1) * ↑↑μ (s ∩ f ⁻¹' I) = ∫⁻ (x : α) in s ∩ f ⁻¹' I, ↑t ^ (n + 1) ∂μ simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) n : ℤ I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1)) M : MeasurableSet (s ∩ f ⁻¹' I) ⊢ ∫⁻ (x : α) in s ∩ f ⁻¹' I, ↑t ^ (n + 1) ∂μ ≤ ∫⁻ (x : α) in s ∩ f ⁻¹' I, ↑t * f x ∂μ apply lintegral_mono_ae ((ae_restrict_iff' M).2 (eventually_of_forall fun x hx => ?_)) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) n : ℤ I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1)) M : MeasurableSet (s ∩ f ⁻¹' I) x : α hx : x ∈ s ∩ f ⁻¹' I ⊢ ↑t ^ (n + 1) ≤ ↑t * f x rw [add_comm, ENNReal.zpow_add t_ne_zero ENNReal.coe_ne_top, zpow_one] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) n : ℤ I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1)) M : MeasurableSet (s ∩ f ⁻¹' I) x : α hx : x ∈ s ∩ f ⁻¹' I ⊢ ↑t * ↑t ^ n ≤ ↑t * f x exact mul_le_mul_left' hx.2.1 _ Qed
VitaliFamily.ae_tendsto_rnDeriv_of_absolutelyContinuous α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv ρ μ x)) have A : (μ.withDensity (v.limRatioMeas hρ)).rnDeriv μ =ᵐ[μ] v.limRatioMeas hρ := rnDeriv_withDensity μ (v.limRatioMeas_measurable hρ) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ A : rnDeriv (withDensity μ (limRatioMeas v hρ)) μ =ᶠ[ae μ] limRatioMeas v hρ ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv ρ μ x)) rw [v.withDensity_limRatioMeas_eq hρ] at A α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ A : rnDeriv ρ μ =ᶠ[ae μ] limRatioMeas v hρ ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv ρ μ x)) filter_upwards [v.ae_tendsto_limRatioMeas hρ, A] with _ _ h'x case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ A : rnDeriv ρ μ =ᶠ[ae μ] limRatioMeas v hρ a✝¹ : α a✝ : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v a✝¹) (𝓝 (limRatioMeas v hρ a✝¹)) h'x : rnDeriv ρ μ a✝¹ = limRatioMeas v hρ a✝¹ ⊢ Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v a✝¹) (𝓝 (rnDeriv ρ μ a✝¹)) rwa [h'x] ** Qed
VitaliFamily.ae_tendsto_rnDeriv α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv ρ μ x)) let t := μ.withDensity (ρ.rnDeriv μ) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ t : Measure α := withDensity μ (rnDeriv ρ μ) ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv ρ μ x)) have eq_add : ρ = ρ.singularPart μ + t := haveLebesgueDecomposition_add _ _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ t : Measure α := withDensity μ (rnDeriv ρ μ) eq_add : ρ = singularPart ρ μ + t ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv ρ μ x)) have A : ∀ᵐ x ∂μ, Tendsto (fun a => ρ.singularPart μ a / μ a) (v.filterAt x) (𝓝 0) := v.ae_eventually_measure_zero_of_singular (mutuallySingular_singularPart ρ μ) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ t : Measure α := withDensity μ (rnDeriv ρ μ) eq_add : ρ = singularPart ρ μ + t A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v x) (𝓝 0) ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv ρ μ x)) have B : ∀ᵐ x ∂μ, t.rnDeriv μ x = ρ.rnDeriv μ x := rnDeriv_withDensity μ (measurable_rnDeriv ρ μ) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ t : Measure α := withDensity μ (rnDeriv ρ μ) eq_add : ρ = singularPart ρ μ + t A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v x) (𝓝 0) B : ∀ᵐ (x : α) ∂μ, rnDeriv t μ x = rnDeriv ρ μ x ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv ρ μ x)) have C : ∀ᵐ x ∂μ, Tendsto (fun a => t a / μ a) (v.filterAt x) (𝓝 (t.rnDeriv μ x)) := v.ae_tendsto_rnDeriv_of_absolutelyContinuous (withDensity_absolutelyContinuous _ _) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ t : Measure α := withDensity μ (rnDeriv ρ μ) eq_add : ρ = singularPart ρ μ + t A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v x) (𝓝 0) B : ∀ᵐ (x : α) ∂μ, rnDeriv t μ x = rnDeriv ρ μ x C : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv t μ x)) ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv ρ μ x)) filter_upwards [A, B, C] with _ Ax Bx Cx case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ t : Measure α := withDensity μ (rnDeriv ρ μ) eq_add : ρ = singularPart ρ μ + t A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v x) (𝓝 0) B : ∀ᵐ (x : α) ∂μ, rnDeriv t μ x = rnDeriv ρ μ x C : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv t μ x)) a✝ : α Ax : Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v a✝) (𝓝 0) Bx : rnDeriv t μ a✝ = rnDeriv ρ μ a✝ Cx : Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v a✝) (𝓝 (rnDeriv t μ a✝)) ⊢ Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v a✝) (𝓝 (rnDeriv ρ μ a✝)) convert Ax.add Cx using 1 case h.e'_3 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ t : Measure α := withDensity μ (rnDeriv ρ μ) eq_add : ρ = singularPart ρ μ + t A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v x) (𝓝 0) B : ∀ᵐ (x : α) ∂μ, rnDeriv t μ x = rnDeriv ρ μ x C : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv t μ x)) a✝ : α Ax : Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v a✝) (𝓝 0) Bx : rnDeriv t μ a✝ = rnDeriv ρ μ a✝ Cx : Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v a✝) (𝓝 (rnDeriv t μ a✝)) ⊢ (fun a => ↑↑ρ a / ↑↑μ a) = fun x => ↑↑(singularPart ρ μ) x / ↑↑μ x + ↑↑t x / ↑↑μ x ext1 a case h.e'_3.h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ t : Measure α := withDensity μ (rnDeriv ρ μ) eq_add : ρ = singularPart ρ μ + t A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v x) (𝓝 0) B : ∀ᵐ (x : α) ∂μ, rnDeriv t μ x = rnDeriv ρ μ x C : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv t μ x)) a✝ : α Ax : Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v a✝) (𝓝 0) Bx : rnDeriv t μ a✝ = rnDeriv ρ μ a✝ Cx : Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v a✝) (𝓝 (rnDeriv t μ a✝)) a : Set α ⊢ ↑↑ρ a / ↑↑μ a = ↑↑(singularPart ρ μ) a / ↑↑μ a + ↑↑t a / ↑↑μ a conv_lhs => rw [eq_add] case h.e'_3.h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ t : Measure α := withDensity μ (rnDeriv ρ μ) eq_add : ρ = singularPart ρ μ + t A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v x) (𝓝 0) B : ∀ᵐ (x : α) ∂μ, rnDeriv t μ x = rnDeriv ρ μ x C : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv t μ x)) a✝ : α Ax : Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v a✝) (𝓝 0) Bx : rnDeriv t μ a✝ = rnDeriv ρ μ a✝ Cx : Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v a✝) (𝓝 (rnDeriv t μ a✝)) a : Set α ⊢ ↑↑(singularPart ρ μ + t) a / ↑↑μ a = ↑↑(singularPart ρ μ) a / ↑↑μ a + ↑↑t a / ↑↑μ a simp only [Pi.add_apply, coe_add, ENNReal.add_div] case h.e'_5 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ t : Measure α := withDensity μ (rnDeriv ρ μ) eq_add : ρ = singularPart ρ μ + t A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v x) (𝓝 0) B : ∀ᵐ (x : α) ∂μ, rnDeriv t μ x = rnDeriv ρ μ x C : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv t μ x)) a✝ : α Ax : Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v a✝) (𝓝 0) Bx : rnDeriv t μ a✝ = rnDeriv ρ μ a✝ Cx : Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v a✝) (𝓝 (rnDeriv t μ a✝)) ⊢ 𝓝 (rnDeriv ρ μ a✝) = 𝓝 (0 + rnDeriv t μ a✝) simp only [Bx, zero_add] ** Qed
VitaliFamily.ae_tendsto_measure_inter_div α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α ⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1) let t := toMeasurable μ s α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α t : Set α := toMeasurable μ s ⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1) have A : ∀ᵐ x ∂μ.restrict s, Tendsto (fun a => μ (t ∩ a) / μ a) (v.filterAt x) (𝓝 (t.indicator 1 x)) := by apply ae_mono restrict_le_self apply ae_tendsto_measure_inter_div_of_measurableSet exact measurableSet_toMeasurable _ _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α t : Set α := toMeasurable μ s A : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x)) ⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1) have B : ∀ᵐ x ∂μ.restrict s, t.indicator 1 x = (1 : ℝ≥0∞) := by refine' ae_restrict_of_ae_restrict_of_subset (subset_toMeasurable μ s) _ filter_upwards [ae_restrict_mem (measurableSet_toMeasurable μ s)] with _ hx simp only [hx, Pi.one_apply, indicator_of_mem] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α t : Set α := toMeasurable μ s A : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x)) B : ∀ᵐ (x : α) ∂Measure.restrict μ s, indicator t 1 x = 1 ⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1) filter_upwards [A, B] with x hx h'x case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α t : Set α := toMeasurable μ s A : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x)) B : ∀ᵐ (x : α) ∂Measure.restrict μ s, indicator t 1 x = 1 x : α hx : Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x)) h'x : indicator t 1 x = 1 ⊢ Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1) rw [h'x] at hx case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α t : Set α := toMeasurable μ s A : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x)) B : ∀ᵐ (x : α) ∂Measure.restrict μ s, indicator t 1 x = 1 x : α hx : Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1) h'x : indicator t 1 x = 1 ⊢ Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1) apply hx.congr' _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α t : Set α := toMeasurable μ s A : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x)) B : ∀ᵐ (x : α) ∂Measure.restrict μ s, indicator t 1 x = 1 x : α hx : Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1) h'x : indicator t 1 x = 1 ⊢ (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) =ᶠ[filterAt v x] fun a => ↑↑μ (s ∩ a) / ↑↑μ a filter_upwards [v.eventually_filterAt_measurableSet x] with _ ha case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α t : Set α := toMeasurable μ s A : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x)) B : ∀ᵐ (x : α) ∂Measure.restrict μ s, indicator t 1 x = 1 x : α hx : Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1) h'x : indicator t 1 x = 1 a✝ : Set α ha : MeasurableSet a✝ ⊢ ↑↑μ (t ∩ a✝) / ↑↑μ a✝ = ↑↑μ (s ∩ a✝) / ↑↑μ a✝ congr 1 case h.e_a α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α t : Set α := toMeasurable μ s A : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x)) B : ∀ᵐ (x : α) ∂Measure.restrict μ s, indicator t 1 x = 1 x : α hx : Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1) h'x : indicator t 1 x = 1 a✝ : Set α ha : MeasurableSet a✝ ⊢ ↑↑μ (t ∩ a✝) = ↑↑μ (s ∩ a✝) exact measure_toMeasurable_inter_of_sigmaFinite ha _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α t : Set α := toMeasurable μ s ⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x)) apply ae_mono restrict_le_self case a α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α t : Set α := toMeasurable μ s ⊢ {x \| (fun x => Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x))) x} ∈ ae μ apply ae_tendsto_measure_inter_div_of_measurableSet case a.hs α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α t : Set α := toMeasurable μ s ⊢ MeasurableSet t exact measurableSet_toMeasurable _ _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α t : Set α := toMeasurable μ s A : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x)) ⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, indicator t 1 x = 1 refine' ae_restrict_of_ae_restrict_of_subset (subset_toMeasurable μ s) _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α t : Set α := toMeasurable μ s A : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x)) ⊢ ∀ᵐ (x : α) ∂Measure.restrict μ (toMeasurable μ s), indicator t 1 x = 1 filter_upwards [ae_restrict_mem (measurableSet_toMeasurable μ s)] with _ hx case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α t : Set α := toMeasurable μ s A : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x)) a✝ : α hx : a✝ ∈ toMeasurable μ s ⊢ indicator t 1 a✝ = 1 simp only [hx, Pi.one_apply, indicator_of_mem] ** Qed
VitaliFamily.ae_tendsto_lintegral_div' α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → ℝ≥0∞ hf : Measurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => (∫⁻ (y : α) in a, f y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (f x)) let ρ := μ.withDensity f α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ✝ : Measure α inst✝ : IsLocallyFiniteMeasure ρ✝ f : α → ℝ≥0∞ hf : Measurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ ρ : Measure α := withDensity μ f ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => (∫⁻ (y : α) in a, f y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (f x)) have : IsFiniteMeasure ρ := isFiniteMeasure_withDensity h'f α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ✝ : Measure α inst✝ : IsLocallyFiniteMeasure ρ✝ f : α → ℝ≥0∞ hf : Measurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ ρ : Measure α := withDensity μ f this : IsFiniteMeasure ρ ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => (∫⁻ (y : α) in a, f y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (f x)) filter_upwards [ae_tendsto_rnDeriv v ρ, rnDeriv_withDensity μ hf] with x hx h'x case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ✝ : Measure α inst✝ : IsLocallyFiniteMeasure ρ✝ f : α → ℝ≥0∞ hf : Measurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ ρ : Measure α := withDensity μ f this : IsFiniteMeasure ρ x : α hx : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv ρ μ x)) h'x : rnDeriv (withDensity μ f) μ x = f x ⊢ Tendsto (fun a => (∫⁻ (y : α) in a, f y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (f x)) rw [← h'x] case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ✝ : Measure α inst✝ : IsLocallyFiniteMeasure ρ✝ f : α → ℝ≥0∞ hf : Measurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ ρ : Measure α := withDensity μ f this : IsFiniteMeasure ρ x : α hx : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv ρ μ x)) h'x : rnDeriv (withDensity μ f) μ x = f x ⊢ Tendsto (fun a => (∫⁻ (y : α) in a, f y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv (withDensity μ f) μ x)) apply hx.congr' _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ✝ : Measure α inst✝ : IsLocallyFiniteMeasure ρ✝ f : α → ℝ≥0∞ hf : Measurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ ρ : Measure α := withDensity μ f this : IsFiniteMeasure ρ x : α hx : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv ρ μ x)) h'x : rnDeriv (withDensity μ f) μ x = f x ⊢ (fun a => ↑↑ρ a / ↑↑μ a) =ᶠ[filterAt v x] fun a => (∫⁻ (y : α) in a, f y ∂μ) / ↑↑μ a filter_upwards [v.eventually_filterAt_measurableSet x] with a ha case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ✝ : Measure α inst✝ : IsLocallyFiniteMeasure ρ✝ f : α → ℝ≥0∞ hf : Measurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ ρ : Measure α := withDensity μ f this : IsFiniteMeasure ρ x : α hx : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv ρ μ x)) h'x : rnDeriv (withDensity μ f) μ x = f x a : Set α ha : MeasurableSet a ⊢ ↑↑ρ a / ↑↑μ a = (∫⁻ (y : α) in a, f y ∂μ) / ↑↑μ a rw [← withDensity_apply f ha] ** Qed
VitaliFamily.ae_tendsto_lintegral_div α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => (∫⁻ (y : α) in a, f y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (f x)) have A : (∫⁻ y, hf.mk f y ∂μ) ≠ ∞ := by convert h'f using 1 apply lintegral_congr_ae exact hf.ae_eq_mk.symm α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ A : ∫⁻ (y : α), AEMeasurable.mk f hf y ∂μ ≠ ⊤ ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => (∫⁻ (y : α) in a, f y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (f x)) filter_upwards [v.ae_tendsto_lintegral_div' hf.measurable_mk A, hf.ae_eq_mk] with x hx h'x case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ A : ∫⁻ (y : α), AEMeasurable.mk f hf y ∂μ ≠ ⊤ x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, AEMeasurable.mk f hf y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (AEMeasurable.mk f hf x)) h'x : f x = AEMeasurable.mk f hf x ⊢ Tendsto (fun a => (∫⁻ (y : α) in a, f y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (f x)) rw [h'x] case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ A : ∫⁻ (y : α), AEMeasurable.mk f hf y ∂μ ≠ ⊤ x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, AEMeasurable.mk f hf y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (AEMeasurable.mk f hf x)) h'x : f x = AEMeasurable.mk f hf x ⊢ Tendsto (fun a => (∫⁻ (y : α) in a, f y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (AEMeasurable.mk f hf x)) convert hx using 1 case h.e'_3 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ A : ∫⁻ (y : α), AEMeasurable.mk f hf y ∂μ ≠ ⊤ x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, AEMeasurable.mk f hf y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (AEMeasurable.mk f hf x)) h'x : f x = AEMeasurable.mk f hf x ⊢ (fun a => (∫⁻ (y : α) in a, f y ∂μ) / ↑↑μ a) = fun a => (∫⁻ (y : α) in a, AEMeasurable.mk f hf y ∂μ) / ↑↑μ a ext1 a case h.e'_3.h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ A : ∫⁻ (y : α), AEMeasurable.mk f hf y ∂μ ≠ ⊤ x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, AEMeasurable.mk f hf y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (AEMeasurable.mk f hf x)) h'x : f x = AEMeasurable.mk f hf x a : Set α ⊢ (∫⁻ (y : α) in a, f y ∂μ) / ↑↑μ a = (∫⁻ (y : α) in a, AEMeasurable.mk f hf y ∂μ) / ↑↑μ a congr 1 case h.e'_3.h.e_a α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ A : ∫⁻ (y : α), AEMeasurable.mk f hf y ∂μ ≠ ⊤ x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, AEMeasurable.mk f hf y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (AEMeasurable.mk f hf x)) h'x : f x = AEMeasurable.mk f hf x a : Set α ⊢ ∫⁻ (y : α) in a, f y ∂μ = ∫⁻ (y : α) in a, AEMeasurable.mk f hf y ∂μ apply lintegral_congr_ae case h.e'_3.h.e_a.h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ A : ∫⁻ (y : α), AEMeasurable.mk f hf y ∂μ ≠ ⊤ x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, AEMeasurable.mk f hf y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (AEMeasurable.mk f hf x)) h'x : f x = AEMeasurable.mk f hf x a : Set α ⊢ (fun a => f a) =ᶠ[ae (Measure.restrict μ a)] fun a => AEMeasurable.mk f hf a exact ae_restrict_of_ae hf.ae_eq_mk α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ ⊢ ∫⁻ (y : α), AEMeasurable.mk f hf y ∂μ ≠ ⊤ convert h'f using 1 case h.e'_2 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ ⊢ ∫⁻ (y : α), AEMeasurable.mk f hf y ∂μ = ∫⁻ (y : α), f y ∂μ apply lintegral_congr_ae case h.e'_2.h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ ⊢ (fun a => AEMeasurable.mk f hf a) =ᶠ[ae μ] fun a => f a exact hf.ae_eq_mk.symm ** Qed
VitaliFamily.ae_tendsto_lintegral_nnnorm_sub_div_of_integrable α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : Integrable f ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) have I : Integrable (hf.1.mk f) μ := hf.congr hf.1.ae_eq_mk α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : Integrable f I : Integrable (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)) ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) filter_upwards [v.ae_tendsto_lintegral_nnnorm_sub_div'_of_integrable I hf.1.stronglyMeasurable_mk, hf.1.ae_eq_mk] with x hx h'x case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : Integrable f I : Integrable (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)) x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y - AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) h'x : f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x ⊢ Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) apply hx.congr _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : Integrable f I : Integrable (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)) x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y - AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) h'x : f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x ⊢ ∀ (x_1 : Set α), (∫⁻ (y : α) in x_1, ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y - AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ) / ↑↑μ x_1 = (∫⁻ (y : α) in x_1, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ x_1 intro a α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : Integrable f I : Integrable (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)) x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y - AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) h'x : f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x a : Set α ⊢ (∫⁻ (y : α) in a, ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y - AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ) / ↑↑μ a = (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a congr 1 case e_a α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : Integrable f I : Integrable (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)) x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y - AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) h'x : f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x a : Set α ⊢ ∫⁻ (y : α) in a, ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y - AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ = ∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ apply lintegral_congr_ae case e_a.h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : Integrable f I : Integrable (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)) x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y - AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) h'x : f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x a : Set α ⊢ (fun a => ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) a - AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊) =ᶠ[ae (Measure.restrict μ a)] fun a => ↑‖f a - f x‖₊ apply ae_restrict_of_ae case e_a.h.h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : Integrable f I : Integrable (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)) x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y - AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) h'x : f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x a : Set α ⊢ ∀ᵐ (x_1 : α) ∂μ, (fun a => ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) a - AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊) x_1 = (fun a => ↑‖f a - f x‖₊) x_1 filter_upwards [hf.1.ae_eq_mk] with y hy case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : Integrable f I : Integrable (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)) x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y - AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) h'x : f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x a : Set α y : α hy : f y = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y ⊢ ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y - AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ = ↑‖f y - f x‖₊ rw [hy, h'x] ** Qed
VitaliFamily.eventually_filterAt_integrableOn α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ x : α f : α → E hf : LocallyIntegrable f ⊢ ∀ᶠ (a : Set α) in filterAt v x, IntegrableOn f a rcases hf x with ⟨w, w_nhds, hw⟩ case intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ x : α f : α → E hf : LocallyIntegrable f w : Set α w_nhds : w ∈ 𝓝 x hw : IntegrableOn f w ⊢ ∀ᶠ (a : Set α) in filterAt v x, IntegrableOn f a filter_upwards [v.eventually_filterAt_subset_of_nhds w_nhds] with a ha case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ x : α f : α → E hf : LocallyIntegrable f w : Set α w_nhds : w ∈ 𝓝 x hw : IntegrableOn f w a : Set α ha : a ⊆ w ⊢ IntegrableOn f a exact hw.mono_set ha ** Qed
VitaliFamily.ae_tendsto_average_norm_sub α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0) filter_upwards [v.ae_tendsto_lintegral_nnnorm_sub_div hf] with x hx case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) ⊢ Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0) have := (ENNReal.tendsto_toReal ENNReal.zero_ne_top).comp hx case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) this : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (ENNReal.toReal 0)) ⊢ Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0) simp only [ENNReal.zero_toReal] at this case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) this : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) ⊢ Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0) apply Tendsto.congr' _ this α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) this : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) ⊢ (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) =ᶠ[filterAt v x] fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ filter_upwards [v.eventually_measure_lt_top x, v.eventually_filterAt_integrableOn x hf] with a h'a h''a case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) this : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) a : Set α h'a : ↑↑μ a < ⊤ h''a : IntegrableOn f a ⊢ (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) a = ⨍ (y : α) in a, ‖f y - f x‖ ∂μ simp only [Function.comp_apply, ENNReal.toReal_div, setAverage_eq, div_eq_inv_mul] ** case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) this : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) a : Set α h'a : ↑↑μ a < ⊤ h''a : IntegrableOn f a ⊢ (ENNReal.toReal (↑↑μ a))⁻¹ * ENNReal.toReal (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) = (ENNReal.toReal (↑↑μ a))⁻¹ • ∫ (y : α) in a, ‖f y - f x‖ ∂μ have A : IntegrableOn (fun y => (‖f y - f x‖₊ : ℝ)) a μ := by simp_rw [coe_nnnorm] exact (h''a.sub (integrableOn_const.2 (Or.inr h'a))).norm case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) this : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) a : Set α h'a : ↑↑μ a < ⊤ h''a : IntegrableOn f a A : IntegrableOn (fun y => ↑‖f y - f x‖₊) a ⊢ (ENNReal.toReal (↑↑μ a))⁻¹ * ENNReal.toReal (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) = (ENNReal.toReal (↑↑μ a))⁻¹ • ∫ (y : α) in a, ‖f y - f x‖ ∂μ rw [lintegral_coe_eq_integral _ A, ENNReal.toReal_ofReal] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) this : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) a : Set α h'a : ↑↑μ a < ⊤ h''a : IntegrableOn f a ⊢ IntegrableOn (fun y => ↑‖f y - f x‖₊) a simp_rw [coe_nnnorm] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) this : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) a : Set α h'a : ↑↑μ a < ⊤ h''a : IntegrableOn f a ⊢ IntegrableOn (fun y => ‖f y - f x‖) a exact (h''a.sub (integrableOn_const.2 (Or.inr h'a))).norm case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) this : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) a : Set α h'a : ↑↑μ a < ⊤ h''a : IntegrableOn f a A : IntegrableOn (fun y => ↑‖f y - f x‖₊) a ⊢ (ENNReal.toReal (↑↑μ a))⁻¹ * ∫ (a : α) in a, ↑‖f a - f x‖₊ ∂μ = (ENNReal.toReal (↑↑μ a))⁻¹ • ∫ (y : α) in a, ‖f y - f x‖ ∂μ simp_rw [coe_nnnorm] case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) this : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) a : Set α h'a : ↑↑μ a < ⊤ h''a : IntegrableOn f a A : IntegrableOn (fun y => ↑‖f y - f x‖₊) a ⊢ (ENNReal.toReal (↑↑μ a))⁻¹ * ∫ (a : α) in a, ‖f a - f x‖ ∂μ = (ENNReal.toReal (↑↑μ a))⁻¹ • ∫ (y : α) in a, ‖f y - f x‖ ∂μ rfl case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) this : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) a : Set α h'a : ↑↑μ a < ⊤ h''a : IntegrableOn f a A : IntegrableOn (fun y => ↑‖f y - f x‖₊) a ⊢ 0 ≤ ∫ (a : α) in a, ↑‖f a - f x‖₊ ∂μ apply integral_nonneg case h.hf α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) this : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) a : Set α h'a : ↑↑μ a < ⊤ h''a : IntegrableOn f a A : IntegrableOn (fun y => ↑‖f y - f x‖₊) a ⊢ 0 ≤ fun a => ↑‖f a - f x‖₊ intro x case h.hf α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f x✝ : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x✝‖₊ ∂μ) / ↑↑μ a) (filterAt v x✝) (𝓝 0) this : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x✝‖₊ ∂μ) / ↑↑μ a) (filterAt v x✝) (𝓝 0) a : Set α h'a : ↑↑μ a < ⊤ h''a : IntegrableOn f a A : IntegrableOn (fun y => ↑‖f y - f x✝‖₊) a x : α ⊢ OfNat.ofNat 0 x ≤ (fun a => ↑‖f a - f x✝‖₊) x exact NNReal.coe_nonneg _ Qed
Turing.BlankExtends.refl Γ : Type u_1 inst✝ : Inhabited Γ l : List Γ ⊢ l = l ++ List.replicate 0 default simp ** Qed
Turing.BlankExtends.trans Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ l₃ : List Γ ⊢ BlankExtends l₁ l₂ → BlankExtends l₂ l₃ → BlankExtends l₁ l₃ rintro ⟨i, rfl⟩ ⟨j, rfl⟩ case intro.intro Γ : Type u_1 inst✝ : Inhabited Γ l₁ : List Γ i j : ℕ ⊢ BlankExtends l₁ (l₁ ++ List.replicate i default ++ List.replicate j default) exact ⟨i + j, by simp [List.replicate_add]⟩ Γ : Type u_1 inst✝ : Inhabited Γ l₁ : List Γ i j : ℕ ⊢ l₁ ++ List.replicate i default ++ List.replicate j default = l₁ ++ List.replicate (i + j) default simp [List.replicate_add] ** Qed
Turing.BlankExtends.below_of_le Γ : Type u_1 inst✝ : Inhabited Γ l l₁ l₂ : List Γ ⊢ BlankExtends l l₁ → BlankExtends l l₂ → List.length l₁ ≤ List.length l₂ → BlankExtends l₁ l₂ rintro ⟨i, rfl⟩ ⟨j, rfl⟩ h case intro.intro Γ : Type u_1 inst✝ : Inhabited Γ l : List Γ i j : ℕ h : List.length (l ++ List.replicate i default) ≤ List.length (l ++ List.replicate j default) ⊢ BlankExtends (l ++ List.replicate i default) (l ++ List.replicate j default) use j - i case h Γ : Type u_1 inst✝ : Inhabited Γ l : List Γ i j : ℕ h : List.length (l ++ List.replicate i default) ≤ List.length (l ++ List.replicate j default) ⊢ l ++ List.replicate j default = l ++ List.replicate i default ++ List.replicate (j - i) default simp only [List.length_append, add_le_add_iff_left, List.length_replicate] at h case h Γ : Type u_1 inst✝ : Inhabited Γ l : List Γ i j : ℕ h : i ≤ j ⊢ l ++ List.replicate j default = l ++ List.replicate i default ++ List.replicate (j - i) default simp only [← List.replicate_add, add_tsub_cancel_of_le h, List.append_assoc] ** Qed
Turing.BlankExtends.above_of_le Γ : Type u_1 inst✝ : Inhabited Γ l l₁ l₂ : List Γ ⊢ BlankExtends l₁ l → BlankExtends l₂ l → List.length l₁ ≤ List.length l₂ → BlankExtends l₁ l₂ rintro ⟨i, rfl⟩ ⟨j, e⟩ h case intro.intro Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ : List Γ i j : ℕ e : l₁ ++ List.replicate i default = l₂ ++ List.replicate j default h : List.length l₁ ≤ List.length l₂ ⊢ BlankExtends l₁ l₂ use i - j case h Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ : List Γ i j : ℕ e : l₁ ++ List.replicate i default = l₂ ++ List.replicate j default h : List.length l₁ ≤ List.length l₂ ⊢ l₂ = l₁ ++ List.replicate (i - j) default refine' List.append_right_cancel (e.symm.trans _) case h Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ : List Γ i j : ℕ e : l₁ ++ List.replicate i default = l₂ ++ List.replicate j default h : List.length l₁ ≤ List.length l₂ ⊢ l₁ ++ List.replicate i default = l₁ ++ List.replicate (i - j) default ++ List.replicate j default rw [List.append_assoc, ← List.replicate_add, tsub_add_cancel_of_le] case h Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ : List Γ i j : ℕ e : l₁ ++ List.replicate i default = l₂ ++ List.replicate j default h : List.length l₁ ≤ List.length l₂ ⊢ j ≤ i apply_fun List.length at e case h Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ : List Γ i j : ℕ h : List.length l₁ ≤ List.length l₂ e : List.length (l₁ ++ List.replicate i default) = List.length (l₂ ++ List.replicate j default) ⊢ j ≤ i simp only [List.length_append, List.length_replicate] at e case h Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ : List Γ i j : ℕ h : List.length l₁ ≤ List.length l₂ e : List.length l₁ + i = List.length l₂ + j ⊢ j ≤ i rwa [← add_le_add_iff_left, e, add_le_add_iff_right] ** Qed
Turing.BlankRel.trans Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ l₃ : List Γ ⊢ BlankRel l₁ l₂ → BlankRel l₂ l₃ → BlankRel l₁ l₃ rintro (h₁ \| h₁) (h₂ \| h₂) case inl.inl Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ l₃ : List Γ h₁ : BlankExtends l₁ l₂ h₂ : BlankExtends l₂ l₃ ⊢ BlankRel l₁ l₃ exact Or.inl (h₁.trans h₂) case inl.inr Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ l₃ : List Γ h₁ : BlankExtends l₁ l₂ h₂ : BlankExtends l₃ l₂ ⊢ BlankRel l₁ l₃ cases' le_total l₁.length l₃.length with h h case inl.inr.inl Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ l₃ : List Γ h₁ : BlankExtends l₁ l₂ h₂ : BlankExtends l₃ l₂ h : List.length l₁ ≤ List.length l₃ ⊢ BlankRel l₁ l₃ exact Or.inl (h₁.above_of_le h₂ h) case inl.inr.inr Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ l₃ : List Γ h₁ : BlankExtends l₁ l₂ h₂ : BlankExtends l₃ l₂ h : List.length l₃ ≤ List.length l₁ ⊢ BlankRel l₁ l₃ exact Or.inr (h₂.above_of_le h₁ h) case inr.inl Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ l₃ : List Γ h₁ : BlankExtends l₂ l₁ h₂ : BlankExtends l₂ l₃ ⊢ BlankRel l₁ l₃ cases' le_total l₁.length l₃.length with h h case inr.inl.inl Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ l₃ : List Γ h₁ : BlankExtends l₂ l₁ h₂ : BlankExtends l₂ l₃ h : List.length l₁ ≤ List.length l₃ ⊢ BlankRel l₁ l₃ exact Or.inl (h₁.below_of_le h₂ h) case inr.inl.inr Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ l₃ : List Γ h₁ : BlankExtends l₂ l₁ h₂ : BlankExtends l₂ l₃ h : List.length l₃ ≤ List.length l₁ ⊢ BlankRel l₁ l₃ exact Or.inr (h₂.below_of_le h₁ h) case inr.inr Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ l₃ : List Γ h₁ : BlankExtends l₂ l₁ h₂ : BlankExtends l₃ l₂ ⊢ BlankRel l₁ l₃ exact Or.inr (h₂.trans h₁) ** Qed
Turing.ListBlank.cons_head_tail Γ : Type u_1 inst✝ : Inhabited Γ ⊢ ∀ (l : ListBlank Γ), cons (head l) (tail l) = l apply Quotient.ind' case h Γ : Type u_1 inst✝ : Inhabited Γ ⊢ ∀ (a : List Γ), cons (head (Quotient.mk'' a)) (tail (Quotient.mk'' a)) = Quotient.mk'' a refine' fun l ↦ Quotient.sound' (Or.inr _) case h Γ : Type u_1 inst✝ : Inhabited Γ l : List Γ ⊢ BlankExtends l (head (Quotient.mk'' l) :: List.tail l) cases l case h.nil Γ : Type u_1 inst✝ : Inhabited Γ ⊢ BlankExtends [] (head (Quotient.mk'' []) :: List.tail []) exact ⟨1, rfl⟩ case h.cons Γ : Type u_1 inst✝ : Inhabited Γ head✝ : Γ tail✝ : List Γ ⊢ BlankExtends (head✝ :: tail✝) (head (Quotient.mk'' (head✝ :: tail✝)) :: List.tail (head✝ :: tail✝)) rfl ** Qed
Turing.ListBlank.nth_zero Γ : Type u_1 inst✝ : Inhabited Γ l : ListBlank Γ ⊢ nth l 0 = head l conv => lhs; rw [← ListBlank.cons_head_tail l] Γ : Type u_1 inst✝ : Inhabited Γ l : ListBlank Γ ⊢ nth (cons (head l) (tail l)) 0 = head l exact Quotient.inductionOn' l.tail fun l ↦ rfl ** Qed
Turing.ListBlank.nth_succ Γ : Type u_1 inst✝ : Inhabited Γ l : ListBlank Γ n : ℕ ⊢ nth l (n + 1) = nth (tail l) n conv => lhs; rw [← ListBlank.cons_head_tail l] Γ : Type u_1 inst✝ : Inhabited Γ l : ListBlank Γ n : ℕ ⊢ nth (cons (head l) (tail l)) (n + 1) = nth (tail l) n exact Quotient.inductionOn' l.tail fun l ↦ rfl ** Qed
Turing.ListBlank.ext Γ : Type u_1 i : Inhabited Γ L₁ L₂ : ListBlank Γ ⊢ (∀ (i_1 : ℕ), nth L₁ i_1 = nth L₂ i_1) → L₁ = L₂ refine' ListBlank.induction_on L₁ fun l₁ ↦ ListBlank.induction_on L₂ fun l₂ H ↦ _ Γ : Type u_1 i : Inhabited Γ L₁ L₂ : ListBlank Γ l₁ l₂ : List Γ H : ∀ (i_1 : ℕ), nth (mk l₁) i_1 = nth (mk l₂) i_1 ⊢ mk l₁ = mk l₂ wlog h : l₁.length ≤ l₂.length Γ : Type u_1 i : Inhabited Γ L₁ L₂ : ListBlank Γ l₁ l₂ : List Γ H : ∀ (i_1 : ℕ), nth (mk l₁) i_1 = nth (mk l₂) i_1 h : List.length l₁ ≤ List.length l₂ ⊢ mk l₁ = mk l₂ refine' Quotient.sound' (Or.inl ⟨l₂.length - l₁.length, _⟩) Γ : Type u_1 i : Inhabited Γ L₁ L₂ : ListBlank Γ l₁ l₂ : List Γ H : ∀ (i_1 : ℕ), nth (mk l₁) i_1 = nth (mk l₂) i_1 h : List.length l₁ ≤ List.length l₂ ⊢ l₂ = l₁ ++ List.replicate (List.length l₂ - List.length l₁) default refine' List.ext_get _ fun i h h₂ ↦ Eq.symm _ case refine'_2 Γ : Type u_1 i✝ : Inhabited Γ L₁ L₂ : ListBlank Γ l₁ l₂ : List Γ H : ∀ (i : ℕ), nth (mk l₁) i = nth (mk l₂) i h✝ : List.length l₁ ≤ List.length l₂ i : ℕ h : i < List.length l₂ h₂ : i < List.length (l₁ ++ List.replicate (List.length l₂ - List.length l₁) default) ⊢ List.get (l₁ ++ List.replicate (List.length l₂ - List.length l₁) default) { val := i, isLt := h₂ } = List.get l₂ { val := i, isLt := h } simp only [ListBlank.nth_mk] at H case refine'_2 Γ : Type u_1 i✝ : Inhabited Γ L₁ L₂ : ListBlank Γ l₁ l₂ : List Γ H : ∀ (i : ℕ), List.getI l₁ i = List.getI l₂ i h✝ : List.length l₁ ≤ List.length l₂ i : ℕ h : i < List.length l₂ h₂ : i < List.length (l₁ ++ List.replicate (List.length l₂ - List.length l₁) default) ⊢ List.get (l₁ ++ List.replicate (List.length l₂ - List.length l₁) default) { val := i, isLt := h₂ } = List.get l₂ { val := i, isLt := h } cases' lt_or_le i l₁.length with h' h' case inr Γ : Type u_1 i : Inhabited Γ L₁ L₂ : ListBlank Γ l₁ l₂ : List Γ H : ∀ (i_1 : ℕ), nth (mk l₁) i_1 = nth (mk l₂) i_1 this : ∀ {Γ : Type u_1} [i : Inhabited Γ] {L₁ L₂ : ListBlank Γ} (l₁ l₂ : List Γ), (∀ (i_1 : ℕ), nth (mk l₁) i_1 = nth (mk l₂) i_1) → List.length l₁ ≤ List.length l₂ → mk l₁ = mk l₂ h : ¬List.length l₁ ≤ List.length l₂ ⊢ mk l₁ = mk l₂ cases le_total l₁.length l₂.length <;> [skip; symm] <;> apply this <;> try assumption case inr.inr.H Γ : Type u_1 i : Inhabited Γ L₁ L₂ : ListBlank Γ l₁ l₂ : List Γ H : ∀ (i_1 : ℕ), nth (mk l₁) i_1 = nth (mk l₂) i_1 this : ∀ {Γ : Type u_1} [i : Inhabited Γ] {L₁ L₂ : ListBlank Γ} (l₁ l₂ : List Γ), (∀ (i_1 : ℕ), nth (mk l₁) i_1 = nth (mk l₂) i_1) → List.length l₁ ≤ List.length l₂ → mk l₁ = mk l₂ h : ¬List.length l₁ ≤ List.length l₂ h✝ : List.length l₂ ≤ List.length l₁ ⊢ ∀ (i_1 : ℕ), nth (mk l₂) i_1 = nth (mk l₁) i_1 intro case inr.inr.H Γ : Type u_1 i : Inhabited Γ L₁ L₂ : ListBlank Γ l₁ l₂ : List Γ H : ∀ (i_1 : ℕ), nth (mk l₁) i_1 = nth (mk l₂) i_1 this : ∀ {Γ : Type u_1} [i : Inhabited Γ] {L₁ L₂ : ListBlank Γ} (l₁ l₂ : List Γ), (∀ (i_1 : ℕ), nth (mk l₁) i_1 = nth (mk l₂) i_1) → List.length l₁ ≤ List.length l₂ → mk l₁ = mk l₂ h : ¬List.length l₁ ≤ List.length l₂ h✝ : List.length l₂ ≤ List.length l₁ i✝ : ℕ ⊢ nth (mk l₂) i✝ = nth (mk l₁) i✝ rw [H] case inr.inr.h Γ : Type u_1 i : Inhabited Γ L₁ L₂ : ListBlank Γ l₁ l₂ : List Γ H : ∀ (i_1 : ℕ), nth (mk l₁) i_1 = nth (mk l₂) i_1 this : ∀ {Γ : Type u_1} [i : Inhabited Γ] {L₁ L₂ : ListBlank Γ} (l₁ l₂ : List Γ), (∀ (i_1 : ℕ), nth (mk l₁) i_1 = nth (mk l₂) i_1) → List.length l₁ ≤ List.length l₂ → mk l₁ = mk l₂ h : ¬List.length l₁ ≤ List.length l₂ h✝ : List.length l₂ ≤ List.length l₁ ⊢ List.length l₂ ≤ List.length l₁ assumption case refine'_1 Γ : Type u_1 i : Inhabited Γ L₁ L₂ : ListBlank Γ l₁ l₂ : List Γ H : ∀ (i_1 : ℕ), nth (mk l₁) i_1 = nth (mk l₂) i_1 h : List.length l₁ ≤ List.length l₂ ⊢ List.length l₂ = List.length (l₁ ++ List.replicate (List.length l₂ - List.length l₁) default) simp only [add_tsub_cancel_of_le h, List.length_append, List.length_replicate] case refine'_2.inl Γ : Type u_1 i✝ : Inhabited Γ L₁ L₂ : ListBlank Γ l₁ l₂ : List Γ H : ∀ (i : ℕ), List.getI l₁ i = List.getI l₂ i h✝ : List.length l₁ ≤ List.length l₂ i : ℕ h : i < List.length l₂ h₂ : i < List.length (l₁ ++ List.replicate (List.length l₂ - List.length l₁) default) h' : i < List.length l₁ ⊢ List.get (l₁ ++ List.replicate (List.length l₂ - List.length l₁) default) { val := i, isLt := h₂ } = List.get l₂ { val := i, isLt := h } simp only [List.get_append _ h', List.get?_eq_get h, List.get?_eq_get h', ← List.getI_eq_get _ h, ← List.getI_eq_get _ h', H] case refine'_2.inr Γ : Type u_1 i✝ : Inhabited Γ L₁ L₂ : ListBlank Γ l₁ l₂ : List Γ H : ∀ (i : ℕ), List.getI l₁ i = List.getI l₂ i h✝ : List.length l₁ ≤ List.length l₂ i : ℕ h : i < List.length l₂ h₂ : i < List.length (l₁ ++ List.replicate (List.length l₂ - List.length l₁) default) h' : List.length l₁ ≤ i ⊢ List.get (l₁ ++ List.replicate (List.length l₂ - List.length l₁) default) { val := i, isLt := h₂ } = List.get l₂ { val := i, isLt := h } simp only [List.get_append_right' h', List.get_replicate, List.get?_eq_get h, List.get?_len_le h', ← List.getI_eq_default _ h', H, List.getI_eq_get _ h] ** Qed
Turing.ListBlank.nth_modifyNth Γ : Type u_1 inst✝ : Inhabited Γ f : Γ → Γ n i : ℕ L : ListBlank Γ ⊢ nth (modifyNth f n L) i = if i = n then f (nth L i) else nth L i induction' n with n IH generalizing i L case zero Γ : Type u_1 inst✝ : Inhabited Γ f : Γ → Γ i✝ : ℕ L✝ : ListBlank Γ i : ℕ L : ListBlank Γ ⊢ nth (modifyNth f Nat.zero L) i = if i = Nat.zero then f (nth L i) else nth L i cases i <;> simp only [ListBlank.nth_zero, if_true, ListBlank.head_cons, ListBlank.modifyNth, ListBlank.nth_succ, if_false, ListBlank.tail_cons, Nat.zero_eq] case succ Γ : Type u_1 inst✝ : Inhabited Γ f : Γ → Γ i✝ : ℕ L✝ : ListBlank Γ n : ℕ IH : ∀ (i : ℕ) (L : ListBlank Γ), nth (modifyNth f n L) i = if i = n then f (nth L i) else nth L i i : ℕ L : ListBlank Γ ⊢ nth (modifyNth f (Nat.succ n) L) i = if i = Nat.succ n then f (nth L i) else nth L i cases i case succ.zero Γ : Type u_1 inst✝ : Inhabited Γ f : Γ → Γ i : ℕ L✝ : ListBlank Γ n : ℕ IH : ∀ (i : ℕ) (L : ListBlank Γ), nth (modifyNth f n L) i = if i = n then f (nth L i) else nth L i L : ListBlank Γ ⊢ nth (modifyNth f (Nat.succ n) L) Nat.zero = if Nat.zero = Nat.succ n then f (nth L Nat.zero) else nth L Nat.zero rw [if_neg (Nat.succ_ne_zero _).symm] case succ.zero Γ : Type u_1 inst✝ : Inhabited Γ f : Γ → Γ i : ℕ L✝ : ListBlank Γ n : ℕ IH : ∀ (i : ℕ) (L : ListBlank Γ), nth (modifyNth f n L) i = if i = n then f (nth L i) else nth L i L : ListBlank Γ ⊢ nth (modifyNth f (Nat.succ n) L) Nat.zero = nth L Nat.zero simp only [ListBlank.nth_zero, ListBlank.head_cons, ListBlank.modifyNth, Nat.zero_eq] case succ.succ Γ : Type u_1 inst✝ : Inhabited Γ f : Γ → Γ i : ℕ L✝ : ListBlank Γ n : ℕ IH : ∀ (i : ℕ) (L : ListBlank Γ), nth (modifyNth f n L) i = if i = n then f (nth L i) else nth L i L : ListBlank Γ n✝ : ℕ ⊢ nth (modifyNth f (Nat.succ n) L) (Nat.succ n✝) = if Nat.succ n✝ = Nat.succ n then f (nth L (Nat.succ n✝)) else nth L (Nat.succ n✝) simp only [IH, ListBlank.modifyNth, ListBlank.nth_succ, ListBlank.tail_cons, Nat.succ.injEq] ** Qed
Turing.ListBlank.head_map Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' l : ListBlank Γ ⊢ head (map f l) = PointedMap.f f (head l) conv => lhs; rw [← ListBlank.cons_head_tail l] Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' l : ListBlank Γ ⊢ head (map f (cons (head l) (tail l))) = PointedMap.f f (head l) exact Quotient.inductionOn' l fun a ↦ rfl ** Qed
Turing.ListBlank.tail_map Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' l : ListBlank Γ ⊢ tail (map f l) = map f (tail l) conv => lhs; rw [← ListBlank.cons_head_tail l] Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' l : ListBlank Γ ⊢ tail (map f (cons (head l) (tail l))) = map f (tail l) exact Quotient.inductionOn' l fun a ↦ rfl ** Qed
Turing.ListBlank.map_cons Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' l : ListBlank Γ a : Γ ⊢ map f (cons a l) = cons (PointedMap.f f a) (map f l) refine' (ListBlank.cons_head_tail _).symm.trans _ Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' l : ListBlank Γ a : Γ ⊢ cons (head (map f (cons a l))) (tail (map f (cons a l))) = cons (PointedMap.f f a) (map f l) simp only [ListBlank.head_map, ListBlank.head_cons, ListBlank.tail_map, ListBlank.tail_cons] ** Qed
Turing.ListBlank.nth_map Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' l : ListBlank Γ n : ℕ ⊢ nth (map f l) n = PointedMap.f f (nth l n) refine' l.inductionOn fun l ↦ _ Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' l✝ : ListBlank Γ n : ℕ l : List Γ ⊢ nth (map f (Quotient.mk (BlankRel.setoid Γ) l)) n = PointedMap.f f (nth (Quotient.mk (BlankRel.setoid Γ) l) n) suffices ((mk l).map f).nth n = f ((mk l).nth n) by exact this Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' l✝ : ListBlank Γ n : ℕ l : List Γ ⊢ nth (map f (mk l)) n = PointedMap.f f (nth (mk l) n) simp only [List.get?_map, ListBlank.map_mk, ListBlank.nth_mk, List.getI_eq_iget_get?] Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' l✝ : ListBlank Γ n : ℕ l : List Γ ⊢ Option.iget (Option.map f.f (List.get? l n)) = PointedMap.f f (Option.iget (List.get? l n)) cases l.get? n Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' l✝ : ListBlank Γ n : ℕ l : List Γ this : nth (map f (mk l)) n = PointedMap.f f (nth (mk l) n) ⊢ nth (map f (Quotient.mk (BlankRel.setoid Γ) l)) n = PointedMap.f f (nth (Quotient.mk (BlankRel.setoid Γ) l) n) exact this case none Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' l✝ : ListBlank Γ n : ℕ l : List Γ ⊢ Option.iget (Option.map f.f none) = PointedMap.f f (Option.iget none) exact f.2.symm case some Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' l✝ : ListBlank Γ n : ℕ l : List Γ val✝ : Γ ⊢ Option.iget (Option.map f.f (some val✝)) = PointedMap.f f (Option.iget (some val✝)) rfl ** Qed
Turing.proj_map_nth ι : Type u_1 Γ : ι → Type u_2 inst✝ : (i : ι) → Inhabited (Γ i) i : ι L : ListBlank ((i : ι) → Γ i) n : ℕ ⊢ ListBlank.nth (ListBlank.map (proj i) L) n = ListBlank.nth L n i rw [ListBlank.nth_map] ι : Type u_1 Γ : ι → Type u_2 inst✝ : (i : ι) → Inhabited (Γ i) i : ι L : ListBlank ((i : ι) → Γ i) n : ℕ ⊢ PointedMap.f (proj i) (ListBlank.nth L n) = ListBlank.nth L n i rfl ** Qed
Turing.ListBlank.map_modifyNth Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' F : PointedMap Γ Γ' f : Γ → Γ f' : Γ' → Γ' H : ∀ (x : Γ), PointedMap.f F (f x) = f' (PointedMap.f F x) n : ℕ L : ListBlank Γ ⊢ map F (modifyNth f n L) = modifyNth f' n (map F L) induction' n with n IH generalizing L <;> simp only [, ListBlank.head_map, ListBlank.modifyNth, ListBlank.map_cons, ListBlank.tail_map] * Qed
Turing.ListBlank.append_mk Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ : List Γ ⊢ append l₁ (mk l₂) = mk (l₁ ++ l₂) induction l₁ <;> simp only [, ListBlank.append, List.nil_append, List.cons_append, ListBlank.cons_mk] * Qed
Turing.ListBlank.append_assoc Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ : List Γ l₃ : ListBlank Γ ⊢ append (l₁ ++ l₂) l₃ = append l₁ (append l₂ l₃) refine' l₃.inductionOn fun l ↦ _ Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ : List Γ l₃ : ListBlank Γ l : List Γ ⊢ append (l₁ ++ l₂) (Quotient.mk (BlankRel.setoid Γ) l) = append l₁ (append l₂ (Quotient.mk (BlankRel.setoid Γ) l)) suffices append (l₁ ++ l₂) (mk l) = append l₁ (append l₂ (mk l)) by exact this Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ : List Γ l₃ : ListBlank Γ l : List Γ ⊢ append (l₁ ++ l₂) (mk l) = append l₁ (append l₂ (mk l)) simp only [ListBlank.append_mk, List.append_assoc] Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ : List Γ l₃ : ListBlank Γ l : List Γ this : append (l₁ ++ l₂) (mk l) = append l₁ (append l₂ (mk l)) ⊢ append (l₁ ++ l₂) (Quotient.mk (BlankRel.setoid Γ) l) = append l₁ (append l₂ (Quotient.mk (BlankRel.setoid Γ) l)) exact this ** Qed
Turing.Tape.move_left_right Γ : Type u_1 inst✝ : Inhabited Γ T : Tape Γ ⊢ move Dir.right (move Dir.left T) = T cases T case mk Γ : Type u_1 inst✝ : Inhabited Γ head✝ : Γ left✝ right✝ : ListBlank Γ ⊢ move Dir.right (move Dir.left { head := head✝, left := left✝, right := right✝ }) = { head := head✝, left := left✝, right := right✝ } simp [Tape.move] ** Qed
Turing.Tape.move_right_left Γ : Type u_1 inst✝ : Inhabited Γ T : Tape Γ ⊢ move Dir.left (move Dir.right T) = T cases T case mk Γ : Type u_1 inst✝ : Inhabited Γ head✝ : Γ left✝ right✝ : ListBlank Γ ⊢ move Dir.left (move Dir.right { head := head✝, left := left✝, right := right✝ }) = { head := head✝, left := left✝, right := right✝ } simp [Tape.move] ** Qed
Turing.Tape.mk'_left_right₀ Γ : Type u_1 inst✝ : Inhabited Γ T : Tape Γ ⊢ mk' T.left (right₀ T) = T cases T case mk Γ : Type u_1 inst✝ : Inhabited Γ head✝ : Γ left✝ right✝ : ListBlank Γ ⊢ mk' { head := head✝, left := left✝, right := right✝ }.left (right₀ { head := head✝, left := left✝, right := right✝ }) = { head := head✝, left := left✝, right := right✝ } simp only [Tape.right₀, Tape.mk', ListBlank.head_cons, ListBlank.tail_cons, eq_self_iff_true, and_self_iff] ** Qed
Turing.Tape.move_left_mk' Γ : Type u_1 inst✝ : Inhabited Γ L R : ListBlank Γ ⊢ move Dir.left (mk' L R) = mk' (ListBlank.tail L) (ListBlank.cons (ListBlank.head L) R) simp only [Tape.move, Tape.mk', ListBlank.head_cons, eq_self_iff_true, ListBlank.cons_head_tail, and_self_iff, ListBlank.tail_cons] ** Qed
Turing.Tape.move_right_mk' Γ : Type u_1 inst✝ : Inhabited Γ L R : ListBlank Γ ⊢ move Dir.right (mk' L R) = mk' (ListBlank.cons (ListBlank.head R) L) (ListBlank.tail R) simp only [Tape.move, Tape.mk', ListBlank.head_cons, eq_self_iff_true, ListBlank.cons_head_tail, and_self_iff, ListBlank.tail_cons] ** Qed
Turing.Tape.right₀_nth Γ : Type u_1 inst✝ : Inhabited Γ T : Tape Γ n : ℕ ⊢ ListBlank.nth (right₀ T) n = nth T ↑n cases n <;> simp only [Tape.nth, Tape.right₀, Int.ofNat_zero, ListBlank.nth_zero, ListBlank.nth_succ, ListBlank.head_cons, ListBlank.tail_cons, Nat.zero_eq] ** Qed
Turing.Tape.move_left_nth Γ : Type u_1 inst✝ : Inhabited Γ a : Γ L R : ListBlank Γ n : ℕ ⊢ nth (move Dir.left { head := a, left := L, right := R }) (↑(n + 1) + 1) = nth { head := a, left := L, right := R } (↑(n + 1) + 1 - 1) rw [add_sub_cancel] Γ : Type u_1 inst✝ : Inhabited Γ a : Γ L R : ListBlank Γ n : ℕ ⊢ nth (move Dir.left { head := a, left := L, right := R }) (↑(n + 1) + 1) = nth { head := a, left := L, right := R } ↑(n + 1) change (R.cons a).nth (n + 1) = R.nth n Γ : Type u_1 inst✝ : Inhabited Γ a : Γ L R : ListBlank Γ n : ℕ ⊢ ListBlank.nth (ListBlank.cons a R) (n + 1) = ListBlank.nth R n rw [ListBlank.nth_succ, ListBlank.tail_cons] ** Qed
Turing.Tape.write_self Γ : Type u_1 inst✝ : Inhabited Γ ⊢ ∀ (T : Tape Γ), write T.head T = T rintro ⟨⟩ case mk Γ : Type u_1 inst✝ : Inhabited Γ head✝ : Γ left✝ right✝ : ListBlank Γ ⊢ write { head := head✝, left := left✝, right := right✝ }.head { head := head✝, left := left✝, right := right✝ } = { head := head✝, left := left✝, right := right✝ } rfl ** Qed
Turing.Tape.write_mk' Γ : Type u_1 inst✝ : Inhabited Γ a b : Γ L R : ListBlank Γ ⊢ write b (mk' L (ListBlank.cons a R)) = mk' L (ListBlank.cons b R) simp only [Tape.write, Tape.mk', ListBlank.head_cons, ListBlank.tail_cons, eq_self_iff_true, and_self_iff] ** Qed
Turing.Tape.map_fst Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' ⊢ ∀ (T : Tape Γ), (map f T).head = PointedMap.f f T.head rintro ⟨⟩ case mk Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' head✝ : Γ left✝ right✝ : ListBlank Γ ⊢ (map f { head := head✝, left := left✝, right := right✝ }).head = PointedMap.f f { head := head✝, left := left✝, right := right✝ }.head rfl ** Qed
Turing.Tape.map_write Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' b : Γ ⊢ ∀ (T : Tape Γ), map f (write b T) = write (PointedMap.f f b) (map f T) rintro ⟨⟩ case mk Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' b head✝ : Γ left✝ right✝ : ListBlank Γ ⊢ map f (write b { head := head✝, left := left✝, right := right✝ }) = write (PointedMap.f f b) (map f { head := head✝, left := left✝, right := right✝ }) rfl ** Qed
Turing.Tape.write_move_right_n Γ : Type u_1 inst✝ : Inhabited Γ f : Γ → Γ L R : ListBlank Γ n : ℕ ⊢ write (f (ListBlank.nth R n)) ((move Dir.right)^[n] (mk' L R)) = (move Dir.right)^[n] (mk' L (ListBlank.modifyNth f n R)) induction' n with n IH generalizing L R case succ Γ : Type u_1 inst✝ : Inhabited Γ f : Γ → Γ L✝ R✝ : ListBlank Γ n : ℕ IH : ∀ (L R : ListBlank Γ), write (f (ListBlank.nth R n)) ((move Dir.right)^[n] (mk' L R)) = (move Dir.right)^[n] (mk' L (ListBlank.modifyNth f n R)) L R : ListBlank Γ ⊢ write (f (ListBlank.nth R (Nat.succ n))) ((move Dir.right)^[Nat.succ n] (mk' L R)) = (move Dir.right)^[Nat.succ n] (mk' L (ListBlank.modifyNth f (Nat.succ n) R)) simp only [ListBlank.head_cons, ListBlank.nth_succ, ListBlank.modifyNth, Tape.move_right_mk', ListBlank.tail_cons, iterate_succ_apply, IH] case zero Γ : Type u_1 inst✝ : Inhabited Γ f : Γ → Γ L✝ R✝ L R : ListBlank Γ ⊢ write (f (ListBlank.nth R Nat.zero)) ((move Dir.right)^[Nat.zero] (mk' L R)) = (move Dir.right)^[Nat.zero] (mk' L (ListBlank.modifyNth f Nat.zero R)) simp only [ListBlank.nth_zero, ListBlank.modifyNth, iterate_zero_apply, Nat.zero_eq] case zero Γ : Type u_1 inst✝ : Inhabited Γ f : Γ → Γ L✝ R✝ L R : ListBlank Γ ⊢ write (f (ListBlank.head R)) (mk' L R) = mk' L (ListBlank.cons (f (ListBlank.head R)) (ListBlank.tail R)) rw [← Tape.write_mk', ListBlank.cons_head_tail] ** Qed
Turing.Tape.map_move Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' T : Tape Γ d : Dir ⊢ map f (move d T) = move d (map f T) cases T case mk Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' d : Dir head✝ : Γ left✝ right✝ : ListBlank Γ ⊢ map f (move d { head := head✝, left := left✝, right := right✝ }) = move d (map f { head := head✝, left := left✝, right := right✝ }) cases d <;> simp only [Tape.move, Tape.map, ListBlank.head_map, eq_self_iff_true, ListBlank.map_cons, and_self_iff, ListBlank.tail_map] ** Qed
Turing.Tape.map_mk' Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' L R : ListBlank Γ ⊢ map f (mk' L R) = mk' (ListBlank.map f L) (ListBlank.map f R) simp only [Tape.mk', Tape.map, ListBlank.head_map, eq_self_iff_true, and_self_iff, ListBlank.tail_map] ** Qed
Turing.reaches₁_eq σ : Type u_1 f : σ → Option σ a b c : σ h : f a = f b ⊢ (∃ b_1, b_1 ∈ f b ∧ ReflTransGen (fun a b => b ∈ f a) b_1 c) ↔ ∃ b, b ∈ f a ∧ ReflTransGen (fun a b => b ∈ f a) b c rw [h] ** Qed
Turing.reaches₁_fwd σ : Type u_1 f : σ → Option σ a b c : σ h₁ : Reaches₁ f a c h₂ : b ∈ f a ⊢ Reaches f b c rcases TransGen.head'_iff.1 h₁ with ⟨b', hab, hbc⟩ case intro.intro σ : Type u_1 f : σ → Option σ a b c : σ h₁ : Reaches₁ f a c h₂ : b ∈ f a b' : σ hab : b' ∈ f a hbc : ReflTransGen (fun a b => b ∈ f a) b' c ⊢ Reaches f b c cases Option.mem_unique hab h₂ case intro.intro.refl σ : Type u_1 f : σ → Option σ a b c : σ h₁ : Reaches₁ f a c h₂ hab : b ∈ f a hbc : ReflTransGen (fun a b => b ∈ f a) b c ⊢ Reaches f b c exact hbc ** Qed
Turing.mem_eval σ : Type u_1 f : σ → Option σ a b : σ ⊢ b ∈ eval f a ↔ Reaches f a b ∧ f b = none refine' ⟨fun h ↦ _, fun ⟨h₁, h₂⟩ ↦ _⟩ case refine'_1 σ : Type u_1 f : σ → Option σ a b : σ h : b ∈ eval f a ⊢ Reaches f a b ∧ f b = none refine' @evalInduction _ _ _ (fun a ↦ Reaches f a b ∧ f b = none) _ h fun a h IH ↦ _ case refine'_1 σ : Type u_1 f : σ → Option σ a✝ b : σ h✝ : b ∈ eval f a✝ a : σ h : b ∈ eval f a IH : ∀ (a' : σ), f a = some a' → (fun a => Reaches f a b ∧ f b = none) a' ⊢ (fun a => Reaches f a b ∧ f b = none) a cases' e : f a with a' case refine'_1.none σ : Type u_1 f : σ → Option σ a✝ b : σ h✝ : b ∈ eval f a✝ a : σ h : b ∈ eval f a IH : ∀ (a' : σ), f a = some a' → (fun a => Reaches f a b ∧ f b = none) a' e : f a = none ⊢ Reaches f a b ∧ f b = none rw [Part.mem_unique h (PFun.mem_fix_iff.2 <\| Or.inl <\| Part.mem_some_iff.2 <\| by rw [e] <;> rfl)] case refine'_1.none σ : Type u_1 f : σ → Option σ a✝ b : σ h✝ : b ∈ eval f a✝ a : σ h : b ∈ eval f a IH : ∀ (a' : σ), f a = some a' → (fun a => Reaches f a b ∧ f b = none) a' e : f a = none ⊢ Reaches f a a ∧ f a = none exact ⟨ReflTransGen.refl, e⟩ σ : Type u_1 f : σ → Option σ a✝ b : σ h✝ : b ∈ eval f a✝ a : σ h : b ∈ eval f a IH : ∀ (a' : σ), f a = some a' → (fun a => Reaches f a b ∧ f b = none) a' e : f a = none ⊢ Sum.inl ?m.63851 = Option.elim (f a) (Sum.inl a) Sum.inr rw [e] <;> rfl case refine'_1.some σ : Type u_1 f : σ → Option σ a✝ b : σ h✝ : b ∈ eval f a✝ a : σ h : b ∈ eval f a IH : ∀ (a' : σ), f a = some a' → (fun a => Reaches f a b ∧ f b = none) a' a' : σ e : f a = some a' ⊢ Reaches f a b ∧ f b = none rcases PFun.mem_fix_iff.1 h with (h \| ⟨_, h, _⟩) <;> rw [e] at h <;> cases Part.mem_some_iff.1 h case refine'_1.some.inr.intro.intro.refl σ : Type u_1 f : σ → Option σ a✝ b : σ h✝¹ : b ∈ eval f a✝ a : σ h✝ : b ∈ eval f a IH : ∀ (a' : σ), f a = some a' → (fun a => Reaches f a b ∧ f b = none) a' a' : σ e : f a = some a' h : Sum.inr a' ∈ Part.some (Option.elim (some a') (Sum.inl a) Sum.inr) right✝ : b ∈ PFun.fix (fun s => Part.some (Option.elim (f s) (Sum.inl s) Sum.inr)) a' ⊢ Reaches f a b ∧ f b = none cases' IH a' e with h₁ h₂ case refine'_1.some.inr.intro.intro.refl.intro σ : Type u_1 f : σ → Option σ a✝ b : σ h✝¹ : b ∈ eval f a✝ a : σ h✝ : b ∈ eval f a IH : ∀ (a' : σ), f a = some a' → (fun a => Reaches f a b ∧ f b = none) a' a' : σ e : f a = some a' h : Sum.inr a' ∈ Part.some (Option.elim (some a') (Sum.inl a) Sum.inr) right✝ : b ∈ PFun.fix (fun s => Part.some (Option.elim (f s) (Sum.inl s) Sum.inr)) a' h₁ : Reaches f a' b h₂ : f b = none ⊢ Reaches f a b ∧ f b = none exact ⟨ReflTransGen.head e h₁, h₂⟩ case refine'_2 σ : Type u_1 f : σ → Option σ a b : σ x✝ : Reaches f a b ∧ f b = none h₁ : Reaches f a b h₂ : f b = none ⊢ b ∈ eval f a refine' ReflTransGen.head_induction_on h₁ _ fun h _ IH ↦ _ case refine'_2.refine'_1 σ : Type u_1 f : σ → Option σ a b : σ x✝ : Reaches f a b ∧ f b = none h₁ : Reaches f a b h₂ : f b = none ⊢ b ∈ eval f b refine' PFun.mem_fix_iff.2 (Or.inl _) case refine'_2.refine'_1 σ : Type u_1 f : σ → Option σ a b : σ x✝ : Reaches f a b ∧ f b = none h₁ : Reaches f a b h₂ : f b = none ⊢ Sum.inl b ∈ Part.some (Option.elim (f b) (Sum.inl b) Sum.inr) rw [h₂] case refine'_2.refine'_1 σ : Type u_1 f : σ → Option σ a b : σ x✝ : Reaches f a b ∧ f b = none h₁ : Reaches f a b h₂ : f b = none ⊢ Sum.inl b ∈ Part.some (Option.elim none (Sum.inl b) Sum.inr) apply Part.mem_some case refine'_2.refine'_2 σ : Type u_1 f : σ → Option σ a b : σ x✝¹ : Reaches f a b ∧ f b = none h₁ : Reaches f a b h₂ : f b = none a✝ c✝ : σ h : c✝ ∈ f a✝ x✝ : ReflTransGen (fun a b => b ∈ f a) c✝ b IH : b ∈ eval f c✝ ⊢ b ∈ eval f a✝ refine' PFun.mem_fix_iff.2 (Or.inr ⟨_, _, IH⟩) case refine'_2.refine'_2 σ : Type u_1 f : σ → Option σ a b : σ x✝¹ : Reaches f a b ∧ f b = none h₁ : Reaches f a b h₂ : f b = none a✝ c✝ : σ h : c✝ ∈ f a✝ x✝ : ReflTransGen (fun a b => b ∈ f a) c✝ b IH : b ∈ eval f c✝ ⊢ Sum.inr c✝ ∈ Part.some (Option.elim (f a✝) (Sum.inl a✝) Sum.inr) rw [h] case refine'_2.refine'_2 σ : Type u_1 f : σ → Option σ a b : σ x✝¹ : Reaches f a b ∧ f b = none h₁ : Reaches f a b h₂ : f b = none a✝ c✝ : σ h : c✝ ∈ f a✝ x✝ : ReflTransGen (fun a b => b ∈ f a) c✝ b IH : b ∈ eval f c✝ ⊢ Sum.inr c✝ ∈ Part.some (Option.elim (some c✝) (Sum.inl a✝) Sum.inr) apply Part.mem_some ** Qed
Turing.eval_maximal₁ σ : Type u_1 f : σ → Option σ a b : σ h : b ∈ eval f a c : σ x✝ : Reaches₁ f b c bc : Reaches₁ f b c := x✝ ⊢ False let ⟨_, b0⟩ := mem_eval.1 h σ : Type u_1 f : σ → Option σ a b : σ h : b ∈ eval f a c : σ x✝ : Reaches₁ f b c bc : Reaches₁ f b c := x✝ left✝ : Reaches f a b b0 : f b = none ⊢ False let ⟨b', h', _⟩ := TransGen.head'_iff.1 bc σ : Type u_1 f : σ → Option σ a b : σ h : b ∈ eval f a c : σ x✝ : Reaches₁ f b c bc : Reaches₁ f b c := x✝ left✝ : Reaches f a b b0 : f b = none b' : σ h' : b' ∈ f b right✝ : ReflTransGen (fun a b => b ∈ f a) b' c ⊢ False cases b0.symm.trans h' ** Qed
Turing.eval_maximal σ : Type u_1 f : σ → Option σ a b : σ h : b ∈ eval f a c : σ left✝ : Reaches f a b b0 : f b = none b' : σ h' : b' ∈ f b ⊢ False cases b0.symm.trans h' ** Qed
Turing.tr_reaches₁ σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₁ : σ₁ ab : Reaches₁ f₁ a₁ b₁ ⊢ ∃ b₂, tr b₁ b₂ ∧ Reaches₁ f₂ a₂ b₂ induction' ab with c₁ ac c₁ d₁ _ cd IH case single σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₁ c₁ : σ₁ ac : c₁ ∈ f₁ a₁ ⊢ ∃ b₂, tr c₁ b₂ ∧ Reaches₁ f₂ a₂ b₂ have := H aa case single σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₁ c₁ : σ₁ ac : c₁ ∈ f₁ a₁ this : match f₁ a₁ with \| some b₁ => ∃ b₂, tr b₁ b₂ ∧ Reaches₁ f₂ a₂ b₂ \| none => f₂ a₂ = none ⊢ ∃ b₂, tr c₁ b₂ ∧ Reaches₁ f₂ a₂ b₂ rwa [show f₁ a₁ = _ from ac] at this case tail σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₁ c₁ d₁ : σ₁ a✝ : TransGen (fun a b => b ∈ f₁ a) a₁ c₁ cd : d₁ ∈ f₁ c₁ IH : ∃ b₂, tr c₁ b₂ ∧ Reaches₁ f₂ a₂ b₂ ⊢ ∃ b₂, tr d₁ b₂ ∧ Reaches₁ f₂ a₂ b₂ rcases IH with ⟨c₂, cc, ac₂⟩ case tail.intro.intro σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₁ c₁ d₁ : σ₁ a✝ : TransGen (fun a b => b ∈ f₁ a) a₁ c₁ cd : d₁ ∈ f₁ c₁ c₂ : σ₂ cc : tr c₁ c₂ ac₂ : Reaches₁ f₂ a₂ c₂ ⊢ ∃ b₂, tr d₁ b₂ ∧ Reaches₁ f₂ a₂ b₂ have := H cc case tail.intro.intro σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₁ c₁ d₁ : σ₁ a✝ : TransGen (fun a b => b ∈ f₁ a) a₁ c₁ cd : d₁ ∈ f₁ c₁ c₂ : σ₂ cc : tr c₁ c₂ ac₂ : Reaches₁ f₂ a₂ c₂ this : match f₁ c₁ with \| some b₁ => ∃ b₂, tr b₁ b₂ ∧ Reaches₁ f₂ c₂ b₂ \| none => f₂ c₂ = none ⊢ ∃ b₂, tr d₁ b₂ ∧ Reaches₁ f₂ a₂ b₂ rw [show f₁ c₁ = _ from cd] at this case tail.intro.intro σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₁ c₁ d₁ : σ₁ a✝ : TransGen (fun a b => b ∈ f₁ a) a₁ c₁ cd : d₁ ∈ f₁ c₁ c₂ : σ₂ cc : tr c₁ c₂ ac₂ : Reaches₁ f₂ a₂ c₂ this : match some d₁ with \| some b₁ => ∃ b₂, tr b₁ b₂ ∧ Reaches₁ f₂ c₂ b₂ \| none => f₂ c₂ = none ⊢ ∃ b₂, tr d₁ b₂ ∧ Reaches₁ f₂ a₂ b₂ rcases this with ⟨d₂, dd, cd₂⟩ case tail.intro.intro.intro.intro σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₁ c₁ d₁ : σ₁ a✝ : TransGen (fun a b => b ∈ f₁ a) a₁ c₁ cd : d₁ ∈ f₁ c₁ c₂ : σ₂ cc : tr c₁ c₂ ac₂ : Reaches₁ f₂ a₂ c₂ d₂ : σ₂ dd : tr d₁ d₂ cd₂ : Reaches₁ f₂ c₂ d₂ ⊢ ∃ b₂, tr d₁ b₂ ∧ Reaches₁ f₂ a₂ b₂ exact ⟨_, dd, ac₂.trans cd₂⟩ ** Qed
Turing.tr_reaches_rev σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ : σ₂ ab : Reaches f₂ a₂ b₂ ⊢ ∃ c₁ c₂, Reaches f₂ b₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ induction' ab with c₂ d₂ _ cd IH case refl σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ : σ₂ ⊢ ∃ c₁ c₂, Reaches f₂ a₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ exact ⟨_, _, ReflTransGen.refl, aa, ReflTransGen.refl⟩ case tail σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ c₂ d₂ : σ₂ a✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂ cd : d₂ ∈ f₂ c₂ IH : ∃ c₁ c₂_1, Reaches f₂ c₂ c₂_1 ∧ tr c₁ c₂_1 ∧ Reaches f₁ a₁ c₁ ⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ rcases IH with ⟨e₁, e₂, ce, ee, ae⟩ case tail.intro.intro.intro.intro σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ c₂ d₂ : σ₂ a✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂ cd : d₂ ∈ f₂ c₂ e₁ : σ₁ e₂ : σ₂ ce : Reaches f₂ c₂ e₂ ee : tr e₁ e₂ ae : Reaches f₁ a₁ e₁ ⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ rcases ReflTransGen.cases_head ce with (rfl \| ⟨d', cd', de⟩) case tail.intro.intro.intro.intro.inl σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ c₂ d₂ : σ₂ a✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂ cd : d₂ ∈ f₂ c₂ e₁ : σ₁ ae : Reaches f₁ a₁ e₁ ce : Reaches f₂ c₂ c₂ ee : tr e₁ c₂ ⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ have := H ee case tail.intro.intro.intro.intro.inl σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ c₂ d₂ : σ₂ a✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂ cd : d₂ ∈ f₂ c₂ e₁ : σ₁ ae : Reaches f₁ a₁ e₁ ce : Reaches f₂ c₂ c₂ ee : tr e₁ c₂ this : match f₁ e₁ with \| some b₁ => ∃ b₂, tr b₁ b₂ ∧ Reaches₁ f₂ c₂ b₂ \| none => f₂ c₂ = none ⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ revert this case tail.intro.intro.intro.intro.inl σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ c₂ d₂ : σ₂ a✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂ cd : d₂ ∈ f₂ c₂ e₁ : σ₁ ae : Reaches f₁ a₁ e₁ ce : Reaches f₂ c₂ c₂ ee : tr e₁ c₂ ⊢ (match f₁ e₁ with \| some b₁ => ∃ b₂, tr b₁ b₂ ∧ Reaches₁ f₂ c₂ b₂ \| none => f₂ c₂ = none) → ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ cases' eg : f₁ e₁ with g₁ <;> simp only [Respects, and_imp, exists_imp] case tail.intro.intro.intro.intro.inl.none σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ c₂ d₂ : σ₂ a✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂ cd : d₂ ∈ f₂ c₂ e₁ : σ₁ ae : Reaches f₁ a₁ e₁ ce : Reaches f₂ c₂ c₂ ee : tr e₁ c₂ eg : f₁ e₁ = none ⊢ f₂ c₂ = none → ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ intro c0 case tail.intro.intro.intro.intro.inl.none σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ c₂ d₂ : σ₂ a✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂ cd : d₂ ∈ f₂ c₂ e₁ : σ₁ ae : Reaches f₁ a₁ e₁ ce : Reaches f₂ c₂ c₂ ee : tr e₁ c₂ eg : f₁ e₁ = none c0 : f₂ c₂ = none ⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ cases cd.symm.trans c0 case tail.intro.intro.intro.intro.inl.some σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ c₂ d₂ : σ₂ a✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂ cd : d₂ ∈ f₂ c₂ e₁ : σ₁ ae : Reaches f₁ a₁ e₁ ce : Reaches f₂ c₂ c₂ ee : tr e₁ c₂ g₁ : σ₁ eg : f₁ e₁ = some g₁ ⊢ ∀ (x : σ₂), tr g₁ x → Reaches₁ f₂ c₂ x → ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ intro g₂ gg cg case tail.intro.intro.intro.intro.inl.some σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ c₂ d₂ : σ₂ a✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂ cd : d₂ ∈ f₂ c₂ e₁ : σ₁ ae : Reaches f₁ a₁ e₁ ce : Reaches f₂ c₂ c₂ ee : tr e₁ c₂ g₁ : σ₁ eg : f₁ e₁ = some g₁ g₂ : σ₂ gg : tr g₁ g₂ cg : Reaches₁ f₂ c₂ g₂ ⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ rcases TransGen.head'_iff.1 cg with ⟨d', cd', dg⟩ case tail.intro.intro.intro.intro.inl.some.intro.intro σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ c₂ d₂ : σ₂ a✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂ cd : d₂ ∈ f₂ c₂ e₁ : σ₁ ae : Reaches f₁ a₁ e₁ ce : Reaches f₂ c₂ c₂ ee : tr e₁ c₂ g₁ : σ₁ eg : f₁ e₁ = some g₁ g₂ : σ₂ gg : tr g₁ g₂ cg : Reaches₁ f₂ c₂ g₂ d' : σ₂ cd' : d' ∈ f₂ c₂ dg : ReflTransGen (fun a b => b ∈ f₂ a) d' g₂ ⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ cases Option.mem_unique cd cd' case tail.intro.intro.intro.intro.inl.some.intro.intro.refl σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ c₂ d₂ : σ₂ a✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂ cd : d₂ ∈ f₂ c₂ e₁ : σ₁ ae : Reaches f₁ a₁ e₁ ce : Reaches f₂ c₂ c₂ ee : tr e₁ c₂ g₁ : σ₁ eg : f₁ e₁ = some g₁ g₂ : σ₂ gg : tr g₁ g₂ cg : Reaches₁ f₂ c₂ g₂ cd' : d₂ ∈ f₂ c₂ dg : ReflTransGen (fun a b => b ∈ f₂ a) d₂ g₂ ⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ exact ⟨_, _, dg, gg, ae.tail eg⟩ case tail.intro.intro.intro.intro.inr.intro.intro σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ c₂ d₂ : σ₂ a✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂ cd : d₂ ∈ f₂ c₂ e₁ : σ₁ e₂ : σ₂ ce : Reaches f₂ c₂ e₂ ee : tr e₁ e₂ ae : Reaches f₁ a₁ e₁ d' : σ₂ cd' : d' ∈ f₂ c₂ de : ReflTransGen (fun a b => b ∈ f₂ a) d' e₂ ⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ cases Option.mem_unique cd cd' case tail.intro.intro.intro.intro.inr.intro.intro.refl σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ c₂ d₂ : σ₂ a✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂ cd : d₂ ∈ f₂ c₂ e₁ : σ₁ e₂ : σ₂ ce : Reaches f₂ c₂ e₂ ee : tr e₁ e₂ ae : Reaches f₁ a₁ e₁ cd' : d₂ ∈ f₂ c₂ de : ReflTransGen (fun a b => b ∈ f₂ a) d₂ e₂ ⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ exact ⟨_, _, de, ee, ae⟩ ** Qed
Turing.frespects_eq σ₁ : Type u_1 σ₂ : Type u_2 f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ a₂ b₂ : σ₂ h : f₂ a₂ = f₂ b₂ ⊢ FRespects f₂ tr a₂ none ↔ FRespects f₂ tr b₂ none unfold FRespects σ₁ : Type u_1 σ₂ : Type u_2 f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ a₂ b₂ : σ₂ h : f₂ a₂ = f₂ b₂ ⊢ (match none with \| some b₁ => Reaches₁ f₂ a₂ (tr b₁) \| none => f₂ a₂ = none) ↔ match none with \| some b₁ => Reaches₁ f₂ b₂ (tr b₁) \| none => f₂ b₂ = none rw [h] ** Qed
Turing.fun_respects σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ a₁ : σ₁ ⊢ (∀ ⦃a₂ : σ₂⦄, (fun a b => tr a = b) a₁ a₂ → match f₁ a₁ with \| some b₁ => ∃ b₂, (fun a b => tr a = b) b₁ b₂ ∧ Reaches₁ f₂ a₂ b₂ \| none => f₂ a₂ = none) ↔ FRespects f₂ tr (tr a₁) (f₁ a₁) cases f₁ a₁ <;> simp only [FRespects, Respects, exists_eq_left', forall_eq'] ** Qed
Turing.tr_eval' σ₁ σ₂ : Type u_1 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ H : Respects f₁ f₂ fun a b => tr a = b a₁ : σ₁ b₂ : σ₂ h : b₂ ∈ tr <$> eval f₁ a₁ ⊢ b₂ ∈ eval f₂ (tr a₁) rcases (Part.mem_map_iff _).1 h with ⟨b₁, ab, bb⟩ case intro.intro σ₁ σ₂ : Type u_1 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ H : Respects f₁ f₂ fun a b => tr a = b a₁ : σ₁ b₂ : σ₂ h : b₂ ∈ tr <$> eval f₁ a₁ b₁ : σ₁ ab : b₁ ∈ eval f₁ a₁ bb : tr b₁ = b₂ ⊢ b₂ ∈ eval f₂ (tr a₁) rcases tr_eval H rfl ab with ⟨_, rfl, h⟩ case intro.intro.intro.intro σ₁ σ₂ : Type u_1 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ H : Respects f₁ f₂ fun a b => tr a = b a₁ : σ₁ b₂ : σ₂ h✝ : b₂ ∈ tr <$> eval f₁ a₁ b₁ : σ₁ ab : b₁ ∈ eval f₁ a₁ bb : tr b₁ = b₂ h : tr b₁ ∈ eval f₂ (tr a₁) ⊢ b₂ ∈ eval f₂ (tr a₁) rwa [bb] at h ** Qed
Turing.TM0.step_supports Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 inst✝ : Inhabited Λ M : Machine₀ S : Set Λ ss : Supports M S ⊢ ∀ {c c' : Cfg₀}, c' ∈ step M c → c.q ∈ S → c'.q ∈ S intro ⟨q, T⟩ c' h₁ h₂ Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 inst✝ : Inhabited Λ M : Machine₀ S : Set Λ ss : Supports M S q : Λ T : Tape Γ c' : Cfg₀ h₁ : c' ∈ step M { q := q, Tape := T } h₂ : { q := q, Tape := T }.q ∈ S ⊢ c'.q ∈ S rcases Option.map_eq_some'.1 h₁ with ⟨⟨q', a⟩, h, rfl⟩ case intro.mk.intro Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 inst✝ : Inhabited Λ M : Machine₀ S : Set Λ ss : Supports M S q : Λ T : Tape Γ h₂ : { q := q, Tape := T }.q ∈ S q' : Λ a : Stmt₀ h : M q T.head = some (q', a) h₁ : (fun x => match x with \| (q', a) => { q := q', Tape := match a with \| Stmt.move d => Tape.move d T \| Stmt.write a => Tape.write a T }) (q', a) ∈ step M { q := q, Tape := T } ⊢ ((fun x => match x with \| (q', a) => { q := q', Tape := match a with \| Stmt.move d => Tape.move d T \| Stmt.write a => Tape.write a T }) (q', a)).q ∈ S exact ss.2 h h₂ ** Qed
Turing.TM0.univ_supports Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 inst✝ : Inhabited Λ M : Machine₀ ⊢ Supports M Set.univ constructor <;> intros <;> apply Set.mem_univ ** Qed
Turing.TM1.stmts₁_self Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q : Stmt₁ ⊢ q ∈ stmts₁ q cases q <;> simp only [stmts₁, Finset.mem_insert_self, Finset.mem_singleton_self] ** Qed
Turing.TM1.stmts₁_trans Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁ q₂ : Stmt₁ ⊢ q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ intro h₁₂ q₀ h₀₁ Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁ q₂ : Stmt₁ h₁₂ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₁ h₀₁ : q₀ ∈ stmts₁ q₁ ⊢ q₀ ∈ stmts₁ q₂ induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;> simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂ case goto Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁ q₂ : Stmt₁ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₁ h₀₁ : q₀ ∈ stmts₁ q₁ a✝ : Γ → σ → Λ h₁₂ : q₁ = goto a✝ ⊢ q₀ ∈ {goto a✝} case halt Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁ q₂ : Stmt₁ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₁ h₀₁ : q₀ ∈ stmts₁ q₁ h₁₂ : q₁ = halt ⊢ q₀ ∈ {halt} case goto l => subst h₁₂; exact h₀₁ case halt Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁ q₂ : Stmt₁ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₁ h₀₁ : q₀ ∈ stmts₁ q₁ h₁₂ : q₁ = halt ⊢ q₀ ∈ {halt} case halt => subst h₁₂; exact h₀₁ case load Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁ q₂ : Stmt₁ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₁ h₀₁ : q₀ ∈ stmts₁ q₁ a✝ : Γ → σ → σ q : Stmt₁ IH : q₁ ∈ stmts₁ q → q₀ ∈ stmts₁ q h₁₂ : q₁ = load a✝ q ∨ q₁ ∈ stmts₁ q ⊢ q₀ ∈ insert (load a✝ q) (stmts₁ q) case branch Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁ q₂ : Stmt₁ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₁ h₀₁ : q₀ ∈ stmts₁ q₁ a✝² : Γ → σ → Bool a✝¹ a✝ : Stmt₁ a_ih✝¹ : q₁ ∈ stmts₁ a✝¹ → q₀ ∈ stmts₁ a✝¹ a_ih✝ : q₁ ∈ stmts₁ a✝ → q₀ ∈ stmts₁ a✝ h₁₂ : q₁ = branch a✝² a✝¹ a✝ ∨ q₁ ∈ stmts₁ a✝¹ ∨ q₁ ∈ stmts₁ a✝ ⊢ q₀ ∈ insert (branch a✝² a✝¹ a✝) (stmts₁ a✝¹ ∪ stmts₁ a✝) case goto Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁ q₂ : Stmt₁ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₁ h₀₁ : q₀ ∈ stmts₁ q₁ a✝ : Γ → σ → Λ h₁₂ : q₁ = goto a✝ ⊢ q₀ ∈ {goto a✝} case halt Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁ q₂ : Stmt₁ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₁ h₀₁ : q₀ ∈ stmts₁ q₁ h₁₂ : q₁ = halt ⊢ q₀ ∈ {halt} rcases h₁₂ with (rfl \| h₁₂) case load.inl Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₂ q₀ : Stmt₁ a✝ : Γ → σ → σ q : Stmt₁ h₁₂ : load a✝ q ∈ stmts₁ q₂ h₀₁ : q₀ ∈ stmts₁ (load a✝ q) IH : load a✝ q ∈ stmts₁ q → q₀ ∈ stmts₁ q ⊢ q₀ ∈ insert (load a✝ q) (stmts₁ q) unfold stmts₁ at h₀₁ case load.inl Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₂ q₀ : Stmt₁ a✝ : Γ → σ → σ q : Stmt₁ h₁₂ : load a✝ q ∈ stmts₁ q₂ IH : load a✝ q ∈ stmts₁ q → q₀ ∈ stmts₁ q h₀₁ : q₀ ∈ insert (load a✝ q) (stmts₁ q) ⊢ q₀ ∈ insert (load a✝ q) (stmts₁ q) exact h₀₁ case load.inr Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁ q₂ : Stmt₁ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₁ h₀₁ : q₀ ∈ stmts₁ q₁ a✝ : Γ → σ → σ q : Stmt₁ IH : q₁ ∈ stmts₁ q → q₀ ∈ stmts₁ q h₁₂ : q₁ ∈ stmts₁ q ⊢ q₀ ∈ insert (load a✝ q) (stmts₁ q) exact Finset.mem_insert_of_mem (IH h₁₂) Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁✝ q₂✝ : Stmt₁ h₁₂✝ : q₁✝ ∈ stmts₁ q₂✝ q₀ : Stmt₁ h₀₁ : q₀ ∈ stmts₁ q₁✝ p : Γ → σ → Bool q₁ q₂ : Stmt₁ IH₁ : q₁✝ ∈ stmts₁ q₁ → q₀ ∈ stmts₁ q₁ IH₂ : q₁✝ ∈ stmts₁ q₂ → q₀ ∈ stmts₁ q₂ h₁₂ : q₁✝ = branch p q₁ q₂ ∨ q₁✝ ∈ stmts₁ q₁ ∨ q₁✝ ∈ stmts₁ q₂ ⊢ q₀ ∈ insert (branch p q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂) rcases h₁₂ with (rfl \| h₁₂ \| h₁₂) case inl Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₂✝ q₀ : Stmt₁ p : Γ → σ → Bool q₁ q₂ : Stmt₁ h₁₂ : branch p q₁ q₂ ∈ stmts₁ q₂✝ h₀₁ : q₀ ∈ stmts₁ (branch p q₁ q₂) IH₁ : branch p q₁ q₂ ∈ stmts₁ q₁ → q₀ ∈ stmts₁ q₁ IH₂ : branch p q₁ q₂ ∈ stmts₁ q₂ → q₀ ∈ stmts₁ q₂ ⊢ q₀ ∈ insert (branch p q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂) unfold stmts₁ at h₀₁ case inl Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₂✝ q₀ : Stmt₁ p : Γ → σ → Bool q₁ q₂ : Stmt₁ h₁₂ : branch p q₁ q₂ ∈ stmts₁ q₂✝ IH₁ : branch p q₁ q₂ ∈ stmts₁ q₁ → q₀ ∈ stmts₁ q₁ IH₂ : branch p q₁ q₂ ∈ stmts₁ q₂ → q₀ ∈ stmts₁ q₂ h₀₁ : q₀ ∈ insert (branch p q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂) ⊢ q₀ ∈ insert (branch p q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂) exact h₀₁ case inr.inl Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁✝ q₂✝ : Stmt₁ h₁₂✝ : q₁✝ ∈ stmts₁ q₂✝ q₀ : Stmt₁ h₀₁ : q₀ ∈ stmts₁ q₁✝ p : Γ → σ → Bool q₁ q₂ : Stmt₁ IH₁ : q₁✝ ∈ stmts₁ q₁ → q₀ ∈ stmts₁ q₁ IH₂ : q₁✝ ∈ stmts₁ q₂ → q₀ ∈ stmts₁ q₂ h₁₂ : q₁✝ ∈ stmts₁ q₁ ⊢ q₀ ∈ insert (branch p q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂) exact Finset.mem_insert_of_mem (Finset.mem_union_left _ <\| IH₁ h₁₂) case inr.inr Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁✝ q₂✝ : Stmt₁ h₁₂✝ : q₁✝ ∈ stmts₁ q₂✝ q₀ : Stmt₁ h₀₁ : q₀ ∈ stmts₁ q₁✝ p : Γ → σ → Bool q₁ q₂ : Stmt₁ IH₁ : q₁✝ ∈ stmts₁ q₁ → q₀ ∈ stmts₁ q₁ IH₂ : q₁✝ ∈ stmts₁ q₂ → q₀ ∈ stmts₁ q₂ h₁₂ : q₁✝ ∈ stmts₁ q₂ ⊢ q₀ ∈ insert (branch p q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂) exact Finset.mem_insert_of_mem (Finset.mem_union_right _ <\| IH₂ h₁₂) Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁ q₂ : Stmt₁ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₁ h₀₁ : q₀ ∈ stmts₁ q₁ l : Γ → σ → Λ h₁₂ : q₁ = goto l ⊢ q₀ ∈ {goto l} subst h₁₂ Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₂ q₀ : Stmt₁ l : Γ → σ → Λ h₁₂ : goto l ∈ stmts₁ q₂ h₀₁ : q₀ ∈ stmts₁ (goto l) ⊢ q₀ ∈ {goto l} exact h₀₁ Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁ q₂ : Stmt₁ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₁ h₀₁ : q₀ ∈ stmts₁ q₁ h₁₂ : q₁ = halt ⊢ q₀ ∈ {halt} subst h₁₂ Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₂ q₀ : Stmt₁ h₁₂ : halt ∈ stmts₁ q₂ h₀₁ : q₀ ∈ stmts₁ halt ⊢ q₀ ∈ {halt} exact h₀₁ ** Qed
Turing.TM1.stmts₁_supportsStmt_mono Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h : q₁ ∈ stmts₁ q₂ hs : SupportsStmt S q₂ ⊢ SupportsStmt S q₁ induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h hs case move Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝ : Dir q : Stmt₁ IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁ hs : SupportsStmt S q h : q₁ = move a✝ q ∨ q₁ ∈ stmts₁ q ⊢ SupportsStmt S q₁ case write Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝ : Γ → σ → Γ q : Stmt₁ IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁ hs : SupportsStmt S q h : q₁ = write a✝ q ∨ q₁ ∈ stmts₁ q ⊢ SupportsStmt S q₁ case load Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝ : Γ → σ → σ q : Stmt₁ IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁ hs : SupportsStmt S q h : q₁ = load a✝ q ∨ q₁ ∈ stmts₁ q ⊢ SupportsStmt S q₁ case branch Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝² : Γ → σ → Bool a✝¹ a✝ : Stmt₁ a_ih✝¹ : q₁ ∈ stmts₁ a✝¹ → SupportsStmt S a✝¹ → SupportsStmt S q₁ a_ih✝ : q₁ ∈ stmts₁ a✝ → SupportsStmt S a✝ → SupportsStmt S q₁ hs : SupportsStmt S a✝¹ ∧ SupportsStmt S a✝ h : q₁ = branch a✝² a✝¹ a✝ ∨ q₁ ∈ stmts₁ a✝¹ ∨ q₁ ∈ stmts₁ a✝ ⊢ SupportsStmt S q₁ case goto Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝ : Γ → σ → Λ hs : ∀ (a : Γ) (v : σ), a✝ a v ∈ S h : q₁ = goto a✝ ⊢ SupportsStmt S q₁ case halt Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ hs : True h : q₁ = halt ⊢ SupportsStmt S q₁ iterate 3 rcases h with (rfl \| h) <;> [exact hs; exact IH h hs] case branch Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝² : Γ → σ → Bool a✝¹ a✝ : Stmt₁ a_ih✝¹ : q₁ ∈ stmts₁ a✝¹ → SupportsStmt S a✝¹ → SupportsStmt S q₁ a_ih✝ : q₁ ∈ stmts₁ a✝ → SupportsStmt S a✝ → SupportsStmt S q₁ hs : SupportsStmt S a✝¹ ∧ SupportsStmt S a✝ h : q₁ = branch a✝² a✝¹ a✝ ∨ q₁ ∈ stmts₁ a✝¹ ∨ q₁ ∈ stmts₁ a✝ ⊢ SupportsStmt S q₁ case goto Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝ : Γ → σ → Λ hs : ∀ (a : Γ) (v : σ), a✝ a v ∈ S h : q₁ = goto a✝ ⊢ SupportsStmt S q₁ case halt Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ hs : True h : q₁ = halt ⊢ SupportsStmt S q₁ case branch p q₁ q₂ IH₁ IH₂ => rcases h with (rfl \| h \| h); exacts [hs, IH₁ h hs.1, IH₂ h hs.2] case goto Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝ : Γ → σ → Λ hs : ∀ (a : Γ) (v : σ), a✝ a v ∈ S h : q₁ = goto a✝ ⊢ SupportsStmt S q₁ case halt Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ hs : True h : q₁ = halt ⊢ SupportsStmt S q₁ case goto l => subst h; exact hs case halt Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ hs : True h : q₁ = halt ⊢ SupportsStmt S q₁ case halt => subst h; trivial case load Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝ : Γ → σ → σ q : Stmt₁ IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁ hs : SupportsStmt S q h : q₁ = load a✝ q ∨ q₁ ∈ stmts₁ q ⊢ SupportsStmt S q₁ case branch Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝² : Γ → σ → Bool a✝¹ a✝ : Stmt₁ a_ih✝¹ : q₁ ∈ stmts₁ a✝¹ → SupportsStmt S a✝¹ → SupportsStmt S q₁ a_ih✝ : q₁ ∈ stmts₁ a✝ → SupportsStmt S a✝ → SupportsStmt S q₁ hs : SupportsStmt S a✝¹ ∧ SupportsStmt S a✝ h : q₁ = branch a✝² a✝¹ a✝ ∨ q₁ ∈ stmts₁ a✝¹ ∨ q₁ ∈ stmts₁ a✝ ⊢ SupportsStmt S q₁ case goto Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝ : Γ → σ → Λ hs : ∀ (a : Γ) (v : σ), a✝ a v ∈ S h : q₁ = goto a✝ ⊢ SupportsStmt S q₁ case halt Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ hs : True h : q₁ = halt ⊢ SupportsStmt S q₁ rcases h with (rfl \| h) <;> [exact hs; exact IH h hs] Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁✝ q₂✝ : Stmt₁ h✝ : q₁✝ ∈ stmts₁ q₂✝ hs✝ : SupportsStmt S q₂✝ p : Γ → σ → Bool q₁ q₂ : Stmt₁ IH₁ : q₁✝ ∈ stmts₁ q₁ → SupportsStmt S q₁ → SupportsStmt S q₁✝ IH₂ : q₁✝ ∈ stmts₁ q₂ → SupportsStmt S q₂ → SupportsStmt S q₁✝ hs : SupportsStmt S q₁ ∧ SupportsStmt S q₂ h : q₁✝ = branch p q₁ q₂ ∨ q₁✝ ∈ stmts₁ q₁ ∨ q₁✝ ∈ stmts₁ q₂ ⊢ SupportsStmt S q₁✝ rcases h with (rfl \| h \| h) case inl Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₂✝ : Stmt₁ hs✝ : SupportsStmt S q₂✝ p : Γ → σ → Bool q₁ q₂ : Stmt₁ hs : SupportsStmt S q₁ ∧ SupportsStmt S q₂ h : branch p q₁ q₂ ∈ stmts₁ q₂✝ IH₁ : branch p q₁ q₂ ∈ stmts₁ q₁ → SupportsStmt S q₁ → SupportsStmt S (branch p q₁ q₂) IH₂ : branch p q₁ q₂ ∈ stmts₁ q₂ → SupportsStmt S q₂ → SupportsStmt S (branch p q₁ q₂) ⊢ SupportsStmt S (branch p q₁ q₂) case inr.inl Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁✝ q₂✝ : Stmt₁ h✝ : q₁✝ ∈ stmts₁ q₂✝ hs✝ : SupportsStmt S q₂✝ p : Γ → σ → Bool q₁ q₂ : Stmt₁ IH₁ : q₁✝ ∈ stmts₁ q₁ → SupportsStmt S q₁ → SupportsStmt S q₁✝ IH₂ : q₁✝ ∈ stmts₁ q₂ → SupportsStmt S q₂ → SupportsStmt S q₁✝ hs : SupportsStmt S q₁ ∧ SupportsStmt S q₂ h : q₁✝ ∈ stmts₁ q₁ ⊢ SupportsStmt S q₁✝ case inr.inr Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁✝ q₂✝ : Stmt₁ h✝ : q₁✝ ∈ stmts₁ q₂✝ hs✝ : SupportsStmt S q₂✝ p : Γ → σ → Bool q₁ q₂ : Stmt₁ IH₁ : q₁✝ ∈ stmts₁ q₁ → SupportsStmt S q₁ → SupportsStmt S q₁✝ IH₂ : q₁✝ ∈ stmts₁ q₂ → SupportsStmt S q₂ → SupportsStmt S q₁✝ hs : SupportsStmt S q₁ ∧ SupportsStmt S q₂ h : q₁✝ ∈ stmts₁ q₂ ⊢ SupportsStmt S q₁✝ exacts [hs, IH₁ h hs.1, IH₂ h hs.2] Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ l : Γ → σ → Λ hs : ∀ (a : Γ) (v : σ), l a v ∈ S h : q₁ = goto l ⊢ SupportsStmt S q₁ subst h Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₂ : Stmt₁ hs✝ : SupportsStmt S q₂ l : Γ → σ → Λ hs : ∀ (a : Γ) (v : σ), l a v ∈ S h : goto l ∈ stmts₁ q₂ ⊢ SupportsStmt S (goto l) exact hs Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ hs : True h : q₁ = halt ⊢ SupportsStmt S q₁ subst h Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₂ : Stmt₁ hs✝ : SupportsStmt S q₂ hs : True h : halt ∈ stmts₁ q₂ ⊢ SupportsStmt S halt trivial ** Qed
Turing.TM1.step_supports Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v : σ T : Tape Γ c' : Cfg₁ h₁ : c' ∈ step M { l := some l₁, var := v, Tape := T } h₂ : { l := some l₁, var := v, Tape := T }.l ∈ ↑Finset.insertNone S ⊢ c'.l ∈ ↑Finset.insertNone S replace h₂ := ss.2 _ (Finset.some_mem_insertNone.1 h₂) Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v : σ T : Tape Γ c' : Cfg₁ h₁ : c' ∈ step M { l := some l₁, var := v, Tape := T } h₂ : SupportsStmt S (M l₁) ⊢ c'.l ∈ ↑Finset.insertNone S simp only [step, Option.mem_def, Option.some.injEq] at h₁ Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v : σ T : Tape Γ c' : Cfg₁ h₂ : SupportsStmt S (M l₁) h₁ : stepAux (M l₁) v T = c' ⊢ c'.l ∈ ↑Finset.insertNone S subst c' Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v : σ T : Tape Γ h₂ : SupportsStmt S (M l₁) ⊢ (stepAux (M l₁) v T).l ∈ ↑Finset.insertNone S revert h₂ Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v : σ T : Tape Γ ⊢ SupportsStmt S (M l₁) → (stepAux (M l₁) v T).l ∈ ↑Finset.insertNone S induction' M l₁ with _ q IH _ q IH _ q IH generalizing v T <;> intro hs case move Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ a✝ : Dir q : Stmt₁ IH : ∀ (v : σ) (T : Tape Γ), SupportsStmt S q → (stepAux q v T).l ∈ ↑Finset.insertNone S v : σ T : Tape Γ hs : SupportsStmt S (move a✝ q) ⊢ (stepAux (move a✝ q) v T).l ∈ ↑Finset.insertNone S case write Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ a✝ : Γ → σ → Γ q : Stmt₁ IH : ∀ (v : σ) (T : Tape Γ), SupportsStmt S q → (stepAux q v T).l ∈ ↑Finset.insertNone S v : σ T : Tape Γ hs : SupportsStmt S (write a✝ q) ⊢ (stepAux (write a✝ q) v T).l ∈ ↑Finset.insertNone S case load Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ a✝ : Γ → σ → σ q : Stmt₁ IH : ∀ (v : σ) (T : Tape Γ), SupportsStmt S q → (stepAux q v T).l ∈ ↑Finset.insertNone S v : σ T : Tape Γ hs : SupportsStmt S (load a✝ q) ⊢ (stepAux (load a✝ q) v T).l ∈ ↑Finset.insertNone S case branch Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ a✝² : Γ → σ → Bool a✝¹ a✝ : Stmt₁ a_ih✝¹ : ∀ (v : σ) (T : Tape Γ), SupportsStmt S a✝¹ → (stepAux a✝¹ v T).l ∈ ↑Finset.insertNone S a_ih✝ : ∀ (v : σ) (T : Tape Γ), SupportsStmt S a✝ → (stepAux a✝ v T).l ∈ ↑Finset.insertNone S v : σ T : Tape Γ hs : SupportsStmt S (branch a✝² a✝¹ a✝) ⊢ (stepAux (branch a✝² a✝¹ a✝) v T).l ∈ ↑Finset.insertNone S case goto Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ a✝ : Γ → σ → Λ v : σ T : Tape Γ hs : SupportsStmt S (goto a✝) ⊢ (stepAux (goto a✝) v T).l ∈ ↑Finset.insertNone S case halt Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ v : σ T : Tape Γ hs : SupportsStmt S halt ⊢ (stepAux halt v T).l ∈ ↑Finset.insertNone S iterate 3 exact IH _ _ hs case goto Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ a✝ : Γ → σ → Λ v : σ T : Tape Γ hs : SupportsStmt S (goto a✝) ⊢ (stepAux (goto a✝) v T).l ∈ ↑Finset.insertNone S case halt Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ v : σ T : Tape Γ hs : SupportsStmt S halt ⊢ (stepAux halt v T).l ∈ ↑Finset.insertNone S case goto => exact Finset.some_mem_insertNone.2 (hs _ _) case halt Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ v : σ T : Tape Γ hs : SupportsStmt S halt ⊢ (stepAux halt v T).l ∈ ↑Finset.insertNone S case halt => apply Multiset.mem_cons_self case load Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ a✝ : Γ → σ → σ q : Stmt₁ IH : ∀ (v : σ) (T : Tape Γ), SupportsStmt S q → (stepAux q v T).l ∈ ↑Finset.insertNone S v : σ T : Tape Γ hs : SupportsStmt S (load a✝ q) ⊢ (stepAux (load a✝ q) v T).l ∈ ↑Finset.insertNone S case branch Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ a✝² : Γ → σ → Bool a✝¹ a✝ : Stmt₁ a_ih✝¹ : ∀ (v : σ) (T : Tape Γ), SupportsStmt S a✝¹ → (stepAux a✝¹ v T).l ∈ ↑Finset.insertNone S a_ih✝ : ∀ (v : σ) (T : Tape Γ), SupportsStmt S a✝ → (stepAux a✝ v T).l ∈ ↑Finset.insertNone S v : σ T : Tape Γ hs : SupportsStmt S (branch a✝² a✝¹ a✝) ⊢ (stepAux (branch a✝² a✝¹ a✝) v T).l ∈ ↑Finset.insertNone S case goto Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ a✝ : Γ → σ → Λ v : σ T : Tape Γ hs : SupportsStmt S (goto a✝) ⊢ (stepAux (goto a✝) v T).l ∈ ↑Finset.insertNone S case halt Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ v : σ T : Tape Γ hs : SupportsStmt S halt ⊢ (stepAux halt v T).l ∈ ↑Finset.insertNone S exact IH _ _ hs Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ p : Γ → σ → Bool q₁' q₂' : Stmt₁ IH₁ : ∀ (v : σ) (T : Tape Γ), SupportsStmt S q₁' → (stepAux q₁' v T).l ∈ ↑Finset.insertNone S IH₂ : ∀ (v : σ) (T : Tape Γ), SupportsStmt S q₂' → (stepAux q₂' v T).l ∈ ↑Finset.insertNone S v : σ T : Tape Γ hs : SupportsStmt S (branch p q₁' q₂') ⊢ (stepAux (branch p q₁' q₂') v T).l ∈ ↑Finset.insertNone S unfold stepAux Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ p : Γ → σ → Bool q₁' q₂' : Stmt₁ IH₁ : ∀ (v : σ) (T : Tape Γ), SupportsStmt S q₁' → (stepAux q₁' v T).l ∈ ↑Finset.insertNone S IH₂ : ∀ (v : σ) (T : Tape Γ), SupportsStmt S q₂' → (stepAux q₂' v T).l ∈ ↑Finset.insertNone S v : σ T : Tape Γ hs : SupportsStmt S (branch p q₁' q₂') ⊢ (bif p T.head v then stepAux q₁' v T else stepAux q₂' v T).l ∈ ↑Finset.insertNone S cases p T.1 v case false Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ p : Γ → σ → Bool q₁' q₂' : Stmt₁ IH₁ : ∀ (v : σ) (T : Tape Γ), SupportsStmt S q₁' → (stepAux q₁' v T).l ∈ ↑Finset.insertNone S IH₂ : ∀ (v : σ) (T : Tape Γ), SupportsStmt S q₂' → (stepAux q₂' v T).l ∈ ↑Finset.insertNone S v : σ T : Tape Γ hs : SupportsStmt S (branch p q₁' q₂') ⊢ (bif false then stepAux q₁' v T else stepAux q₂' v T).l ∈ ↑Finset.insertNone S exact IH₂ _ _ hs.2 case true Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ p : Γ → σ → Bool q₁' q₂' : Stmt₁ IH₁ : ∀ (v : σ) (T : Tape Γ), SupportsStmt S q₁' → (stepAux q₁' v T).l ∈ ↑Finset.insertNone S IH₂ : ∀ (v : σ) (T : Tape Γ), SupportsStmt S q₂' → (stepAux q₂' v T).l ∈ ↑Finset.insertNone S v : σ T : Tape Γ hs : SupportsStmt S (branch p q₁' q₂') ⊢ (bif true then stepAux q₁' v T else stepAux q₂' v T).l ∈ ↑Finset.insertNone S exact IH₁ _ _ hs.1 Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ a✝ : Γ → σ → Λ v : σ T : Tape Γ hs : SupportsStmt S (goto a✝) ⊢ (stepAux (goto a✝) v T).l ∈ ↑Finset.insertNone S exact Finset.some_mem_insertNone.2 (hs _ _) Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ v : σ T : Tape Γ hs : SupportsStmt S halt ⊢ (stepAux halt v T).l ∈ ↑Finset.insertNone S apply Multiset.mem_cons_self ** Qed
Turing.TM1to1.stepAux_move Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ d : Dir q : Stmt Bool Λ' σ v : σ T : Tape Bool ⊢ stepAux (move d q) v T = stepAux q v ((Tape.move d)^[n] T) suffices ∀ i, stepAux ((Stmt.move d)^[i] q) v T = stepAux q v ((Tape.move d)^[i] T) from this n Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ d : Dir q : Stmt Bool Λ' σ v : σ T : Tape Bool ⊢ ∀ (i : ℕ), stepAux ((Stmt.move d)^[i] q) v T = stepAux q v ((Tape.move d)^[i] T) intro i Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ d : Dir q : Stmt Bool Λ' σ v : σ T : Tape Bool i : ℕ ⊢ stepAux ((Stmt.move d)^[i] q) v T = stepAux q v ((Tape.move d)^[i] T) induction' i with i IH generalizing T case succ Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ d : Dir q : Stmt Bool Λ' σ v : σ T✝ : Tape Bool i : ℕ IH : ∀ (T : Tape Bool), stepAux ((Stmt.move d)^[i] q) v T = stepAux q v ((Tape.move d)^[i] T) T : Tape Bool ⊢ stepAux ((Stmt.move d)^[Nat.succ i] q) v T = stepAux q v ((Tape.move d)^[Nat.succ i] T) rw [iterate_succ', iterate_succ] case succ Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ d : Dir q : Stmt Bool Λ' σ v : σ T✝ : Tape Bool i : ℕ IH : ∀ (T : Tape Bool), stepAux ((Stmt.move d)^[i] q) v T = stepAux q v ((Tape.move d)^[i] T) T : Tape Bool ⊢ stepAux ((Stmt.move d ∘ (Stmt.move d)^[i]) q) v T = stepAux q v (((Tape.move d)^[i] ∘ Tape.move d) T) simp only [stepAux, Function.comp_apply] case succ Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ d : Dir q : Stmt Bool Λ' σ v : σ T✝ : Tape Bool i : ℕ IH : ∀ (T : Tape Bool), stepAux ((Stmt.move d)^[i] q) v T = stepAux q v ((Tape.move d)^[i] T) T : Tape Bool ⊢ stepAux ((Stmt.move d)^[i] q) v (Tape.move d T) = stepAux q v ((Tape.move d)^[i] (Tape.move d T)) rw [IH] case zero Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ d : Dir q : Stmt Bool Λ' σ v : σ T✝ T : Tape Bool ⊢ stepAux ((Stmt.move d)^[Nat.zero] q) v T = stepAux q v ((Tape.move d)^[Nat.zero] T) rfl ** Qed
Turing.TM1to1.supportsStmt_move Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ S : Finset Λ' d : Dir q : Stmt Bool Λ' σ ⊢ SupportsStmt S (move d q) = SupportsStmt S q suffices ∀ {i}, SupportsStmt S ((Stmt.move d)^[i] q) = _ from this Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ S : Finset Λ' d : Dir q : Stmt Bool Λ' σ ⊢ ∀ {i : ℕ}, SupportsStmt S ((Stmt.move d)^[i] q) = SupportsStmt S q intro i Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ S : Finset Λ' d : Dir q : Stmt Bool Λ' σ i : ℕ ⊢ SupportsStmt S ((Stmt.move d)^[i] q) = SupportsStmt S q induction i generalizing q <;> simp only [, iterate] * case succ Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ S : Finset Λ' d : Dir n✝ : ℕ n_ih✝ : ∀ {q : Stmt Bool Λ' σ}, SupportsStmt S ((Stmt.move d)^[n✝] q) = SupportsStmt S q q : Stmt Bool Λ' σ ⊢ SupportsStmt S (Stmt.move d q) = SupportsStmt S q rfl Qed
Turing.TM1to1.supportsStmt_write Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ S : Finset Λ' l : List Bool q : Stmt Bool Λ' σ ⊢ SupportsStmt S (write l q) = SupportsStmt S q induction' l with _ l IH <;> simp only [write, SupportsStmt, ] * Qed
Turing.TM1to1.supportsStmt_read Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ S : Finset Λ' f✝ : Γ → Stmt Bool Λ' σ this : ∀ (i : ℕ) (f : Vector Bool i → Stmt Bool Λ' σ), (∀ (v : Vector Bool i), SupportsStmt S (f v)) → SupportsStmt S (readAux i f) hf : ∀ (a : Γ), SupportsStmt S (f✝ a) ⊢ ∀ (v : Vector Bool n), SupportsStmt S (move Dir.left (f✝ (dec v))) intro Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ S : Finset Λ' f✝ : Γ → Stmt Bool Λ' σ this : ∀ (i : ℕ) (f : Vector Bool i → Stmt Bool Λ' σ), (∀ (v : Vector Bool i), SupportsStmt S (f v)) → SupportsStmt S (readAux i f) hf : ∀ (a : Γ), SupportsStmt S (f✝ a) v✝ : Vector Bool n ⊢ SupportsStmt S (move Dir.left (f✝ (dec v✝))) simp only [supportsStmt_move, hf] Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ S : Finset Λ' f✝ : Γ → Stmt Bool Λ' σ i : ℕ f : Vector Bool i → Stmt Bool Λ' σ hf : ∀ (v : Vector Bool i), SupportsStmt S (f v) ⊢ SupportsStmt S (readAux i f) induction' i with i IH case succ Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ S : Finset Λ' f✝¹ : Γ → Stmt Bool Λ' σ i✝ : ℕ f✝ : Vector Bool i✝ → Stmt Bool Λ' σ hf✝ : ∀ (v : Vector Bool i✝), SupportsStmt S (f✝ v) i : ℕ IH : ∀ (f : Vector Bool i → Stmt Bool Λ' σ), (∀ (v : Vector Bool i), SupportsStmt S (f v)) → SupportsStmt S (readAux i f) f : Vector Bool (Nat.succ i) → Stmt Bool Λ' σ hf : ∀ (v : Vector Bool (Nat.succ i)), SupportsStmt S (f v) ⊢ SupportsStmt S (readAux (Nat.succ i) f) constructor <;> apply IH <;> intro <;> apply hf case zero Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ S : Finset Λ' f✝¹ : Γ → Stmt Bool Λ' σ i : ℕ f✝ : Vector Bool i → Stmt Bool Λ' σ hf✝ : ∀ (v : Vector Bool i), SupportsStmt S (f✝ v) f : Vector Bool Nat.zero → Stmt Bool Λ' σ hf : ∀ (v : Vector Bool Nat.zero), SupportsStmt S (f v) ⊢ SupportsStmt S (readAux Nat.zero f) exact hf _ ** Qed
Turing.TM1to1.trTape'_move_left Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ L R : ListBlank Γ ⊢ (Tape.move Dir.left)^[n] (trTape' enc0 L R) = trTape' enc0 (ListBlank.tail L) (ListBlank.cons (ListBlank.head L) R) obtain ⟨a, L, rfl⟩ := L.exists_cons case intro.intro Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ R : ListBlank Γ a : Γ L : ListBlank Γ ⊢ (Tape.move Dir.left)^[n] (trTape' enc0 (ListBlank.cons a L) R) = trTape' enc0 (ListBlank.tail (ListBlank.cons a L)) (ListBlank.cons (ListBlank.head (ListBlank.cons a L)) R) simp only [trTape', ListBlank.cons_bind, ListBlank.head_cons, ListBlank.tail_cons] case intro.intro Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ R : ListBlank Γ a : Γ L : ListBlank Γ ⊢ (Tape.move Dir.left)^[n] (Tape.mk' (ListBlank.append (List.reverse (Vector.toList (enc a))) (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x))) (_ : ∃ n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default))) (ListBlank.bind R (fun x => Vector.toList (enc x)) (_ : ∃ n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default))) = Tape.mk' (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x))) (_ : ∃ n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default)) (ListBlank.append (Vector.toList (enc a)) (ListBlank.bind R (fun x => Vector.toList (enc x)) (_ : ∃ n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default))) suffices ∀ {L' R' l₁ l₂} (_ : Vector.toList (enc a) = List.reverseAux l₁ l₂), (Tape.move Dir.left)^[l₁.length] (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) = Tape.mk' L' (ListBlank.append (Vector.toList (enc a)) R') by simpa only [List.length_reverse, Vector.toList_length] using this (List.reverse_reverse _).symm case intro.intro Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ R : ListBlank Γ a : Γ L : ListBlank Γ ⊢ ∀ {L' R' : ListBlank Bool} {l₁ l₂ : List Bool}, Vector.toList (enc a) = List.reverseAux l₁ l₂ → (Tape.move Dir.left)^[List.length l₁] (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) = Tape.mk' L' (ListBlank.append (Vector.toList (enc a)) R') intro _ _ l₁ l₂ e case intro.intro Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ R : ListBlank Γ a : Γ L : ListBlank Γ L'✝ R'✝ : ListBlank Bool l₁ l₂ : List Bool e : Vector.toList (enc a) = List.reverseAux l₁ l₂ ⊢ (Tape.move Dir.left)^[List.length l₁] (Tape.mk' (ListBlank.append l₁ L'✝) (ListBlank.append l₂ R'✝)) = Tape.mk' L'✝ (ListBlank.append (Vector.toList (enc a)) R'✝) induction' l₁ with b l₁ IH generalizing l₂ case intro.intro.cons Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ R : ListBlank Γ a : Γ L : ListBlank Γ L'✝ R'✝ : ListBlank Bool l₁✝ l₂✝ : List Bool e✝ : Vector.toList (enc a) = List.reverseAux l₁✝ l₂✝ b : Bool l₁ : List Bool IH : ∀ {l₂ : List Bool}, Vector.toList (enc a) = List.reverseAux l₁ l₂ → (Tape.move Dir.left)^[List.length l₁] (Tape.mk' (ListBlank.append l₁ L'✝) (ListBlank.append l₂ R'✝)) = Tape.mk' L'✝ (ListBlank.append (Vector.toList (enc a)) R'✝) l₂ : List Bool e : Vector.toList (enc a) = List.reverseAux (b :: l₁) l₂ ⊢ (Tape.move Dir.left)^[List.length (b :: l₁)] (Tape.mk' (ListBlank.append (b :: l₁) L'✝) (ListBlank.append l₂ R'✝)) = Tape.mk' L'✝ (ListBlank.append (Vector.toList (enc a)) R'✝) simp only [List.length, List.cons_append, iterate_succ_apply] case intro.intro.cons Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ R : ListBlank Γ a : Γ L : ListBlank Γ L'✝ R'✝ : ListBlank Bool l₁✝ l₂✝ : List Bool e✝ : Vector.toList (enc a) = List.reverseAux l₁✝ l₂✝ b : Bool l₁ : List Bool IH : ∀ {l₂ : List Bool}, Vector.toList (enc a) = List.reverseAux l₁ l₂ → (Tape.move Dir.left)^[List.length l₁] (Tape.mk' (ListBlank.append l₁ L'✝) (ListBlank.append l₂ R'✝)) = Tape.mk' L'✝ (ListBlank.append (Vector.toList (enc a)) R'✝) l₂ : List Bool e : Vector.toList (enc a) = List.reverseAux (b :: l₁) l₂ ⊢ (Tape.move Dir.left)^[List.length l₁] (Tape.move Dir.left (Tape.mk' (ListBlank.append (b :: l₁) L'✝) (ListBlank.append l₂ R'✝))) = Tape.mk' L'✝ (ListBlank.append (Vector.toList (enc a)) R'✝) convert IH e case h.e'_2.h.e'_4 Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ R : ListBlank Γ a : Γ L : ListBlank Γ L'✝ R'✝ : ListBlank Bool l₁✝ l₂✝ : List Bool e✝ : Vector.toList (enc a) = List.reverseAux l₁✝ l₂✝ b : Bool l₁ : List Bool IH : ∀ {l₂ : List Bool}, Vector.toList (enc a) = List.reverseAux l₁ l₂ → (Tape.move Dir.left)^[List.length l₁] (Tape.mk' (ListBlank.append l₁ L'✝) (ListBlank.append l₂ R'✝)) = Tape.mk' L'✝ (ListBlank.append (Vector.toList (enc a)) R'✝) l₂ : List Bool e : Vector.toList (enc a) = List.reverseAux (b :: l₁) l₂ ⊢ Tape.move Dir.left (Tape.mk' (ListBlank.append (b :: l₁) L'✝) (ListBlank.append l₂ R'✝)) = Tape.mk' (ListBlank.append l₁ L'✝) (ListBlank.append (b :: l₂) R'✝) simp only [ListBlank.tail_cons, ListBlank.append, Tape.move_left_mk', ListBlank.head_cons] Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ R : ListBlank Γ a : Γ L : ListBlank Γ this : ∀ {L' R' : ListBlank Bool} {l₁ l₂ : List Bool}, Vector.toList (enc a) = List.reverseAux l₁ l₂ → (Tape.move Dir.left)^[List.length l₁] (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) = Tape.mk' L' (ListBlank.append (Vector.toList (enc a)) R') ⊢ (Tape.move Dir.left)^[n] (Tape.mk' (ListBlank.append (List.reverse (Vector.toList (enc a))) (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x))) (_ : ∃ n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default))) (ListBlank.bind R (fun x => Vector.toList (enc x)) (_ : ∃ n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default))) = Tape.mk' (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x))) (_ : ∃ n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default)) (ListBlank.append (Vector.toList (enc a)) (ListBlank.bind R (fun x => Vector.toList (enc x)) (_ : ∃ n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default))) simpa only [List.length_reverse, Vector.toList_length] using this (List.reverse_reverse _).symm case intro.intro.nil Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ R : ListBlank Γ a : Γ L : ListBlank Γ L'✝ R'✝ : ListBlank Bool l₁ l₂✝ : List Bool e✝ : Vector.toList (enc a) = List.reverseAux l₁ l₂✝ l₂ : List Bool e : Vector.toList (enc a) = List.reverseAux [] l₂ ⊢ (Tape.move Dir.left)^[List.length []] (Tape.mk' (ListBlank.append [] L'✝) (ListBlank.append l₂ R'✝)) = Tape.mk' L'✝ (ListBlank.append (Vector.toList (enc a)) R'✝) cases e case intro.intro.nil.refl Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ R : ListBlank Γ a : Γ L : ListBlank Γ L'✝ R'✝ : ListBlank Bool l₁ l₂ : List Bool e : Vector.toList (enc a) = List.reverseAux l₁ l₂ ⊢ (Tape.move Dir.left)^[List.length []] (Tape.mk' (ListBlank.append [] L'✝) (ListBlank.append (enc a).1 R'✝)) = Tape.mk' L'✝ (ListBlank.append (Vector.toList (enc a)) R'✝) rfl ** Qed
Turing.TM1to1.trTape'_move_right Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ L R : ListBlank Γ ⊢ (Tape.move Dir.right)^[n] (trTape' enc0 L R) = trTape' enc0 (ListBlank.cons (ListBlank.head R) L) (ListBlank.tail R) suffices ∀ i L, (Tape.move Dir.right)^[i] ((Tape.move Dir.left)^[i] L) = L by refine' (Eq.symm _).trans (this n _) simp only [trTape'_move_left, ListBlank.cons_head_tail, ListBlank.head_cons, ListBlank.tail_cons] Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ L R : ListBlank Γ ⊢ ∀ (i : ℕ) (L : Tape Bool), (Tape.move Dir.right)^[i] ((Tape.move Dir.left)^[i] L) = L intro i _ Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ L R : ListBlank Γ i : ℕ L✝ : Tape Bool ⊢ (Tape.move Dir.right)^[i] ((Tape.move Dir.left)^[i] L✝) = L✝ induction' i with i IH case succ Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ L R : ListBlank Γ L✝ : Tape Bool i : ℕ IH : (Tape.move Dir.right)^[i] ((Tape.move Dir.left)^[i] L✝) = L✝ ⊢ (Tape.move Dir.right)^[Nat.succ i] ((Tape.move Dir.left)^[Nat.succ i] L✝) = L✝ rw [iterate_succ_apply, iterate_succ_apply', Tape.move_left_right, IH] Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ L R : ListBlank Γ this : ∀ (i : ℕ) (L : Tape (?m.376782 i)), (Tape.move Dir.right)^[i] ((Tape.move Dir.left)^[i] L) = L ⊢ (Tape.move Dir.right)^[n] (trTape' enc0 L R) = trTape' enc0 (ListBlank.cons (ListBlank.head R) L) (ListBlank.tail R) refine' (Eq.symm _).trans (this n _) Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ L R : ListBlank Γ this : ∀ (i : ℕ) (L : Tape Bool), (Tape.move Dir.right)^[i] ((Tape.move Dir.left)^[i] L) = L ⊢ (Tape.move Dir.right)^[n] ((Tape.move Dir.left)^[n] (trTape' enc0 (ListBlank.cons (ListBlank.head R) L) (ListBlank.tail R))) = (Tape.move Dir.right)^[n] (trTape' enc0 L R) simp only [trTape'_move_left, ListBlank.cons_head_tail, ListBlank.head_cons, ListBlank.tail_cons] case zero Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ L R : ListBlank Γ L✝ : Tape Bool ⊢ (Tape.move Dir.right)^[Nat.zero] ((Tape.move Dir.left)^[Nat.zero] L✝) = L✝ rfl ** Qed
Turing.TM1to1.stepAux_write Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ q : Stmt Bool Λ' σ v : σ a b : Γ L R : ListBlank Γ ⊢ stepAux (write (Vector.toList (enc a)) q) v (trTape' enc0 L (ListBlank.cons b R)) = stepAux q v (trTape' enc0 (ListBlank.cons a L) R) simp only [trTape', ListBlank.cons_bind] Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ q : Stmt Bool Λ' σ v : σ a b : Γ L R : ListBlank Γ ⊢ stepAux (write (Vector.toList (enc a)) q) v (Tape.mk' (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x))) (_ : ∃ n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default)) (ListBlank.append (Vector.toList (enc b)) (ListBlank.bind R (fun x => Vector.toList (enc x)) (_ : ∃ n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))) = stepAux q v (Tape.mk' (ListBlank.append (List.reverse (Vector.toList (enc a))) (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x))) (_ : ∃ n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default))) (ListBlank.bind R (fun x => Vector.toList (enc x)) (_ : ∃ n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default))) suffices ∀ {L' R'} (l₁ l₂ l₂' : List Bool) (_ : l₂'.length = l₂.length), stepAux (write l₂ q) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂' R')) = stepAux q v (Tape.mk' (L'.append (List.reverseAux l₂ l₁)) R') by refine' this [] _ _ ((enc b).2.trans (enc a).2.symm) Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ q : Stmt Bool Λ' σ v : σ a b : Γ L R : ListBlank Γ ⊢ ∀ {L' R' : ListBlank Bool} (l₁ l₂ l₂' : List Bool), List.length l₂' = List.length l₂ → stepAux (write l₂ q) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂' R')) = stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R') clear a b L R Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ q : Stmt Bool Λ' σ v : σ ⊢ ∀ {L' R' : ListBlank Bool} (l₁ l₂ l₂' : List Bool), List.length l₂' = List.length l₂ → stepAux (write l₂ q) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂' R')) = stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R') intro L' R' l₁ l₂ l₂' e Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ q : Stmt Bool Λ' σ v : σ L' R' : ListBlank Bool l₁ l₂ l₂' : List Bool e : List.length l₂' = List.length l₂ ⊢ stepAux (write l₂ q) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂' R')) = stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R') induction' l₂ with a l₂ IH generalizing l₁ l₂' case cons Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ q : Stmt Bool Λ' σ v : σ L' R' : ListBlank Bool l₁✝ l₂✝ l₂'✝ : List Bool e✝ : List.length l₂'✝ = List.length l₂✝ a : Bool l₂ : List Bool IH : ∀ (l₁ l₂' : List Bool), List.length l₂' = List.length l₂ → stepAux (write l₂ q) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂' R')) = stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R') l₁ l₂' : List Bool e : List.length l₂' = List.length (a :: l₂) ⊢ stepAux (write (a :: l₂) q) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂' R')) = stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux (a :: l₂) l₁) L') R') cases' l₂' with b l₂' <;> simp only [List.length_nil, List.length_cons, Nat.succ_inj'] at e case cons.cons Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ q : Stmt Bool Λ' σ v : σ L' R' : ListBlank Bool l₁✝ l₂✝ l₂'✝ : List Bool e✝ : List.length l₂'✝ = List.length l₂✝ a : Bool l₂ : List Bool IH : ∀ (l₁ l₂' : List Bool), List.length l₂' = List.length l₂ → stepAux (write l₂ q) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂' R')) = stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R') l₁ : List Bool b : Bool l₂' : List Bool e : List.length l₂' = List.length l₂ ⊢ stepAux (write (a :: l₂) q) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append (b :: l₂') R')) = stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux (a :: l₂) l₁) L') R') rw [List.reverseAux, ← IH (a :: l₁) l₂' e] case cons.cons Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ q : Stmt Bool Λ' σ v : σ L' R' : ListBlank Bool l₁✝ l₂✝ l₂'✝ : List Bool e✝ : List.length l₂'✝ = List.length l₂✝ a : Bool l₂ : List Bool IH : ∀ (l₁ l₂' : List Bool), List.length l₂' = List.length l₂ → stepAux (write l₂ q) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂' R')) = stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R') l₁ : List Bool b : Bool l₂' : List Bool e : List.length l₂' = List.length l₂ ⊢ stepAux (write (a :: l₂) q) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append (b :: l₂') R')) = stepAux (write l₂ q) v (Tape.mk' (ListBlank.append (a :: l₁) L') (ListBlank.append l₂' R')) simp only [stepAux, ListBlank.append, Tape.write_mk', Tape.move_right_mk', ListBlank.head_cons, ListBlank.tail_cons] Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ q : Stmt Bool Λ' σ v : σ a b : Γ L R : ListBlank Γ this : ∀ {L' R' : ListBlank Bool} (l₁ l₂ l₂' : List Bool), List.length l₂' = List.length l₂ → stepAux (write l₂ q) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂' R')) = stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R') ⊢ stepAux (write (Vector.toList (enc a)) q) v (Tape.mk' (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x))) (_ : ∃ n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default)) (ListBlank.append (Vector.toList (enc b)) (ListBlank.bind R (fun x => Vector.toList (enc x)) (_ : ∃ n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))) = stepAux q v (Tape.mk' (ListBlank.append (List.reverse (Vector.toList (enc a))) (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x))) (_ : ∃ n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default))) (ListBlank.bind R (fun x => Vector.toList (enc x)) (_ : ∃ n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default))) refine' this [] _ _ ((enc b).2.trans (enc a).2.symm) case nil Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ q : Stmt Bool Λ' σ v : σ L' R' : ListBlank Bool l₁✝ l₂ l₂'✝ : List Bool e✝ : List.length l₂'✝ = List.length l₂ l₁ l₂' : List Bool e : List.length l₂' = List.length [] ⊢ stepAux (write [] q) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂' R')) = stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux [] l₁) L') R') cases List.length_eq_zero.1 e case nil.refl Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ q : Stmt Bool Λ' σ v : σ L' R' : ListBlank Bool l₁✝ l₂ l₂' : List Bool e✝ : List.length l₂' = List.length l₂ l₁ : List Bool e : List.length [] = List.length [] ⊢ stepAux (write [] q) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append [] R')) = stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux [] l₁) L') R') rfl ** Qed
Turing.TM2.stmts₁_self K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q : Stmt₂ ⊢ q ∈ stmts₁ q cases q <;> simp only [Finset.mem_insert_self, Finset.mem_singleton_self, stmts₁] ** Qed
Turing.TM2.stmts₁_trans K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁ q₂ : Stmt₂ ⊢ q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ intro h₁₂ q₀ h₀₁ K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁ q₂ : Stmt₂ h₁₂ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₂ h₀₁ : q₀ ∈ stmts₁ q₁ ⊢ q₀ ∈ stmts₁ q₂ induction' q₂ with _ _ q IH _ _ q IH _ _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;> simp only [Finset.mem_insert, Finset.mem_singleton, Finset.mem_union] at h₁₂ case goto K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁ q₂ : Stmt₂ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₂ h₀₁ : q₀ ∈ stmts₁ q₁ a✝ : σ → Λ h₁₂ : q₁ = goto a✝ ⊢ q₀ ∈ {goto a✝} case halt K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁ q₂ : Stmt₂ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₂ h₀₁ : q₀ ∈ stmts₁ q₁ h₁₂ : q₁ = halt ⊢ q₀ ∈ {halt} case goto l => subst h₁₂; exact h₀₁ case halt K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁ q₂ : Stmt₂ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₂ h₀₁ : q₀ ∈ stmts₁ q₁ h₁₂ : q₁ = halt ⊢ q₀ ∈ {halt} case halt => subst h₁₂; exact h₀₁ case load K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁ q₂ : Stmt₂ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₂ h₀₁ : q₀ ∈ stmts₁ q₁ a✝ : σ → σ q : Stmt₂ IH : q₁ ∈ stmts₁ q → q₀ ∈ stmts₁ q h₁₂ : q₁ = load a✝ q ∨ q₁ ∈ stmts₁ q ⊢ q₀ ∈ insert (load a✝ q) (stmts₁ q) case branch K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁ q₂ : Stmt₂ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₂ h₀₁ : q₀ ∈ stmts₁ q₁ a✝² : σ → Bool a✝¹ a✝ : Stmt₂ a_ih✝¹ : q₁ ∈ stmts₁ a✝¹ → q₀ ∈ stmts₁ a✝¹ a_ih✝ : q₁ ∈ stmts₁ a✝ → q₀ ∈ stmts₁ a✝ h₁₂ : q₁ = branch a✝² a✝¹ a✝ ∨ q₁ ∈ stmts₁ a✝¹ ∨ q₁ ∈ stmts₁ a✝ ⊢ q₀ ∈ insert (branch a✝² a✝¹ a✝) (stmts₁ a✝¹ ∪ stmts₁ a✝) case goto K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁ q₂ : Stmt₂ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₂ h₀₁ : q₀ ∈ stmts₁ q₁ a✝ : σ → Λ h₁₂ : q₁ = goto a✝ ⊢ q₀ ∈ {goto a✝} case halt K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁ q₂ : Stmt₂ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₂ h₀₁ : q₀ ∈ stmts₁ q₁ h₁₂ : q₁ = halt ⊢ q₀ ∈ {halt} rcases h₁₂ with (rfl \| h₁₂) case load.inl K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₂ q₀ : Stmt₂ a✝ : σ → σ q : Stmt₂ h₁₂ : load a✝ q ∈ stmts₁ q₂ h₀₁ : q₀ ∈ stmts₁ (load a✝ q) IH : load a✝ q ∈ stmts₁ q → q₀ ∈ stmts₁ q ⊢ q₀ ∈ insert (load a✝ q) (stmts₁ q) unfold stmts₁ at h₀₁ case load.inl K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₂ q₀ : Stmt₂ a✝ : σ → σ q : Stmt₂ h₁₂ : load a✝ q ∈ stmts₁ q₂ IH : load a✝ q ∈ stmts₁ q → q₀ ∈ stmts₁ q h₀₁ : q₀ ∈ insert (load a✝ q) (stmts₁ q) ⊢ q₀ ∈ insert (load a✝ q) (stmts₁ q) exact h₀₁ case load.inr K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁ q₂ : Stmt₂ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₂ h₀₁ : q₀ ∈ stmts₁ q₁ a✝ : σ → σ q : Stmt₂ IH : q₁ ∈ stmts₁ q → q₀ ∈ stmts₁ q h₁₂ : q₁ ∈ stmts₁ q ⊢ q₀ ∈ insert (load a✝ q) (stmts₁ q) exact Finset.mem_insert_of_mem (IH h₁₂) K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁✝ q₂✝ : Stmt₂ h₁₂✝ : q₁✝ ∈ stmts₁ q₂✝ q₀ : Stmt₂ h₀₁ : q₀ ∈ stmts₁ q₁✝ f : σ → Bool q₁ q₂ : Stmt₂ IH₁ : q₁✝ ∈ stmts₁ q₁ → q₀ ∈ stmts₁ q₁ IH₂ : q₁✝ ∈ stmts₁ q₂ → q₀ ∈ stmts₁ q₂ h₁₂ : q₁✝ = branch f q₁ q₂ ∨ q₁✝ ∈ stmts₁ q₁ ∨ q₁✝ ∈ stmts₁ q₂ ⊢ q₀ ∈ insert (branch f q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂) rcases h₁₂ with (rfl \| h₁₂ \| h₁₂) case inl K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₂✝ q₀ : Stmt₂ f : σ → Bool q₁ q₂ : Stmt₂ h₁₂ : branch f q₁ q₂ ∈ stmts₁ q₂✝ h₀₁ : q₀ ∈ stmts₁ (branch f q₁ q₂) IH₁ : branch f q₁ q₂ ∈ stmts₁ q₁ → q₀ ∈ stmts₁ q₁ IH₂ : branch f q₁ q₂ ∈ stmts₁ q₂ → q₀ ∈ stmts₁ q₂ ⊢ q₀ ∈ insert (branch f q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂) unfold stmts₁ at h₀₁ case inl K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₂✝ q₀ : Stmt₂ f : σ → Bool q₁ q₂ : Stmt₂ h₁₂ : branch f q₁ q₂ ∈ stmts₁ q₂✝ IH₁ : branch f q₁ q₂ ∈ stmts₁ q₁ → q₀ ∈ stmts₁ q₁ IH₂ : branch f q₁ q₂ ∈ stmts₁ q₂ → q₀ ∈ stmts₁ q₂ h₀₁ : q₀ ∈ insert (branch f q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂) ⊢ q₀ ∈ insert (branch f q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂) exact h₀₁ case inr.inl K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁✝ q₂✝ : Stmt₂ h₁₂✝ : q₁✝ ∈ stmts₁ q₂✝ q₀ : Stmt₂ h₀₁ : q₀ ∈ stmts₁ q₁✝ f : σ → Bool q₁ q₂ : Stmt₂ IH₁ : q₁✝ ∈ stmts₁ q₁ → q₀ ∈ stmts₁ q₁ IH₂ : q₁✝ ∈ stmts₁ q₂ → q₀ ∈ stmts₁ q₂ h₁₂ : q₁✝ ∈ stmts₁ q₁ ⊢ q₀ ∈ insert (branch f q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂) exact Finset.mem_insert_of_mem (Finset.mem_union_left _ (IH₁ h₁₂)) case inr.inr K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁✝ q₂✝ : Stmt₂ h₁₂✝ : q₁✝ ∈ stmts₁ q₂✝ q₀ : Stmt₂ h₀₁ : q₀ ∈ stmts₁ q₁✝ f : σ → Bool q₁ q₂ : Stmt₂ IH₁ : q₁✝ ∈ stmts₁ q₁ → q₀ ∈ stmts₁ q₁ IH₂ : q₁✝ ∈ stmts₁ q₂ → q₀ ∈ stmts₁ q₂ h₁₂ : q₁✝ ∈ stmts₁ q₂ ⊢ q₀ ∈ insert (branch f q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂) exact Finset.mem_insert_of_mem (Finset.mem_union_right _ (IH₂ h₁₂)) K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁ q₂ : Stmt₂ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₂ h₀₁ : q₀ ∈ stmts₁ q₁ l : σ → Λ h₁₂ : q₁ = goto l ⊢ q₀ ∈ {goto l} subst h₁₂ K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₂ q₀ : Stmt₂ l : σ → Λ h₁₂ : goto l ∈ stmts₁ q₂ h₀₁ : q₀ ∈ stmts₁ (goto l) ⊢ q₀ ∈ {goto l} exact h₀₁ K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁ q₂ : Stmt₂ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₂ h₀₁ : q₀ ∈ stmts₁ q₁ h₁₂ : q₁ = halt ⊢ q₀ ∈ {halt} subst h₁₂ K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₂ q₀ : Stmt₂ h₁₂ : halt ∈ stmts₁ q₂ h₀₁ : q₀ ∈ stmts₁ halt ⊢ q₀ ∈ {halt} exact h₀₁ ** Qed
Turing.TM2.stmts₁_supportsStmt_mono K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h : q₁ ∈ stmts₁ q₂ hs : SupportsStmt S q₂ ⊢ SupportsStmt S q₁ induction' q₂ with _ _ q IH _ _ q IH _ _ q IH _ q IH <;> simp only [stmts₁, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h hs case push K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ k✝ : K a✝ : σ → Γ k✝ q : Stmt₂ IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁ hs : SupportsStmt S q h : q₁ = push k✝ a✝ q ∨ q₁ ∈ stmts₁ q ⊢ SupportsStmt S q₁ case peek K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ k✝ : K a✝ : σ → Option (Γ k✝) → σ q : Stmt₂ IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁ hs : SupportsStmt S q h : q₁ = peek k✝ a✝ q ∨ q₁ ∈ stmts₁ q ⊢ SupportsStmt S q₁ case pop K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ k✝ : K a✝ : σ → Option (Γ k✝) → σ q : Stmt₂ IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁ hs : SupportsStmt S q h : q₁ = pop k✝ a✝ q ∨ q₁ ∈ stmts₁ q ⊢ SupportsStmt S q₁ case load K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝ : σ → σ q : Stmt₂ IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁ hs : SupportsStmt S q h : q₁ = load a✝ q ∨ q₁ ∈ stmts₁ q ⊢ SupportsStmt S q₁ case branch K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝² : σ → Bool a✝¹ a✝ : Stmt₂ a_ih✝¹ : q₁ ∈ stmts₁ a✝¹ → SupportsStmt S a✝¹ → SupportsStmt S q₁ a_ih✝ : q₁ ∈ stmts₁ a✝ → SupportsStmt S a✝ → SupportsStmt S q₁ hs : SupportsStmt S a✝¹ ∧ SupportsStmt S a✝ h : q₁ = branch a✝² a✝¹ a✝ ∨ q₁ ∈ stmts₁ a✝¹ ∨ q₁ ∈ stmts₁ a✝ ⊢ SupportsStmt S q₁ case goto K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝ : σ → Λ hs : ∀ (v : σ), a✝ v ∈ S h : q₁ = goto a✝ ⊢ SupportsStmt S q₁ case halt K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ hs : True h : q₁ = halt ⊢ SupportsStmt S q₁ iterate 4 rcases h with (rfl \| h) <;> [exact hs; exact IH h hs] case branch K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝² : σ → Bool a✝¹ a✝ : Stmt₂ a_ih✝¹ : q₁ ∈ stmts₁ a✝¹ → SupportsStmt S a✝¹ → SupportsStmt S q₁ a_ih✝ : q₁ ∈ stmts₁ a✝ → SupportsStmt S a✝ → SupportsStmt S q₁ hs : SupportsStmt S a✝¹ ∧ SupportsStmt S a✝ h : q₁ = branch a✝² a✝¹ a✝ ∨ q₁ ∈ stmts₁ a✝¹ ∨ q₁ ∈ stmts₁ a✝ ⊢ SupportsStmt S q₁ case goto K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝ : σ → Λ hs : ∀ (v : σ), a✝ v ∈ S h : q₁ = goto a✝ ⊢ SupportsStmt S q₁ case halt K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ hs : True h : q₁ = halt ⊢ SupportsStmt S q₁ case branch f q₁ q₂ IH₁ IH₂ => rcases h with (rfl \| h \| h); exacts [hs, IH₁ h hs.1, IH₂ h hs.2] case goto K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝ : σ → Λ hs : ∀ (v : σ), a✝ v ∈ S h : q₁ = goto a✝ ⊢ SupportsStmt S q₁ case halt K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ hs : True h : q₁ = halt ⊢ SupportsStmt S q₁ case goto l => subst h; exact hs case halt K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ hs : True h : q₁ = halt ⊢ SupportsStmt S q₁ case halt => subst h; trivial case load K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝ : σ → σ q : Stmt₂ IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁ hs : SupportsStmt S q h : q₁ = load a✝ q ∨ q₁ ∈ stmts₁ q ⊢ SupportsStmt S q₁ case branch K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝² : σ → Bool a✝¹ a✝ : Stmt₂ a_ih✝¹ : q₁ ∈ stmts₁ a✝¹ → SupportsStmt S a✝¹ → SupportsStmt S q₁ a_ih✝ : q₁ ∈ stmts₁ a✝ → SupportsStmt S a✝ → SupportsStmt S q₁ hs : SupportsStmt S a✝¹ ∧ SupportsStmt S a✝ h : q₁ = branch a✝² a✝¹ a✝ ∨ q₁ ∈ stmts₁ a✝¹ ∨ q₁ ∈ stmts₁ a✝ ⊢ SupportsStmt S q₁ case goto K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝ : σ → Λ hs : ∀ (v : σ), a✝ v ∈ S h : q₁ = goto a✝ ⊢ SupportsStmt S q₁ case halt K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ hs : True h : q₁ = halt ⊢ SupportsStmt S q₁ rcases h with (rfl \| h) <;> [exact hs; exact IH h hs] K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁✝ q₂✝ : Stmt₂ h✝ : q₁✝ ∈ stmts₁ q₂✝ hs✝ : SupportsStmt S q₂✝ f : σ → Bool q₁ q₂ : Stmt₂ IH₁ : q₁✝ ∈ stmts₁ q₁ → SupportsStmt S q₁ → SupportsStmt S q₁✝ IH₂ : q₁✝ ∈ stmts₁ q₂ → SupportsStmt S q₂ → SupportsStmt S q₁✝ hs : SupportsStmt S q₁ ∧ SupportsStmt S q₂ h : q₁✝ = branch f q₁ q₂ ∨ q₁✝ ∈ stmts₁ q₁ ∨ q₁✝ ∈ stmts₁ q₂ ⊢ SupportsStmt S q₁✝ rcases h with (rfl \| h \| h) case inl K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₂✝ : Stmt₂ hs✝ : SupportsStmt S q₂✝ f : σ → Bool q₁ q₂ : Stmt₂ hs : SupportsStmt S q₁ ∧ SupportsStmt S q₂ h : branch f q₁ q₂ ∈ stmts₁ q₂✝ IH₁ : branch f q₁ q₂ ∈ stmts₁ q₁ → SupportsStmt S q₁ → SupportsStmt S (branch f q₁ q₂) IH₂ : branch f q₁ q₂ ∈ stmts₁ q₂ → SupportsStmt S q₂ → SupportsStmt S (branch f q₁ q₂) ⊢ SupportsStmt S (branch f q₁ q₂) case inr.inl K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁✝ q₂✝ : Stmt₂ h✝ : q₁✝ ∈ stmts₁ q₂✝ hs✝ : SupportsStmt S q₂✝ f : σ → Bool q₁ q₂ : Stmt₂ IH₁ : q₁✝ ∈ stmts₁ q₁ → SupportsStmt S q₁ → SupportsStmt S q₁✝ IH₂ : q₁✝ ∈ stmts₁ q₂ → SupportsStmt S q₂ → SupportsStmt S q₁✝ hs : SupportsStmt S q₁ ∧ SupportsStmt S q₂ h : q₁✝ ∈ stmts₁ q₁ ⊢ SupportsStmt S q₁✝ case inr.inr K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁✝ q₂✝ : Stmt₂ h✝ : q₁✝ ∈ stmts₁ q₂✝ hs✝ : SupportsStmt S q₂✝ f : σ → Bool q₁ q₂ : Stmt₂ IH₁ : q₁✝ ∈ stmts₁ q₁ → SupportsStmt S q₁ → SupportsStmt S q₁✝ IH₂ : q₁✝ ∈ stmts₁ q₂ → SupportsStmt S q₂ → SupportsStmt S q₁✝ hs : SupportsStmt S q₁ ∧ SupportsStmt S q₂ h : q₁✝ ∈ stmts₁ q₂ ⊢ SupportsStmt S q₁✝ exacts [hs, IH₁ h hs.1, IH₂ h hs.2] K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ l : σ → Λ hs : ∀ (v : σ), l v ∈ S h : q₁ = goto l ⊢ SupportsStmt S q₁ subst h K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₂ : Stmt₂ hs✝ : SupportsStmt S q₂ l : σ → Λ hs : ∀ (v : σ), l v ∈ S h : goto l ∈ stmts₁ q₂ ⊢ SupportsStmt S (goto l) exact hs K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ hs : True h : q₁ = halt ⊢ SupportsStmt S q₁ subst h K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₂ : Stmt₂ hs✝ : SupportsStmt S q₂ hs : True h : halt ∈ stmts₁ q₂ ⊢ SupportsStmt S halt trivial ** Qed
Turing.TM2.step_supports K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v : σ T : (k : K) → List (Γ k) c' : Cfg₂ h₁ : c' ∈ step M { l := some l₁, var := v, stk := T } h₂ : { l := some l₁, var := v, stk := T }.l ∈ ↑Finset.insertNone S ⊢ c'.l ∈ ↑Finset.insertNone S replace h₂ := ss.2 _ (Finset.some_mem_insertNone.1 h₂) K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v : σ T : (k : K) → List (Γ k) c' : Cfg₂ h₁ : c' ∈ step M { l := some l₁, var := v, stk := T } h₂ : SupportsStmt S (M l₁) ⊢ c'.l ∈ ↑Finset.insertNone S simp only [step, Option.mem_def, Option.some.injEq] at h₁ K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v : σ T : (k : K) → List (Γ k) c' : Cfg₂ h₂ : SupportsStmt S (M l₁) h₁ : stepAux (M l₁) v T = c' ⊢ c'.l ∈ ↑Finset.insertNone S subst c' K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v : σ T : (k : K) → List (Γ k) h₂ : SupportsStmt S (M l₁) ⊢ (stepAux (M l₁) v T).l ∈ ↑Finset.insertNone S revert h₂ K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v : σ T : (k : K) → List (Γ k) ⊢ SupportsStmt S (M l₁) → (stepAux (M l₁) v T).l ∈ ↑Finset.insertNone S induction' M l₁ with _ _ q IH _ _ q IH _ _ q IH _ q IH generalizing v T <;> intro hs case push K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) k✝ : K a✝ : σ → Γ k✝ q : Stmt₂ IH : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S q → (stepAux q v T).l ∈ ↑Finset.insertNone S v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (push k✝ a✝ q) ⊢ (stepAux (push k✝ a✝ q) v T).l ∈ ↑Finset.insertNone S case peek K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) k✝ : K a✝ : σ → Option (Γ k✝) → σ q : Stmt₂ IH : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S q → (stepAux q v T).l ∈ ↑Finset.insertNone S v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (peek k✝ a✝ q) ⊢ (stepAux (peek k✝ a✝ q) v T).l ∈ ↑Finset.insertNone S case pop K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) k✝ : K a✝ : σ → Option (Γ k✝) → σ q : Stmt₂ IH : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S q → (stepAux q v T).l ∈ ↑Finset.insertNone S v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (pop k✝ a✝ q) ⊢ (stepAux (pop k✝ a✝ q) v T).l ∈ ↑Finset.insertNone S case load K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) a✝ : σ → σ q : Stmt₂ IH : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S q → (stepAux q v T).l ∈ ↑Finset.insertNone S v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (load a✝ q) ⊢ (stepAux (load a✝ q) v T).l ∈ ↑Finset.insertNone S case branch K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) a✝² : σ → Bool a✝¹ a✝ : Stmt₂ a_ih✝¹ : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S a✝¹ → (stepAux a✝¹ v T).l ∈ ↑Finset.insertNone S a_ih✝ : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S a✝ → (stepAux a✝ v T).l ∈ ↑Finset.insertNone S v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (branch a✝² a✝¹ a✝) ⊢ (stepAux (branch a✝² a✝¹ a✝) v T).l ∈ ↑Finset.insertNone S case goto K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) a✝ : σ → Λ v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (goto a✝) ⊢ (stepAux (goto a✝) v T).l ∈ ↑Finset.insertNone S case halt K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S halt ⊢ (stepAux halt v T).l ∈ ↑Finset.insertNone S iterate 4 exact IH _ _ hs case goto K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) a✝ : σ → Λ v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (goto a✝) ⊢ (stepAux (goto a✝) v T).l ∈ ↑Finset.insertNone S case halt K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S halt ⊢ (stepAux halt v T).l ∈ ↑Finset.insertNone S case goto => exact Finset.some_mem_insertNone.2 (hs _) case halt K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S halt ⊢ (stepAux halt v T).l ∈ ↑Finset.insertNone S case halt => apply Multiset.mem_cons_self case load K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) a✝ : σ → σ q : Stmt₂ IH : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S q → (stepAux q v T).l ∈ ↑Finset.insertNone S v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (load a✝ q) ⊢ (stepAux (load a✝ q) v T).l ∈ ↑Finset.insertNone S case branch K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) a✝² : σ → Bool a✝¹ a✝ : Stmt₂ a_ih✝¹ : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S a✝¹ → (stepAux a✝¹ v T).l ∈ ↑Finset.insertNone S a_ih✝ : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S a✝ → (stepAux a✝ v T).l ∈ ↑Finset.insertNone S v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (branch a✝² a✝¹ a✝) ⊢ (stepAux (branch a✝² a✝¹ a✝) v T).l ∈ ↑Finset.insertNone S case goto K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) a✝ : σ → Λ v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (goto a✝) ⊢ (stepAux (goto a✝) v T).l ∈ ↑Finset.insertNone S case halt K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S halt ⊢ (stepAux halt v T).l ∈ ↑Finset.insertNone S exact IH _ _ hs K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) p : σ → Bool q₁' q₂' : Stmt₂ IH₁ : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S q₁' → (stepAux q₁' v T).l ∈ ↑Finset.insertNone S IH₂ : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S q₂' → (stepAux q₂' v T).l ∈ ↑Finset.insertNone S v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (branch p q₁' q₂') ⊢ (stepAux (branch p q₁' q₂') v T).l ∈ ↑Finset.insertNone S unfold stepAux K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) p : σ → Bool q₁' q₂' : Stmt₂ IH₁ : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S q₁' → (stepAux q₁' v T).l ∈ ↑Finset.insertNone S IH₂ : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S q₂' → (stepAux q₂' v T).l ∈ ↑Finset.insertNone S v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (branch p q₁' q₂') ⊢ (bif p v then stepAux q₁' v T else stepAux q₂' v T).l ∈ ↑Finset.insertNone S cases p v case false K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) p : σ → Bool q₁' q₂' : Stmt₂ IH₁ : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S q₁' → (stepAux q₁' v T).l ∈ ↑Finset.insertNone S IH₂ : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S q₂' → (stepAux q₂' v T).l ∈ ↑Finset.insertNone S v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (branch p q₁' q₂') ⊢ (bif false then stepAux q₁' v T else stepAux q₂' v T).l ∈ ↑Finset.insertNone S exact IH₂ _ _ hs.2 case true K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) p : σ → Bool q₁' q₂' : Stmt₂ IH₁ : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S q₁' → (stepAux q₁' v T).l ∈ ↑Finset.insertNone S IH₂ : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S q₂' → (stepAux q₂' v T).l ∈ ↑Finset.insertNone S v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (branch p q₁' q₂') ⊢ (bif true then stepAux q₁' v T else stepAux q₂' v T).l ∈ ↑Finset.insertNone S exact IH₁ _ _ hs.1 K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) a✝ : σ → Λ v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (goto a✝) ⊢ (stepAux (goto a✝) v T).l ∈ ↑Finset.insertNone S exact Finset.some_mem_insertNone.2 (hs _) K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S halt ⊢ (stepAux halt v T).l ∈ ↑Finset.insertNone S apply Multiset.mem_cons_self ** Qed
Turing.TM2to1.stk_nth_val K : Type u_1 Γ : K → Type u_2 L : ListBlank ((k : K) → Option (Γ k)) k : K S : List (Γ k) n : ℕ hL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S)) ⊢ ListBlank.nth L n k = List.get? (List.reverse S) n rw [← proj_map_nth, hL, ← List.map_reverse, ListBlank.nth_mk, List.getI_eq_iget_get?, List.get?_map] K : Type u_1 Γ : K → Type u_2 L : ListBlank ((k : K) → Option (Γ k)) k : K S : List (Γ k) n : ℕ hL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S)) ⊢ Option.iget (Option.map some (List.get? (List.reverse S) n)) = List.get? (List.reverse S) n cases S.reverse.get? n <;> rfl ** Qed
Turing.TM2to1.addBottom_map K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ L : ListBlank ((k : K) → Option (Γ k)) ⊢ default.2 = default rfl K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ L : ListBlank ((k : K) → Option (Γ k)) ⊢ ListBlank.map { f := Prod.snd, map_pt' := (_ : default.2 = default.2) } (addBottom L) = L simp only [addBottom, ListBlank.map_cons] K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ L : ListBlank ((k : K) → Option (Γ k)) ⊢ ListBlank.cons (ListBlank.head L) (ListBlank.map { f := Prod.snd, map_pt' := (_ : default.2 = default.2) } (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } (ListBlank.tail L))) = L convert ListBlank.cons_head_tail L case h.e'_2.h.e'_4 K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ L : ListBlank ((k : K) → Option (Γ k)) ⊢ ListBlank.map { f := Prod.snd, map_pt' := (_ : default.2 = default.2) } (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } (ListBlank.tail L)) = ListBlank.tail L generalize ListBlank.tail L = L' case h.e'_2.h.e'_4 K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ L L' : ListBlank ((k : K) → Option (Γ k)) ⊢ ListBlank.map { f := Prod.snd, map_pt' := (_ : default.2 = default.2) } (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } L') = L' refine' L'.induction_on fun l ↦ _ case h.e'_2.h.e'_4 K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ L L' : ListBlank ((k : K) → Option (Γ k)) l : List ((k : K) → Option (Γ k)) ⊢ ListBlank.map { f := Prod.snd, map_pt' := (_ : default.2 = default.2) } (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } (ListBlank.mk l)) = ListBlank.mk l simp ** Qed
Turing.TM2to1.addBottom_nth_succ_fst K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ L : ListBlank ((k : K) → Option (Γ k)) n : ℕ ⊢ (ListBlank.nth (addBottom L) (n + 1)).1 = false rw [ListBlank.nth_succ, addBottom, ListBlank.tail_cons, ListBlank.nth_map] ** Qed
Turing.TM2to1.addBottom_head_fst K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ L : ListBlank ((k : K) → Option (Γ k)) ⊢ (ListBlank.head (addBottom L)).1 = true rw [addBottom, ListBlank.head_cons] ** Qed
Turing.TM2to1.supports_run K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ S : Finset Λ k : K s : StAct k q : Stmt₂ ⊢ TM2.SupportsStmt S (stRun s q) ↔ TM2.SupportsStmt S q cases s <;> rfl ** Qed
Turing.TM2to1.trStmts₁_run K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K s : StAct k q : Stmt₂ ⊢ trStmts₁ (stRun s q) = {go k s q, ret q} ∪ trStmts₁ q cases s <;> simp only [trStmts₁] ** Qed
Turing.TM2to1.tr_respects_aux₂ K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) o : StAct k ⊢ let v' := stVar v (S k) o; let Sk' := stWrite v (S k) o; let S' := update S k Sk'; ∃ L', (∀ (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S' k)))) ∧ TM1.stepAux (trStAct q o) v ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))) = TM1.stepAux q v' ((Tape.move Dir.right)^[List.length (S' k)] (Tape.mk' ∅ (addBottom L'))) dsimp only K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) o : StAct k ⊢ ∃ L', (∀ (k_1 : K), ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (stWrite v (S k) o) k_1)))) ∧ TM1.stepAux (trStAct q o) v ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))) = TM1.stepAux q (stVar v (S k) o) ((Tape.move Dir.right)^[List.length (update S k (stWrite v (S k) o) k)] (Tape.mk' ∅ (addBottom L'))) simp K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) o : StAct k ⊢ ∃ L', (∀ (k_1 : K), ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (stWrite v (S k) o) k_1)))) ∧ TM1.stepAux (trStAct q o) v ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))) = TM1.stepAux q (stVar v (S k) o) ((Tape.move Dir.right)^[List.length (stWrite v (S k) o)] (Tape.mk' ∅ (addBottom L'))) cases o <;> simp only [stWrite, stVar, trStAct, TM1.stepAux] K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k ⊢ ∃ L', (∀ (k_1 : K), ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k_1)))) ∧ TM1.stepAux q v (Tape.move Dir.right (Tape.write (((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))).head.1, update ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))).head.2 k (some (f v))) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))))) = TM1.stepAux q v ((Tape.move Dir.right)^[List.length (f v :: S k)] (Tape.mk' ∅ (addBottom L'))) have := Tape.write_move_right_n fun a : Γ' ↦ (a.1, update a.2 k (some (f v))) K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) ⊢ ∃ L', (∀ (k_1 : K), ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k_1)))) ∧ TM1.stepAux q v (Tape.move Dir.right (Tape.write (((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))).head.1, update ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))).head.2 k (some (f v))) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))))) = TM1.stepAux q v ((Tape.move Dir.right)^[List.length (f v :: S k)] (Tape.mk' ∅ (addBottom L'))) refine' ⟨_, fun k' ↦ _, by erw [Tape.move_right_n_head, List.length, Tape.mk'_nth_nat, this, addBottom_modifyNth fun a ↦ update a k (some (f v)), Nat.add_one, iterate_succ'] rfl⟩ K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) k' : K ⊢ ListBlank.map (proj k') (ListBlank.modifyNth (fun a => update a k (some (f v))) (List.length (S k)) L) = ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k'))) refine' ListBlank.ext fun i ↦ _ K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) k' : K i : ℕ ⊢ ListBlank.nth (ListBlank.map (proj k') (ListBlank.modifyNth (fun a => update a k (some (f v))) (List.length (S k)) L)) i = ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k')))) i rw [ListBlank.nth_map, ListBlank.nth_modifyNth, proj, PointedMap.mk_val] K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) k' : K i : ℕ ⊢ ite (i = List.length (S k)) (update (ListBlank.nth L i) k (some (f v))) (ListBlank.nth L i) k' = ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k')))) i by_cases h' : k' = k K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) ⊢ TM1.stepAux q v (Tape.move Dir.right (Tape.write (((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))).head.1, update ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))).head.2 k (some (f v))) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))))) = TM1.stepAux q v ((Tape.move Dir.right)^[List.length (f v :: S k)] (Tape.mk' ∅ (addBottom ?m.673260))) erw [Tape.move_right_n_head, List.length, Tape.mk'_nth_nat, this, addBottom_modifyNth fun a ↦ update a k (some (f v)), Nat.add_one, iterate_succ'] K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) ⊢ TM1.stepAux q v (Tape.move Dir.right ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom (ListBlank.modifyNth (fun a => update a k (some (f v))) (List.length (S k)) L))))) = TM1.stepAux q v ((Tape.move Dir.right ∘ (Tape.move Dir.right)^[List.length (S k)]) (Tape.mk' ∅ (addBottom ?m.673260))) K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) ⊢ ListBlank ((k : K) → Option (Γ k)) K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) ⊢ ListBlank ((k : K) → Option (Γ k)) K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) ⊢ ListBlank ((k : K) → Option (Γ k)) K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) ⊢ ListBlank ((k : K) → Option (Γ k)) K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) ⊢ ListBlank ((k : K) → Option (Γ k)) K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) ⊢ ListBlank ((k : K) → Option (Γ k)) K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) ⊢ ListBlank ((k : K) → Option (Γ k)) rfl case pos K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) k' : K i : ℕ h' : k' = k ⊢ ite (i = List.length (S k)) (update (ListBlank.nth L i) k (some (f v))) (ListBlank.nth L i) k' = ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k')))) i subst k' case pos K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) i : ℕ ⊢ ite (i = List.length (S k)) (update (ListBlank.nth L i) k (some (f v))) (ListBlank.nth L i) k = ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k)))) i split_ifs with h <;> simp only [List.reverse_cons, Function.update_same, ListBlank.nth_mk, List.map] case neg K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) i : ℕ h : ¬i = List.length (S k) ⊢ ListBlank.nth L i k = List.getI (List.reverse (List.map some (S k)) ++ [some (f v)]) i rw [← proj_map_nth, hL, ListBlank.nth_mk] case neg K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) i : ℕ h : ¬i = List.length (S k) ⊢ List.getI (List.reverse (List.map some (S k))) i = List.getI (List.reverse (List.map some (S k)) ++ [some (f v)]) i cases' lt_or_gt_of_ne h with h h case pos K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) i : ℕ h : i = List.length (S k) ⊢ some (f v) = List.getI (List.reverse (List.map some (S k)) ++ [some (f v)]) i rw [List.getI_eq_get, List.get_append_right'] <;> simp only [h, List.get_singleton, List.length_map, List.length_reverse, Nat.succ_pos', List.length_append, lt_add_iff_pos_right, List.length, le_refl] case neg.inl K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) i : ℕ h✝ : ¬i = List.length (S k) h : i < List.length (S k) ⊢ List.getI (List.reverse (List.map some (S k))) i = List.getI (List.reverse (List.map some (S k)) ++ [some (f v)]) i rw [List.getI_append] case neg.inl.h K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) i : ℕ h✝ : ¬i = List.length (S k) h : i < List.length (S k) ⊢ i < List.length (List.reverse (List.map some (S k))) simpa only [List.length_map, List.length_reverse] using h case neg.inr K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) i : ℕ h✝ : ¬i = List.length (S k) h : i > List.length (S k) ⊢ List.getI (List.reverse (List.map some (S k))) i = List.getI (List.reverse (List.map some (S k)) ++ [some (f v)]) i rw [gt_iff_lt] at h case neg.inr K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) i : ℕ h✝ : ¬i = List.length (S k) h : List.length (S k) < i ⊢ List.getI (List.reverse (List.map some (S k))) i = List.getI (List.reverse (List.map some (S k)) ++ [some (f v)]) i rw [List.getI_eq_default, List.getI_eq_default] <;> simp only [Nat.add_one_le_iff, h, List.length, le_of_lt, List.length_reverse, List.length_append, List.length_map] case neg K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) k' : K i : ℕ h' : ¬k' = k ⊢ ite (i = List.length (S k)) (update (ListBlank.nth L i) k (some (f v))) (ListBlank.nth L i) k' = ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k')))) i split_ifs <;> rw [Function.update_noteq h', ← proj_map_nth, hL] case pos K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) k' : K i : ℕ h' : ¬k' = k h✝ : i = List.length (S k) ⊢ ListBlank.nth (ListBlank.mk (List.reverse (List.map some (S k')))) i = ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k')))) i rw [Function.update_noteq h'] K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ ⊢ ∃ L', (∀ (k_1 : K), ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (S k) k_1)))) ∧ TM1.stepAux q (f v ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L)))).head.2 k)) (Tape.move Dir.right (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))))) = TM1.stepAux q (f v (List.head? (S k))) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L'))) rw [Function.update_eq_self] K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ ⊢ ∃ L', (∀ (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S k)))) ∧ TM1.stepAux q (f v ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L)))).head.2 k)) (Tape.move Dir.right (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))))) = TM1.stepAux q (f v (List.head? (S k))) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L'))) use L, hL case right K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ ⊢ TM1.stepAux q (f v ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L)))).head.2 k)) (Tape.move Dir.right (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))))) = TM1.stepAux q (f v (List.head? (S k))) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))) rw [Tape.move_left_right] case right K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ ⊢ TM1.stepAux q (f v ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L)))).head.2 k)) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))) = TM1.stepAux q (f v (List.head? (S k))) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))) congr case right.e_a.e_a K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ ⊢ (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L)))).head.2 k = List.head? (S k) cases e : S k case right.e_a.e_a.cons K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ head✝ : Γ k tail✝ : List (Γ k) e : S k = head✝ :: tail✝ ⊢ (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (head✝ :: tail✝)] (Tape.mk' ∅ (addBottom L)))).head.2 k = List.head? (head✝ :: tail✝) rw [List.length_cons, iterate_succ', Function.comp, Tape.move_right_left, Tape.move_right_n_head, Tape.mk'_nth_nat, addBottom_nth_snd, stk_nth_val _ (hL k), e, List.reverse_cons, ← List.length_reverse, List.get?_concat_length] case right.e_a.e_a.cons K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ head✝ : Γ k tail✝ : List (Γ k) e : S k = head✝ :: tail✝ ⊢ some head✝ = List.head? (head✝ :: tail✝) rfl case right.e_a.e_a.nil K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ e : S k = [] ⊢ (Tape.move Dir.left ((Tape.move Dir.right)^[List.length []] (Tape.mk' ∅ (addBottom L)))).head.2 k = List.head? [] rfl K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ ⊢ ∃ L', (∀ (k_1 : K), ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (List.tail (S k)) k_1)))) ∧ (bif ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))).head.1 then TM1.stepAux q (f v none) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))) else TM1.stepAux q (f v ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L)))).head.2 k)) (Tape.write ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L)))).head.1, update (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L)))).head.2 k none) (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L)))))) = TM1.stepAux q (f v (List.head? (S k))) ((Tape.move Dir.right)^[List.length (List.tail (S k))] (Tape.mk' ∅ (addBottom L'))) cases' e : S k with hd tl case nil K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ e : S k = [] ⊢ ∃ L', (∀ (k_1 : K), ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (List.tail []) k_1)))) ∧ (bif ((Tape.move Dir.right)^[List.length []] (Tape.mk' ∅ (addBottom L))).head.1 then TM1.stepAux q (f v none) ((Tape.move Dir.right)^[List.length []] (Tape.mk' ∅ (addBottom L))) else TM1.stepAux q (f v ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length []] (Tape.mk' ∅ (addBottom L)))).head.2 k)) (Tape.write ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length []] (Tape.mk' ∅ (addBottom L)))).head.1, update (Tape.move Dir.left ((Tape.move Dir.right)^[List.length []] (Tape.mk' ∅ (addBottom L)))).head.2 k none) (Tape.move Dir.left ((Tape.move Dir.right)^[List.length []] (Tape.mk' ∅ (addBottom L)))))) = TM1.stepAux q (f v (List.head? [])) ((Tape.move Dir.right)^[List.length (List.tail [])] (Tape.mk' ∅ (addBottom L'))) simp only [Tape.mk'_head, ListBlank.head_cons, Tape.move_left_mk', List.length, Tape.write_mk', List.head?, iterate_zero_apply, List.tail_nil] case nil K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ e : S k = [] ⊢ ∃ L', (∀ (k_1 : K), ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k [] k_1)))) ∧ (bif (ListBlank.head (addBottom L)).1 then TM1.stepAux q (f v none) (Tape.mk' ∅ (addBottom L)) else TM1.stepAux q (f v ((ListBlank.head ∅).2 k)) (Tape.mk' (ListBlank.tail ∅) (ListBlank.cons ((ListBlank.head ∅).1, update (ListBlank.head ∅).2 k none) (addBottom L)))) = TM1.stepAux q (f v none) (Tape.mk' ∅ (addBottom L')) rw [← e, Function.update_eq_self] case nil K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ e : S k = [] ⊢ ∃ L', (∀ (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S k)))) ∧ (bif (ListBlank.head (addBottom L)).1 then TM1.stepAux q (f v none) (Tape.mk' ∅ (addBottom L)) else TM1.stepAux q (f v ((ListBlank.head ∅).2 k)) (Tape.mk' (ListBlank.tail ∅) (ListBlank.cons ((ListBlank.head ∅).1, update (ListBlank.head ∅).2 k none) (addBottom L)))) = TM1.stepAux q (f v none) (Tape.mk' ∅ (addBottom L')) exact ⟨L, hL, by rw [addBottom_head_fst, cond]⟩ K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ e : S k = [] ⊢ (bif (ListBlank.head (addBottom L)).1 then TM1.stepAux q (f v none) (Tape.mk' ∅ (addBottom L)) else TM1.stepAux q (f v ((ListBlank.head ∅).2 k)) (Tape.mk' (ListBlank.tail ∅) (ListBlank.cons ((ListBlank.head ∅).1, update (ListBlank.head ∅).2 k none) (addBottom L)))) = TM1.stepAux q (f v none) (Tape.mk' ∅ (addBottom L)) rw [addBottom_head_fst, cond] case cons K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl ⊢ ∃ L', (∀ (k_1 : K), ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (List.tail (hd :: tl)) k_1)))) ∧ (bif ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' ∅ (addBottom L))).head.1 then TM1.stepAux q (f v none) ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' ∅ (addBottom L))) else TM1.stepAux q (f v ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' ∅ (addBottom L)))).head.2 k)) (Tape.write ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' ∅ (addBottom L)))).head.1, update (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' ∅ (addBottom L)))).head.2 k none) (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' ∅ (addBottom L)))))) = TM1.stepAux q (f v (List.head? (hd :: tl))) ((Tape.move Dir.right)^[List.length (List.tail (hd :: tl))] (Tape.mk' ∅ (addBottom L'))) refine' ⟨_, fun k' ↦ _, by erw [List.length_cons, Tape.move_right_n_head, Tape.mk'_nth_nat, addBottom_nth_succ_fst, cond, iterate_succ', Function.comp, Tape.move_right_left, Tape.move_right_n_head, Tape.mk'_nth_nat, Tape.write_move_right_n fun a : Γ' ↦ (a.1, update a.2 k none), addBottom_modifyNth fun a ↦ update a k none, addBottom_nth_snd, stk_nth_val _ (hL k), e, show (List.cons hd tl).reverse.get? tl.length = some hd by rw [List.reverse_cons, ← List.length_reverse, List.get?_concat_length], List.head?, List.tail]⟩ case cons K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl k' : K ⊢ ListBlank.map (proj k') (ListBlank.modifyNth (fun a => update a k none) (List.length tl) L) = ListBlank.mk (List.reverse (List.map some (update S k (List.tail (hd :: tl)) k'))) refine' ListBlank.ext fun i ↦ _ case cons K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl k' : K i : ℕ ⊢ ListBlank.nth (ListBlank.map (proj k') (ListBlank.modifyNth (fun a => update a k none) (List.length tl) L)) i = ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (List.tail (hd :: tl)) k')))) i rw [ListBlank.nth_map, ListBlank.nth_modifyNth, proj, PointedMap.mk_val] case cons K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl k' : K i : ℕ ⊢ ite (i = List.length tl) (update (ListBlank.nth L i) k none) (ListBlank.nth L i) k' = ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (List.tail (hd :: tl)) k')))) i by_cases h' : k' = k K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl ⊢ (bif ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' ∅ (addBottom L))).head.1 then TM1.stepAux q (f v none) ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' ∅ (addBottom L))) else TM1.stepAux q (f v ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' ∅ (addBottom L)))).head.2 k)) (Tape.write ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' ∅ (addBottom L)))).head.1, update (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' ∅ (addBottom L)))).head.2 k none) (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' ∅ (addBottom L)))))) = TM1.stepAux q (f v (List.head? (hd :: tl))) ((Tape.move Dir.right)^[List.length (List.tail (hd :: tl))] (Tape.mk' ∅ (addBottom ?m.690014))) erw [List.length_cons, Tape.move_right_n_head, Tape.mk'_nth_nat, addBottom_nth_succ_fst, cond, iterate_succ', Function.comp, Tape.move_right_left, Tape.move_right_n_head, Tape.mk'_nth_nat, Tape.write_move_right_n fun a : Γ' ↦ (a.1, update a.2 k none), addBottom_modifyNth fun a ↦ update a k none, addBottom_nth_snd, stk_nth_val _ (hL k), e, show (List.cons hd tl).reverse.get? tl.length = some hd by rw [List.reverse_cons, ← List.length_reverse, List.get?_concat_length], List.head?, List.tail] K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl ⊢ List.get? (List.reverse (hd :: tl)) (List.length tl) = some hd rw [List.reverse_cons, ← List.length_reverse, List.get?_concat_length] case pos K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl k' : K i : ℕ h' : k' = k ⊢ ite (i = List.length tl) (update (ListBlank.nth L i) k none) (ListBlank.nth L i) k' = ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (List.tail (hd :: tl)) k')))) i subst k' case pos K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl i : ℕ ⊢ ite (i = List.length tl) (update (ListBlank.nth L i) k none) (ListBlank.nth L i) k = ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (List.tail (hd :: tl)) k)))) i split_ifs with h <;> simp only [Function.update_same, ListBlank.nth_mk, List.tail] case neg K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl i : ℕ h : ¬i = List.length tl ⊢ ListBlank.nth L i k = List.getI (List.reverse (List.map some tl)) i rw [← proj_map_nth, hL, ListBlank.nth_mk, e, List.map, List.reverse_cons] case neg K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl i : ℕ h : ¬i = List.length tl ⊢ List.getI (List.reverse (List.map some tl) ++ [some hd]) i = List.getI (List.reverse (List.map some tl)) i cases' lt_or_gt_of_ne h with h h case pos K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl i : ℕ h : i = List.length tl ⊢ none = List.getI (List.reverse (List.map some tl)) i rw [List.getI_eq_default] case pos.hn K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl i : ℕ h : i = List.length tl ⊢ List.length (List.reverse (List.map some tl)) ≤ i rw [h, List.length_reverse, List.length_map] case pos K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl i : ℕ h : i = List.length tl ⊢ none = default rfl case neg.inl K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl i : ℕ h✝ : ¬i = List.length tl h : i < List.length tl ⊢ List.getI (List.reverse (List.map some tl) ++ [some hd]) i = List.getI (List.reverse (List.map some tl)) i rw [List.getI_append] case neg.inl.h K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl i : ℕ h✝ : ¬i = List.length tl h : i < List.length tl ⊢ i < List.length (List.reverse (List.map some tl)) simpa only [List.length_map, List.length_reverse] using h case neg.inr K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl i : ℕ h✝ : ¬i = List.length tl h : i > List.length tl ⊢ List.getI (List.reverse (List.map some tl) ++ [some hd]) i = List.getI (List.reverse (List.map some tl)) i rw [gt_iff_lt] at h case neg.inr K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl i : ℕ h✝ : ¬i = List.length tl h : List.length tl < i ⊢ List.getI (List.reverse (List.map some tl) ++ [some hd]) i = List.getI (List.reverse (List.map some tl)) i rw [List.getI_eq_default, List.getI_eq_default] <;> simp only [Nat.add_one_le_iff, h, List.length, le_of_lt, List.length_reverse, List.length_append, List.length_map] case neg K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl k' : K i : ℕ h' : ¬k' = k ⊢ ite (i = List.length tl) (update (ListBlank.nth L i) k none) (ListBlank.nth L i) k' = ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (List.tail (hd :: tl)) k')))) i split_ifs <;> rw [Function.update_noteq h', ← proj_map_nth, hL] case pos K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl k' : K i : ℕ h' : ¬k' = k h✝ : i = List.length tl ⊢ ListBlank.nth (ListBlank.mk (List.reverse (List.map some (S k')))) i = ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (List.tail (hd :: tl)) k')))) i rw [Function.update_noteq h'] ** Qed
Turing.TM2to1.tr_eval K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ M : Λ → Stmt₂ k : K L : List (Γ k) L₁ : ListBlank Γ' L₂ : List (Γ k) H₁ : L₁ ∈ TM1.eval (tr M) (trInit k L) H₂ : L₂ ∈ TM2.eval M k L ⊢ ∃ S L', addBottom L' = L₁ ∧ (∀ (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S k)))) ∧ S k = L₂ obtain ⟨c₁, h₁, rfl⟩ := (Part.mem_map_iff _).1 H₁ case intro.intro K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ M : Λ → Stmt₂ k : K L L₂ : List (Γ k) H₂ : L₂ ∈ TM2.eval M k L c₁ : TM1.Cfg Γ' Λ' σ h₁ : c₁ ∈ eval (TM1.step (tr M)) (TM1.init (trInit k L)) H₁ : Tape.right₀ c₁.Tape ∈ TM1.eval (tr M) (trInit k L) ⊢ ∃ S L', addBottom L' = Tape.right₀ c₁.Tape ∧ (∀ (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S k)))) ∧ S k = L₂ obtain ⟨c₂, h₂, rfl⟩ := (Part.mem_map_iff _).1 H₂ case intro.intro.intro.intro K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ M : Λ → Stmt₂ k : K L : List (Γ k) c₁ : TM1.Cfg Γ' Λ' σ h₁ : c₁ ∈ eval (TM1.step (tr M)) (TM1.init (trInit k L)) H₁ : Tape.right₀ c₁.Tape ∈ TM1.eval (tr M) (trInit k L) c₂ : Cfg₂ h₂ : c₂ ∈ eval (TM2.step M) (TM2.init k L) H₂ : TM2.Cfg.stk c₂ k ∈ TM2.eval M k L ⊢ ∃ S L', addBottom L' = Tape.right₀ c₁.Tape ∧ (∀ (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S k)))) ∧ S k = TM2.Cfg.stk c₂ k obtain ⟨_, ⟨L', hT⟩, h₃⟩ := Turing.tr_eval (tr_respects M) (trCfg_init k L) h₂ case intro.intro.intro.intro.intro.intro.mk K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ M : Λ → Stmt₂ k : K L : List (Γ k) c₁ : TM1.Cfg Γ' Λ' σ h₁ : c₁ ∈ eval (TM1.step (tr M)) (TM1.init (trInit k L)) H₁ : Tape.right₀ c₁.Tape ∈ TM1.eval (tr M) (trInit k L) q✝ : Option Λ v✝ : σ S✝ : (k : K) → List (Γ k) L' : ListBlank ((k : K) → Option (Γ k)) hT : ∀ (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S✝ k))) h₂ : { l := q✝, var := v✝, stk := S✝ } ∈ eval (TM2.step M) (TM2.init k L) H₂ : TM2.Cfg.stk { l := q✝, var := v✝, stk := S✝ } k ∈ TM2.eval M k L h₃ : { l := Option.map normal q✝, var := v✝, Tape := Tape.mk' ∅ (addBottom L') } ∈ eval (TM1.step (tr M)) (TM1.init (trInit k L)) ⊢ ∃ S L', addBottom L' = Tape.right₀ c₁.Tape ∧ (∀ (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S k)))) ∧ S k = TM2.Cfg.stk { l := q✝, var := v✝, stk := S✝ } k cases Part.mem_unique h₁ h₃ case intro.intro.intro.intro.intro.intro.mk.refl K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ M : Λ → Stmt₂ k : K L : List (Γ k) q✝ : Option Λ v✝ : σ S✝ : (k : K) → List (Γ k) L' : ListBlank ((k : K) → Option (Γ k)) hT : ∀ (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S✝ k))) h₂ : { l := q✝, var := v✝, stk := S✝ } ∈ eval (TM2.step M) (TM2.init k L) H₂ : TM2.Cfg.stk { l := q✝, var := v✝, stk := S✝ } k ∈ TM2.eval M k L h₃ : { l := Option.map normal q✝, var := v✝, Tape := Tape.mk' ∅ (addBottom L') } ∈ eval (TM1.step (tr M)) (TM1.init (trInit k L)) h₁ : { l := Option.map normal q✝, var := v✝, Tape := Tape.mk' ∅ (addBottom L') } ∈ eval (TM1.step (tr M)) (TM1.init (trInit k L)) H₁ : Tape.right₀ { l := Option.map normal q✝, var := v✝, Tape := Tape.mk' ∅ (addBottom L') }.Tape ∈ TM1.eval (tr M) (trInit k L) ⊢ ∃ S L'_1, addBottom L'_1 = Tape.right₀ { l := Option.map normal q✝, var := v✝, Tape := Tape.mk' ∅ (addBottom L') }.Tape ∧ (∀ (k : K), ListBlank.map (proj k) L'_1 = ListBlank.mk (List.reverse (List.map some (S k)))) ∧ S k = TM2.Cfg.stk { l := q✝, var := v✝, stk := S✝ } k exact ⟨_, L', by simp only [Tape.mk'_right₀], hT, rfl⟩ K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ M : Λ → Stmt₂ k : K L : List (Γ k) q✝ : Option Λ v✝ : σ S✝ : (k : K) → List (Γ k) L' : ListBlank ((k : K) → Option (Γ k)) hT : ∀ (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S✝ k))) h₂ : { l := q✝, var := v✝, stk := S✝ } ∈ eval (TM2.step M) (TM2.init k L) H₂ : TM2.Cfg.stk { l := q✝, var := v✝, stk := S✝ } k ∈ TM2.eval M k L h₃ : { l := Option.map normal q✝, var := v✝, Tape := Tape.mk' ∅ (addBottom L') } ∈ eval (TM1.step (tr M)) (TM1.init (trInit k L)) h₁ : { l := Option.map normal q✝, var := v✝, Tape := Tape.mk' ∅ (addBottom L') } ∈ eval (TM1.step (tr M)) (TM1.init (trInit k L)) H₁ : Tape.right₀ { l := Option.map normal q✝, var := v✝, Tape := Tape.mk' ∅ (addBottom L') }.Tape ∈ TM1.eval (tr M) (trInit k L) ⊢ addBottom L' = Tape.right₀ { l := Option.map normal q✝, var := v✝, Tape := Tape.mk' ∅ (addBottom L') }.Tape simp only [Tape.mk'_right₀] ** Qed
borel_eq_top_of_countable α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝² : TopologicalSpace α inst✝¹ : T1Space α inst✝ : Countable α ⊢ borel α = ⊤ refine' top_le_iff.1 fun s _ => biUnion_of_singleton s ▸ _ α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝² : TopologicalSpace α inst✝¹ : T1Space α inst✝ : Countable α s : Set α x✝ : MeasurableSet s ⊢ MeasurableSet (⋃ x ∈ s, {x}) apply MeasurableSet.biUnion s.to_countable α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝² : TopologicalSpace α inst✝¹ : T1Space α inst✝ : Countable α s : Set α x✝ : MeasurableSet s ⊢ ∀ (b : α), b ∈ s → MeasurableSet {b} intro x _ α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝² : TopologicalSpace α inst✝¹ : T1Space α inst✝ : Countable α s : Set α x✝ : MeasurableSet s x : α a✝ : x ∈ s ⊢ MeasurableSet {x} apply MeasurableSet.of_compl case h α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝² : TopologicalSpace α inst✝¹ : T1Space α inst✝ : Countable α s : Set α x✝ : MeasurableSet s x : α a✝ : x ∈ s ⊢ MeasurableSet {x}ᶜ apply GenerateMeasurable.basic case h.a α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝² : TopologicalSpace α inst✝¹ : T1Space α inst✝ : Countable α s : Set α x✝ : MeasurableSet s x : α a✝ : x ∈ s ⊢ {x}ᶜ ∈ {s \| IsOpen s} exact isClosed_singleton.isOpen_compl ** Qed
borel_eq_generateFrom_Iio α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α ⊢ borel α = MeasurableSpace.generateFrom (range Iio) refine' le_antisymm _ (generateFrom_le _) case refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α ⊢ borel α ≤ MeasurableSpace.generateFrom (range Iio) rw [borel_eq_generateFrom_of_subbasis (@OrderTopology.topology_eq_generate_intervals α _ _ _)] case refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α ⊢ MeasurableSpace.generateFrom {s \| ∃ a, s = Ioi a ∨ s = Iio a} ≤ MeasurableSpace.generateFrom (range Iio) letI : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) case refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) ⊢ MeasurableSpace.generateFrom {s \| ∃ a, s = Ioi a ∨ s = Iio a} ≤ MeasurableSpace.generateFrom (range Iio) have H : ∀ a : α, MeasurableSet (Iio a) := fun a => GenerateMeasurable.basic _ ⟨_, rfl⟩ case refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) ⊢ MeasurableSpace.generateFrom {s \| ∃ a, s = Ioi a ∨ s = Iio a} ≤ MeasurableSpace.generateFrom (range Iio) refine' generateFrom_le _ case refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) ⊢ ∀ (t : Set α), t ∈ {s \| ∃ a, s = Ioi a ∨ s = Iio a} → MeasurableSet t rintro _ ⟨a, rfl \| rfl⟩ <;> [skip; apply H] case refine'_1.intro.inl α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a : α ⊢ MeasurableSet (Ioi a) by_cases h : ∃ a', ∀ b, a < b ↔ a' ≤ b case pos α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a : α h : ∃ a', ∀ (b : α), a < b ↔ a' ≤ b ⊢ MeasurableSet (Ioi a) rcases h with ⟨a', ha'⟩ case pos.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a a' : α ha' : ∀ (b : α), a < b ↔ a' ≤ b ⊢ MeasurableSet (Ioi a) rw [(_ : Ioi a = (Iio a')ᶜ)] α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a a' : α ha' : ∀ (b : α), a < b ↔ a' ≤ b ⊢ Ioi a = (Iio a')ᶜ simp [Set.ext_iff, ha'] case pos.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a a' : α ha' : ∀ (b : α), a < b ↔ a' ≤ b ⊢ MeasurableSet (Iio a')ᶜ exact (H _).compl case neg α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a : α h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b ⊢ MeasurableSet (Ioi a) rcases isOpen_iUnion_countable (fun a' : { a' : α // a < a' } => { b \| a'.1 < b }) fun a' => isOpen_lt' _ with ⟨v, ⟨hv⟩, vu⟩ case neg.intro.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a : α h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b v : Set { a' // a < a' } vu : ⋃ i ∈ v, {b \| ↑i < b} = ⋃ i, {b \| ↑i < b} hv : Encodable ↑v ⊢ MeasurableSet (Ioi a) simp [Set.ext_iff] at vu case neg.intro.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this✝ : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a : α h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b v : Set { a' // a < a' } hv : Encodable ↑v vu : ∀ (x : α), (∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 < x) ↔ ∃ a_1, a < a_1 ∧ a_1 < x this : Ioi a = ⋃ x, (Iio ↑↑x)ᶜ ⊢ MeasurableSet (Ioi a) rw [this] case neg.intro.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this✝ : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a : α h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b v : Set { a' // a < a' } hv : Encodable ↑v vu : ∀ (x : α), (∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 < x) ↔ ∃ a_1, a < a_1 ∧ a_1 < x this : Ioi a = ⋃ x, (Iio ↑↑x)ᶜ ⊢ MeasurableSet (⋃ x, (Iio ↑↑x)ᶜ) apply MeasurableSet.iUnion case neg.intro.intro.intro.h α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this✝ : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a : α h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b v : Set { a' // a < a' } hv : Encodable ↑v vu : ∀ (x : α), (∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 < x) ↔ ∃ a_1, a < a_1 ∧ a_1 < x this : Ioi a = ⋃ x, (Iio ↑↑x)ᶜ ⊢ ∀ (b : ↑v), MeasurableSet (Iio ↑↑b)ᶜ exact fun _ => (H _).compl α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a : α h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b v : Set { a' // a < a' } hv : Encodable ↑v vu : ∀ (x : α), (∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 < x) ↔ ∃ a_1, a < a_1 ∧ a_1 < x ⊢ Ioi a = ⋃ x, (Iio ↑↑x)ᶜ simp only [compl_Iio, iUnion_coe_set, Set.ext_iff, mem_Ioi, mem_iUnion, mem_Ici, exists_prop, Subtype.exists, exists_and_right] α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a : α h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b v : Set { a' // a < a' } hv : Encodable ↑v vu : ∀ (x : α), (∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 < x) ↔ ∃ a_1, a < a_1 ∧ a_1 < x ⊢ ∀ (x : α), a < x ↔ ∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 ≤ x refine' fun x => ⟨fun ax => _, fun ⟨a', ⟨h, _⟩, ax⟩ => lt_of_lt_of_le h ax⟩ α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a : α h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b v : Set { a' // a < a' } hv : Encodable ↑v vu : ∀ (x : α), (∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 < x) ↔ ∃ a_1, a < a_1 ∧ a_1 < x x : α ax : a < x ⊢ ∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 ≤ x rcases (vu x).2 (by refine' not_imp_comm.1 (fun h => _) h exact ⟨x, fun b => ⟨fun ab => le_of_not_lt fun h' => h ⟨b, ab, h'⟩, lt_of_lt_of_le ax⟩⟩) with ⟨a', h₁, h₂⟩ α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a : α h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b v : Set { a' // a < a' } hv : Encodable ↑v vu : ∀ (x : α), (∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 < x) ↔ ∃ a_1, a < a_1 ∧ a_1 < x x : α ax : a < x ⊢ ∃ a_1, a < a_1 ∧ a_1 < x refine' not_imp_comm.1 (fun h => _) h α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a : α h✝ : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b v : Set { a' // a < a' } hv : Encodable ↑v vu : ∀ (x : α), (∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 < x) ↔ ∃ a_1, a < a_1 ∧ a_1 < x x : α ax : a < x h : ¬∃ a_1, a < a_1 ∧ a_1 < x ⊢ ∃ a', ∀ (b : α), a < b ↔ a' ≤ b exact ⟨x, fun b => ⟨fun ab => le_of_not_lt fun h' => h ⟨b, ab, h'⟩, lt_of_lt_of_le ax⟩⟩ case intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a : α h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b v : Set { a' // a < a' } hv : Encodable ↑v vu : ∀ (x : α), (∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 < x) ↔ ∃ a_1, a < a_1 ∧ a_1 < x x : α ax : a < x a' : α h₁ : ∃ x, { val := a', property := (_ : a < a') } ∈ v h₂ : a' < x ⊢ ∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 ≤ x exact ⟨a', h₁, le_of_lt h₂⟩ case refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α ⊢ ∀ (t : Set α), t ∈ range Iio → MeasurableSet t rw [forall_range_iff] case refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α ⊢ ∀ (i : α), MeasurableSet (Iio i) intro a case refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α a : α ⊢ MeasurableSet (Iio a) exact GenerateMeasurable.basic _ isOpen_Iio ** Qed
borel_eq_generateFrom_Ioi α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α ⊢ SecondCountableTopology α infer_instance ** Qed
borel_eq_generateFrom_Iic α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α ⊢ borel α = MeasurableSpace.generateFrom (range Iic) rw [borel_eq_generateFrom_Ioi] α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α ⊢ MeasurableSpace.generateFrom (range Ioi) = MeasurableSpace.generateFrom (range Iic) refine' le_antisymm _ _ case refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α ⊢ MeasurableSpace.generateFrom (range Ioi) ≤ MeasurableSpace.generateFrom (range Iic) refine' MeasurableSpace.generateFrom_le fun t ht => _ case refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t✝ u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α t : Set α ht : t ∈ range Ioi ⊢ MeasurableSet t obtain ⟨u, rfl⟩ := ht case refine'_1.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u✝ : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α u : α ⊢ MeasurableSet (Ioi u) rw [← compl_Iic] case refine'_1.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u✝ : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α u : α ⊢ MeasurableSet (Iic u)ᶜ exact (MeasurableSpace.measurableSet_generateFrom (mem_range.mpr ⟨u, rfl⟩)).compl case refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α ⊢ MeasurableSpace.generateFrom (range Iic) ≤ MeasurableSpace.generateFrom (range Ioi) refine' MeasurableSpace.generateFrom_le fun t ht => _ case refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t✝ u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α t : Set α ht : t ∈ range Iic ⊢ MeasurableSet t obtain ⟨u, rfl⟩ := ht case refine'_2.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u✝ : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α u : α ⊢ MeasurableSet (Iic u) rw [← compl_Ioi] case refine'_2.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u✝ : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α u : α ⊢ MeasurableSet (Ioi u)ᶜ exact (MeasurableSpace.measurableSet_generateFrom (mem_range.mpr ⟨u, rfl⟩)).compl ** Qed
IsGδ.measurableSet α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ h : IsGδ s ⊢ MeasurableSet s rcases h with ⟨S, hSo, hSc, rfl⟩ case intro.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ S : Set (Set α) hSo : ∀ (t : Set α), t ∈ S → IsOpen t hSc : Set.Countable S ⊢ MeasurableSet (⋂₀ S) exact MeasurableSet.sInter hSc fun t ht => (hSo t ht).measurableSet ** Qed
measurable_of_isOpen α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ f : δ → γ hf : ∀ (s : Set γ), IsOpen s → MeasurableSet (f ⁻¹' s) ⊢ Measurable f exact measurable_generateFrom hf ** Qed
measurable_of_isClosed α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ f : δ → γ hf : ∀ (s : Set γ), IsClosed s → MeasurableSet (f ⁻¹' s) ⊢ Measurable f apply measurable_of_isOpen case hf α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ f : δ → γ hf : ∀ (s : Set γ), IsClosed s → MeasurableSet (f ⁻¹' s) ⊢ ∀ (s : Set γ), IsOpen s → MeasurableSet (f ⁻¹' s) intro s hs case hf α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ f : δ → γ hf : ∀ (s : Set γ), IsClosed s → MeasurableSet (f ⁻¹' s) s : Set γ hs : IsOpen s ⊢ MeasurableSet (f ⁻¹' s) rw [← MeasurableSet.compl_iff, ← preimage_compl] case hf α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ f : δ → γ hf : ∀ (s : Set γ), IsClosed s → MeasurableSet (f ⁻¹' s) s : Set γ hs : IsOpen s ⊢ MeasurableSet (f ⁻¹' sᶜ) apply hf case hf.a α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ f : δ → γ hf : ∀ (s : Set γ), IsClosed s → MeasurableSet (f ⁻¹' s) s : Set γ hs : IsOpen s ⊢ IsClosed sᶜ rw [isClosed_compl_iff] case hf.a α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ f : δ → γ hf : ∀ (s : Set γ), IsClosed s → MeasurableSet (f ⁻¹' s) s : Set γ hs : IsOpen s ⊢ IsOpen s exact hs ** Qed
measurable_of_is_closed' α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ f : δ → γ hf : ∀ (s : Set γ), IsClosed s → Set.Nonempty s → s ≠ univ → MeasurableSet (f ⁻¹' s) ⊢ Measurable f apply measurable_of_isClosed case hf α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ f : δ → γ hf : ∀ (s : Set γ), IsClosed s → Set.Nonempty s → s ≠ univ → MeasurableSet (f ⁻¹' s) ⊢ ∀ (s : Set γ), IsClosed s → MeasurableSet (f ⁻¹' s) intro s hs case hf α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ f : δ → γ hf : ∀ (s : Set γ), IsClosed s → Set.Nonempty s → s ≠ univ → MeasurableSet (f ⁻¹' s) s : Set γ hs : IsClosed s ⊢ MeasurableSet (f ⁻¹' s) cases' eq_empty_or_nonempty s with h1 h1 case hf.inr α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ f : δ → γ hf : ∀ (s : Set γ), IsClosed s → Set.Nonempty s → s ≠ univ → MeasurableSet (f ⁻¹' s) s : Set γ hs : IsClosed s h1 : Set.Nonempty s ⊢ MeasurableSet (f ⁻¹' s) by_cases h2 : s = univ case neg α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ f : δ → γ hf : ∀ (s : Set γ), IsClosed s → Set.Nonempty s → s ≠ univ → MeasurableSet (f ⁻¹' s) s : Set γ hs : IsClosed s h1 : Set.Nonempty s h2 : ¬s = univ ⊢ MeasurableSet (f ⁻¹' s) exact hf s hs h1 h2 case hf.inl α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ f : δ → γ hf : ∀ (s : Set γ), IsClosed s → Set.Nonempty s → s ≠ univ → MeasurableSet (f ⁻¹' s) s : Set γ hs : IsClosed s h1 : s = ∅ ⊢ MeasurableSet (f ⁻¹' s) simp [h1] case pos α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ f : δ → γ hf : ∀ (s : Set γ), IsClosed s → Set.Nonempty s → s ≠ univ → MeasurableSet (f ⁻¹' s) s : Set γ hs : IsClosed s h1 : Set.Nonempty s h2 : s = univ ⊢ MeasurableSet (f ⁻¹' s) simp [h2] ** Qed
Set.OrdConnected.measurableSet α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹⁶ : TopologicalSpace α inst✝¹⁵ : MeasurableSpace α inst✝¹⁴ : OpensMeasurableSpace α inst✝¹³ : TopologicalSpace β inst✝¹² : MeasurableSpace β inst✝¹¹ : OpensMeasurableSpace β inst✝¹⁰ : TopologicalSpace γ inst✝⁹ : MeasurableSpace γ inst✝⁸ : BorelSpace γ inst✝⁷ : TopologicalSpace γ₂ inst✝⁶ : MeasurableSpace γ₂ inst✝⁵ : BorelSpace γ₂ inst✝⁴ : MeasurableSpace δ α' : Type u_6 inst✝³ : TopologicalSpace α' inst✝² : MeasurableSpace α' inst✝¹ : LinearOrder α inst✝ : OrderClosedTopology α a b x : α h : OrdConnected s ⊢ MeasurableSet s let u := ⋃ (x ∈ s) (y ∈ s), Ioo x y α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u✝ : Set α inst✝¹⁶ : TopologicalSpace α inst✝¹⁵ : MeasurableSpace α inst✝¹⁴ : OpensMeasurableSpace α inst✝¹³ : TopologicalSpace β inst✝¹² : MeasurableSpace β inst✝¹¹ : OpensMeasurableSpace β inst✝¹⁰ : TopologicalSpace γ inst✝⁹ : MeasurableSpace γ inst✝⁸ : BorelSpace γ inst✝⁷ : TopologicalSpace γ₂ inst✝⁶ : MeasurableSpace γ₂ inst✝⁵ : BorelSpace γ₂ inst✝⁴ : MeasurableSpace δ α' : Type u_6 inst✝³ : TopologicalSpace α' inst✝² : MeasurableSpace α' inst✝¹ : LinearOrder α inst✝ : OrderClosedTopology α a b x : α h : OrdConnected s u : Set α := ⋃ x ∈ s, ⋃ y ∈ s, Ioo x y ⊢ MeasurableSet s have huopen : IsOpen u := isOpen_biUnion fun _ _ => isOpen_biUnion fun _ _ => isOpen_Ioo α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u✝ : Set α inst✝¹⁶ : TopologicalSpace α inst✝¹⁵ : MeasurableSpace α inst✝¹⁴ : OpensMeasurableSpace α inst✝¹³ : TopologicalSpace β inst✝¹² : MeasurableSpace β inst✝¹¹ : OpensMeasurableSpace β inst✝¹⁰ : TopologicalSpace γ inst✝⁹ : MeasurableSpace γ inst✝⁸ : BorelSpace γ inst✝⁷ : TopologicalSpace γ₂ inst✝⁶ : MeasurableSpace γ₂ inst✝⁵ : BorelSpace γ₂ inst✝⁴ : MeasurableSpace δ α' : Type u_6 inst✝³ : TopologicalSpace α' inst✝² : MeasurableSpace α' inst✝¹ : LinearOrder α inst✝ : OrderClosedTopology α a b x : α h : OrdConnected s u : Set α := ⋃ x ∈ s, ⋃ y ∈ s, Ioo x y huopen : IsOpen u ⊢ MeasurableSet s have humeas : MeasurableSet u := huopen.measurableSet α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u✝ : Set α inst✝¹⁶ : TopologicalSpace α inst✝¹⁵ : MeasurableSpace α inst✝¹⁴ : OpensMeasurableSpace α inst✝¹³ : TopologicalSpace β inst✝¹² : MeasurableSpace β inst✝¹¹ : OpensMeasurableSpace β inst✝¹⁰ : TopologicalSpace γ inst✝⁹ : MeasurableSpace γ inst✝⁸ : BorelSpace γ inst✝⁷ : TopologicalSpace γ₂ inst✝⁶ : MeasurableSpace γ₂ inst✝⁵ : BorelSpace γ₂ inst✝⁴ : MeasurableSpace δ α' : Type u_6 inst✝³ : TopologicalSpace α' inst✝² : MeasurableSpace α' inst✝¹ : LinearOrder α inst✝ : OrderClosedTopology α a b x : α h : OrdConnected s u : Set α := ⋃ x ∈ s, ⋃ y ∈ s, Ioo x y huopen : IsOpen u humeas : MeasurableSet u ⊢ MeasurableSet s have hfinite : (s \ u).Finite := s.finite_diff_iUnion_Ioo α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u✝ : Set α inst✝¹⁶ : TopologicalSpace α inst✝¹⁵ : MeasurableSpace α inst✝¹⁴ : OpensMeasurableSpace α inst✝¹³ : TopologicalSpace β inst✝¹² : MeasurableSpace β inst✝¹¹ : OpensMeasurableSpace β inst✝¹⁰ : TopologicalSpace γ inst✝⁹ : MeasurableSpace γ inst✝⁸ : BorelSpace γ inst✝⁷ : TopologicalSpace γ₂ inst✝⁶ : MeasurableSpace γ₂ inst✝⁵ : BorelSpace γ₂ inst✝⁴ : MeasurableSpace δ α' : Type u_6 inst✝³ : TopologicalSpace α' inst✝² : MeasurableSpace α' inst✝¹ : LinearOrder α inst✝ : OrderClosedTopology α a b x : α h : OrdConnected s u : Set α := ⋃ x ∈ s, ⋃ y ∈ s, Ioo x y huopen : IsOpen u humeas : MeasurableSet u hfinite : Set.Finite (s \ u) ⊢ MeasurableSet s have : u ⊆ s := iUnion₂_subset fun x hx => iUnion₂_subset fun y hy => Ioo_subset_Icc_self.trans (h.out hx hy) α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u✝ : Set α inst✝¹⁶ : TopologicalSpace α inst✝¹⁵ : MeasurableSpace α inst✝¹⁴ : OpensMeasurableSpace α inst✝¹³ : TopologicalSpace β inst✝¹² : MeasurableSpace β inst✝¹¹ : OpensMeasurableSpace β inst✝¹⁰ : TopologicalSpace γ inst✝⁹ : MeasurableSpace γ inst✝⁸ : BorelSpace γ inst✝⁷ : TopologicalSpace γ₂ inst✝⁶ : MeasurableSpace γ₂ inst✝⁵ : BorelSpace γ₂ inst✝⁴ : MeasurableSpace δ α' : Type u_6 inst✝³ : TopologicalSpace α' inst✝² : MeasurableSpace α' inst✝¹ : LinearOrder α inst✝ : OrderClosedTopology α a b x : α h : OrdConnected s u : Set α := ⋃ x ∈ s, ⋃ y ∈ s, Ioo x y huopen : IsOpen u humeas : MeasurableSet u hfinite : Set.Finite (s \ u) this : u ⊆ s ⊢ MeasurableSet s rw [← union_diff_cancel this] α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u✝ : Set α inst✝¹⁶ : TopologicalSpace α inst✝¹⁵ : MeasurableSpace α inst✝¹⁴ : OpensMeasurableSpace α inst✝¹³ : TopologicalSpace β inst✝¹² : MeasurableSpace β inst✝¹¹ : OpensMeasurableSpace β inst✝¹⁰ : TopologicalSpace γ inst✝⁹ : MeasurableSpace γ inst✝⁸ : BorelSpace γ inst✝⁷ : TopologicalSpace γ₂ inst✝⁶ : MeasurableSpace γ₂ inst✝⁵ : BorelSpace γ₂ inst✝⁴ : MeasurableSpace δ α' : Type u_6 inst✝³ : TopologicalSpace α' inst✝² : MeasurableSpace α' inst✝¹ : LinearOrder α inst✝ : OrderClosedTopology α a b x : α h : OrdConnected s u : Set α := ⋃ x ∈ s, ⋃ y ∈ s, Ioo x y huopen : IsOpen u humeas : MeasurableSet u hfinite : Set.Finite (s \ u) this : u ⊆ s ⊢ MeasurableSet (u ∪ s \ u) exact humeas.union hfinite.measurableSet ** Qed
generateFrom_Ico_mem_le_borel α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t✝ u : Set α✝ inst✝¹⁹ : TopologicalSpace α✝ inst✝¹⁸ : MeasurableSpace α✝ inst✝¹⁷ : OpensMeasurableSpace α✝ inst✝¹⁶ : TopologicalSpace β inst✝¹⁵ : MeasurableSpace β inst✝¹⁴ : OpensMeasurableSpace β inst✝¹³ : TopologicalSpace γ inst✝¹² : MeasurableSpace γ inst✝¹¹ : BorelSpace γ inst✝¹⁰ : TopologicalSpace γ₂ inst✝⁹ : MeasurableSpace γ₂ inst✝⁸ : BorelSpace γ₂ inst✝⁷ : MeasurableSpace δ α' : Type u_6 inst✝⁶ : TopologicalSpace α' inst✝⁵ : MeasurableSpace α' inst✝⁴ : LinearOrder α✝ inst✝³ : OrderClosedTopology α✝ a b x : α✝ α : Type u_7 inst✝² : TopologicalSpace α inst✝¹ : LinearOrder α inst✝ : OrderClosedTopology α s t : Set α ⊢ MeasurableSpace.generateFrom {S \| ∃ l, l ∈ s ∧ ∃ u, u ∈ t ∧ l < u ∧ Ico l u = S} ≤ borel α apply generateFrom_le case h α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t✝ u : Set α✝ inst✝¹⁹ : TopologicalSpace α✝ inst✝¹⁸ : MeasurableSpace α✝ inst✝¹⁷ : OpensMeasurableSpace α✝ inst✝¹⁶ : TopologicalSpace β inst✝¹⁵ : MeasurableSpace β inst✝¹⁴ : OpensMeasurableSpace β inst✝¹³ : TopologicalSpace γ inst✝¹² : MeasurableSpace γ inst✝¹¹ : BorelSpace γ inst✝¹⁰ : TopologicalSpace γ₂ inst✝⁹ : MeasurableSpace γ₂ inst✝⁸ : BorelSpace γ₂ inst✝⁷ : MeasurableSpace δ α' : Type u_6 inst✝⁶ : TopologicalSpace α' inst✝⁵ : MeasurableSpace α' inst✝⁴ : LinearOrder α✝ inst✝³ : OrderClosedTopology α✝ a b x : α✝ α : Type u_7 inst✝² : TopologicalSpace α inst✝¹ : LinearOrder α inst✝ : OrderClosedTopology α s t : Set α ⊢ ∀ (t_1 : Set α), t_1 ∈ {S \| ∃ l, l ∈ s ∧ ∃ u, u ∈ t ∧ l < u ∧ Ico l u = S} → MeasurableSet t_1 borelize α case h α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t✝ u : Set α✝ inst✝¹⁹ : TopologicalSpace α✝ inst✝¹⁸ : MeasurableSpace α✝ inst✝¹⁷ : OpensMeasurableSpace α✝ inst✝¹⁶ : TopologicalSpace β inst✝¹⁵ : MeasurableSpace β inst✝¹⁴ : OpensMeasurableSpace β inst✝¹³ : TopologicalSpace γ inst✝¹² : MeasurableSpace γ inst✝¹¹ : BorelSpace γ inst✝¹⁰ : TopologicalSpace γ₂ inst✝⁹ : MeasurableSpace γ₂ inst✝⁸ : BorelSpace γ₂ inst✝⁷ : MeasurableSpace δ α' : Type u_6 inst✝⁶ : TopologicalSpace α' inst✝⁵ : MeasurableSpace α' inst✝⁴ : LinearOrder α✝ inst✝³ : OrderClosedTopology α✝ a b x : α✝ α : Type u_7 inst✝² : TopologicalSpace α inst✝¹ : LinearOrder α inst✝ : OrderClosedTopology α s t : Set α this✝¹ : MeasurableSpace α := borel α this✝ : BorelSpace α ⊢ ∀ (t_1 : Set α), t_1 ∈ {S \| ∃ l, l ∈ s ∧ ∃ u, u ∈ t ∧ l < u ∧ Ico l u = S} → MeasurableSet t_1 rintro _ ⟨a, -, b, -, -, rfl⟩ case h.intro.intro.intro.intro.intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t✝ u : Set α✝ inst✝¹⁹ : TopologicalSpace α✝ inst✝¹⁸ : MeasurableSpace α✝ inst✝¹⁷ : OpensMeasurableSpace α✝ inst✝¹⁶ : TopologicalSpace β inst✝¹⁵ : MeasurableSpace β inst✝¹⁴ : OpensMeasurableSpace β inst✝¹³ : TopologicalSpace γ inst✝¹² : MeasurableSpace γ inst✝¹¹ : BorelSpace γ inst✝¹⁰ : TopologicalSpace γ₂ inst✝⁹ : MeasurableSpace γ₂ inst✝⁸ : BorelSpace γ₂ inst✝⁷ : MeasurableSpace δ α' : Type u_6 inst✝⁶ : TopologicalSpace α' inst✝⁵ : MeasurableSpace α' inst✝⁴ : LinearOrder α✝ inst✝³ : OrderClosedTopology α✝ a✝ b✝ x : α✝ α : Type u_7 inst✝² : TopologicalSpace α inst✝¹ : LinearOrder α inst✝ : OrderClosedTopology α s t : Set α this✝¹ : MeasurableSpace α := borel α this✝ : BorelSpace α a b : α ⊢ MeasurableSet (Ico a b) exact measurableSet_Ico ** Qed
Dense.borel_eq_generateFrom_Ico_mem α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α✝ inst✝²² : TopologicalSpace α✝ inst✝²¹ : MeasurableSpace α✝ inst✝²⁰ : OpensMeasurableSpace α✝ inst✝¹⁹ : TopologicalSpace β inst✝¹⁸ : MeasurableSpace β inst✝¹⁷ : OpensMeasurableSpace β inst✝¹⁶ : TopologicalSpace γ inst✝¹⁵ : MeasurableSpace γ inst✝¹⁴ : BorelSpace γ inst✝¹³ : TopologicalSpace γ₂ inst✝¹² : MeasurableSpace γ₂ inst✝¹¹ : BorelSpace γ₂ inst✝¹⁰ : MeasurableSpace δ α' : Type u_6 inst✝⁹ : TopologicalSpace α' inst✝⁸ : MeasurableSpace α' inst✝⁷ : LinearOrder α✝ inst✝⁶ : OrderClosedTopology α✝ a b x : α✝ α : Type u_7 inst✝⁵ : TopologicalSpace α inst✝⁴ : LinearOrder α inst✝³ : OrderTopology α inst✝² : SecondCountableTopology α inst✝¹ : DenselyOrdered α inst✝ : NoMinOrder α s : Set α hd : Dense s ⊢ ∀ (x : α), IsBot x → x ∈ s simp ** Qed

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essSup_map_measure_of_measurable α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α inst✝⁵ : CompleteLattice β γ : Type u_3 mγ : MeasurableSpace γ f : α → γ g : γ → β inst✝⁴ : MeasurableSpace β inst✝³ : TopologicalSpace β inst✝² : SecondCountableTopology β inst✝¹ : OrderClosedTopology β inst✝ : OpensMeasurableSpace β hg : Measurable g hf : AEMeasurable f ⊢ essSup g (Measure.map f μ) = essSup (g ∘ f) μ refine' le_antisymm _ (essSup_comp_le_essSup_map_measure hf) α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α inst✝⁵ : CompleteLattice β γ : Type u_3 mγ : MeasurableSpace γ f : α → γ g : γ → β inst✝⁴ : MeasurableSpace β inst✝³ : TopologicalSpace β inst✝² : SecondCountableTopology β inst✝¹ : OrderClosedTopology β inst✝ : OpensMeasurableSpace β hg : Measurable g hf : AEMeasurable f ⊢ essSup g (Measure.map f μ) ≤ essSup (g ∘ f) μ refine' limsSup_le_limsSup (by isBoundedDefault) (by isBoundedDefault) (fun c h_le => _) α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α inst✝⁵ : CompleteLattice β γ : Type u_3 mγ : MeasurableSpace γ f : α → γ g : γ → β inst✝⁴ : MeasurableSpace β inst✝³ : TopologicalSpace β inst✝² : SecondCountableTopology β inst✝¹ : OrderClosedTopology β inst✝ : OpensMeasurableSpace β hg : Measurable g hf : AEMeasurable f c : β h_le : ∀ᶠ (n : β) in map (g ∘ f) (Measure.ae μ), n ≤ c ⊢ ∀ᶠ (n : β) in map g (Measure.ae (Measure.map f μ)), n ≤ c rw [eventually_map] at h_le ⊢ α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α inst✝⁵ : CompleteLattice β γ : Type u_3 mγ : MeasurableSpace γ f : α → γ g : γ → β inst✝⁴ : MeasurableSpace β inst✝³ : TopologicalSpace β inst✝² : SecondCountableTopology β inst✝¹ : OrderClosedTopology β inst✝ : OpensMeasurableSpace β hg : Measurable g hf : AEMeasurable f c : β h_le : ∀ᵐ (a : α) ∂μ, (g ∘ f) a ≤ c ⊢ ∀ᵐ (a : γ) ∂Measure.map f μ, g a ≤ c rw [ae_map_iff hf (measurableSet_le hg measurable_const)] α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α inst✝⁵ : CompleteLattice β γ : Type u_3 mγ : MeasurableSpace γ f : α → γ g : γ → β inst✝⁴ : MeasurableSpace β inst✝³ : TopologicalSpace β inst✝² : SecondCountableTopology β inst✝¹ : OrderClosedTopology β inst✝ : OpensMeasurableSpace β hg : Measurable g hf : AEMeasurable f c : β h_le : ∀ᵐ (a : α) ∂μ, (g ∘ f) a ≤ c ⊢ ∀ᵐ (x : α) ∂μ, g (f x) ≤ c exact h_le α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α inst✝⁵ : CompleteLattice β γ : Type u_3 mγ : MeasurableSpace γ f : α → γ g : γ → β inst✝⁴ : MeasurableSpace β inst✝³ : TopologicalSpace β inst✝² : SecondCountableTopology β inst✝¹ : OrderClosedTopology β inst✝ : OpensMeasurableSpace β hg : Measurable g hf : AEMeasurable f ⊢ IsCobounded (fun x x_1 => x ≤ x_1) (map g (Measure.ae (Measure.map f μ))) isBoundedDefault α : Type u_1 β : Type u_2 m : MeasurableSpace α μ ν : Measure α inst✝⁵ : CompleteLattice β γ : Type u_3 mγ : MeasurableSpace γ f : α → γ g : γ → β inst✝⁴ : MeasurableSpace β inst✝³ : TopologicalSpace β inst✝² : SecondCountableTopology β inst✝¹ : OrderClosedTopology β inst✝ : OpensMeasurableSpace β hg : Measurable g hf : AEMeasurable f ⊢ IsBounded (fun x x_1 => x ≤ x_1) (map (g ∘ f) (Measure.ae μ)) isBoundedDefault ** Qed
VitaliFamily.eventually_measure_lt_top α : Type u_1 inst✝² : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝¹ : NormedAddCommGroup E inst✝ : IsLocallyFiniteMeasure μ x : α ⊢ ∀ᶠ (a : Set α) in filterAt v x, ↑↑μ a < ⊤ obtain ⟨ε, εpos, με⟩ : ∃ (ε : ℝ), 0 < ε ∧ μ (closedBall x ε) < ∞ := (μ.finiteAt_nhds x).exists_mem_basis nhds_basis_closedBall case intro.intro α : Type u_1 inst✝² : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝¹ : NormedAddCommGroup E inst✝ : IsLocallyFiniteMeasure μ x : α ε : ℝ εpos : 0 < ε με : ↑↑μ (closedBall x ε) < ⊤ ⊢ ∀ᶠ (a : Set α) in filterAt v x, ↑↑μ a < ⊤ exact v.eventually_filterAt_iff.2 ⟨ε, εpos, fun a _ haε => (measure_mono haε).trans_lt με⟩ ** Qed
VitaliFamily.measure_le_of_frequently_le α : Type u_1 inst✝⁴ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝³ : NormedAddCommGroup E inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α ρ ν : Measure α inst✝ : IsLocallyFiniteMeasure ν hρ : ρ ≪ μ s : Set α hs : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑ν a ⊢ ↑↑ρ s ≤ ↑↑ν s apply ENNReal.le_of_forall_pos_le_add fun ε εpos _ => ?_ α : Type u_1 inst✝⁴ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝³ : NormedAddCommGroup E inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α ρ ν : Measure α inst✝ : IsLocallyFiniteMeasure ν hρ : ρ ≪ μ s : Set α hs : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑ν a ε : ℝ≥0 εpos : 0 < ε x✝ : ↑↑ν s < ⊤ ⊢ ↑↑ρ s ≤ ↑↑ν s + ↑ε obtain ⟨U, sU, U_open, νU⟩ : ∃ (U : Set α), s ⊆ U ∧ IsOpen U ∧ ν U ≤ ν s + ε := exists_isOpen_le_add s ν (ENNReal.coe_pos.2 εpos).ne' case intro.intro.intro α : Type u_1 inst✝⁴ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝³ : NormedAddCommGroup E inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α ρ ν : Measure α inst✝ : IsLocallyFiniteMeasure ν hρ : ρ ≪ μ s : Set α hs : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑ν a ε : ℝ≥0 εpos : 0 < ε x✝ : ↑↑ν s < ⊤ U : Set α sU : s ⊆ U U_open : IsOpen U νU : ↑↑ν U ≤ ↑↑ν s + ↑ε ⊢ ↑↑ρ s ≤ ↑↑ν s + ↑ε let f : α → Set (Set α) := fun _ => {a \| ρ a ≤ ν a ∧ a ⊆ U} case intro.intro.intro α : Type u_1 inst✝⁴ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝³ : NormedAddCommGroup E inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α ρ ν : Measure α inst✝ : IsLocallyFiniteMeasure ν hρ : ρ ≪ μ s : Set α hs : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑ν a ε : ℝ≥0 εpos : 0 < ε x✝ : ↑↑ν s < ⊤ U : Set α sU : s ⊆ U U_open : IsOpen U νU : ↑↑ν U ≤ ↑↑ν s + ↑ε f : α → Set (Set α) := fun x => {a \| ↑↑ρ a ≤ ↑↑ν a ∧ a ⊆ U} ⊢ ↑↑ρ s ≤ ↑↑ν s + ↑ε have h : v.FineSubfamilyOn f s := by apply v.fineSubfamilyOn_of_frequently f s fun x hx => ?_ have := (hs x hx).and_eventually ((v.eventually_filterAt_mem_sets x).and (v.eventually_filterAt_subset_of_nhds (U_open.mem_nhds (sU hx)))) apply Frequently.mono this rintro a ⟨ρa, _, aU⟩ exact ⟨ρa, aU⟩ case intro.intro.intro α : Type u_1 inst✝⁴ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝³ : NormedAddCommGroup E inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α ρ ν : Measure α inst✝ : IsLocallyFiniteMeasure ν hρ : ρ ≪ μ s : Set α hs : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑ν a ε : ℝ≥0 εpos : 0 < ε x✝ : ↑↑ν s < ⊤ U : Set α sU : s ⊆ U U_open : IsOpen U νU : ↑↑ν U ≤ ↑↑ν s + ↑ε f : α → Set (Set α) := fun x => {a \| ↑↑ρ a ≤ ↑↑ν a ∧ a ⊆ U} h : FineSubfamilyOn v f s ⊢ ↑↑ρ s ≤ ↑↑ν s + ↑ε haveI : Encodable h.index := h.index_countable.toEncodable case intro.intro.intro α : Type u_1 inst✝⁴ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝³ : NormedAddCommGroup E inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α ρ ν : Measure α inst✝ : IsLocallyFiniteMeasure ν hρ : ρ ≪ μ s : Set α hs : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑ν a ε : ℝ≥0 εpos : 0 < ε x✝ : ↑↑ν s < ⊤ U : Set α sU : s ⊆ U U_open : IsOpen U νU : ↑↑ν U ≤ ↑↑ν s + ↑ε f : α → Set (Set α) := fun x => {a \| ↑↑ρ a ≤ ↑↑ν a ∧ a ⊆ U} h : FineSubfamilyOn v f s this : Encodable ↑(FineSubfamilyOn.index h) ⊢ ↑↑ρ s ≤ ↑↑ν s + ↑ε calc ρ s ≤ ∑' x : h.index, ρ (h.covering x) := h.measure_le_tsum_of_absolutelyContinuous hρ _ ≤ ∑' x : h.index, ν (h.covering x) := (ENNReal.tsum_le_tsum fun x => (h.covering_mem x.2).1) _ = ν (⋃ x : h.index, h.covering x) := by rw [measure_iUnion h.covering_disjoint_subtype fun i => h.measurableSet_u i.2] _ ≤ ν U := (measure_mono (iUnion_subset fun i => (h.covering_mem i.2).2)) _ ≤ ν s + ε := νU α : Type u_1 inst✝⁴ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝³ : NormedAddCommGroup E inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α ρ ν : Measure α inst✝ : IsLocallyFiniteMeasure ν hρ : ρ ≪ μ s : Set α hs : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑ν a ε : ℝ≥0 εpos : 0 < ε x✝ : ↑↑ν s < ⊤ U : Set α sU : s ⊆ U U_open : IsOpen U νU : ↑↑ν U ≤ ↑↑ν s + ↑ε f : α → Set (Set α) := fun x => {a \| ↑↑ρ a ≤ ↑↑ν a ∧ a ⊆ U} ⊢ FineSubfamilyOn v f s apply v.fineSubfamilyOn_of_frequently f s fun x hx => ?_ α : Type u_1 inst✝⁴ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝³ : NormedAddCommGroup E inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α ρ ν : Measure α inst✝ : IsLocallyFiniteMeasure ν hρ : ρ ≪ μ s : Set α hs : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑ν a ε : ℝ≥0 εpos : 0 < ε x✝ : ↑↑ν s < ⊤ U : Set α sU : s ⊆ U U_open : IsOpen U νU : ↑↑ν U ≤ ↑↑ν s + ↑ε f : α → Set (Set α) := fun x => {a \| ↑↑ρ a ≤ ↑↑ν a ∧ a ⊆ U} x : α hx : x ∈ s ⊢ ∃ᶠ (a : Set α) in filterAt v x, a ∈ f x have := (hs x hx).and_eventually ((v.eventually_filterAt_mem_sets x).and (v.eventually_filterAt_subset_of_nhds (U_open.mem_nhds (sU hx)))) α : Type u_1 inst✝⁴ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝³ : NormedAddCommGroup E inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α ρ ν : Measure α inst✝ : IsLocallyFiniteMeasure ν hρ : ρ ≪ μ s : Set α hs : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑ν a ε : ℝ≥0 εpos : 0 < ε x✝ : ↑↑ν s < ⊤ U : Set α sU : s ⊆ U U_open : IsOpen U νU : ↑↑ν U ≤ ↑↑ν s + ↑ε f : α → Set (Set α) := fun x => {a \| ↑↑ρ a ≤ ↑↑ν a ∧ a ⊆ U} x : α hx : x ∈ s this : ∃ᶠ (x_1 : Set α) in filterAt v x, ↑↑ρ x_1 ≤ ↑↑ν x_1 ∧ x_1 ∈ setsAt v x ∧ x_1 ⊆ U ⊢ ∃ᶠ (a : Set α) in filterAt v x, a ∈ f x apply Frequently.mono this α : Type u_1 inst✝⁴ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝³ : NormedAddCommGroup E inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α ρ ν : Measure α inst✝ : IsLocallyFiniteMeasure ν hρ : ρ ≪ μ s : Set α hs : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑ν a ε : ℝ≥0 εpos : 0 < ε x✝ : ↑↑ν s < ⊤ U : Set α sU : s ⊆ U U_open : IsOpen U νU : ↑↑ν U ≤ ↑↑ν s + ↑ε f : α → Set (Set α) := fun x => {a \| ↑↑ρ a ≤ ↑↑ν a ∧ a ⊆ U} x : α hx : x ∈ s this : ∃ᶠ (x_1 : Set α) in filterAt v x, ↑↑ρ x_1 ≤ ↑↑ν x_1 ∧ x_1 ∈ setsAt v x ∧ x_1 ⊆ U ⊢ ∀ (x_1 : Set α), ↑↑ρ x_1 ≤ ↑↑ν x_1 ∧ x_1 ∈ setsAt v x ∧ x_1 ⊆ U → x_1 ∈ f x rintro a ⟨ρa, _, aU⟩ case intro.intro α : Type u_1 inst✝⁴ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝³ : NormedAddCommGroup E inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α ρ ν : Measure α inst✝ : IsLocallyFiniteMeasure ν hρ : ρ ≪ μ s : Set α hs : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑ν a ε : ℝ≥0 εpos : 0 < ε x✝ : ↑↑ν s < ⊤ U : Set α sU : s ⊆ U U_open : IsOpen U νU : ↑↑ν U ≤ ↑↑ν s + ↑ε f : α → Set (Set α) := fun x => {a \| ↑↑ρ a ≤ ↑↑ν a ∧ a ⊆ U} x : α hx : x ∈ s this : ∃ᶠ (x_1 : Set α) in filterAt v x, ↑↑ρ x_1 ≤ ↑↑ν x_1 ∧ x_1 ∈ setsAt v x ∧ x_1 ⊆ U a : Set α ρa : ↑↑ρ a ≤ ↑↑ν a left✝ : a ∈ setsAt v x aU : a ⊆ U ⊢ a ∈ f x exact ⟨ρa, aU⟩ α : Type u_1 inst✝⁴ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝³ : NormedAddCommGroup E inst✝² : SecondCountableTopology α inst✝¹ : BorelSpace α ρ ν : Measure α inst✝ : IsLocallyFiniteMeasure ν hρ : ρ ≪ μ s : Set α hs : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑ν a ε : ℝ≥0 εpos : 0 < ε x✝ : ↑↑ν s < ⊤ U : Set α sU : s ⊆ U U_open : IsOpen U νU : ↑↑ν U ≤ ↑↑ν s + ↑ε f : α → Set (Set α) := fun x => {a \| ↑↑ρ a ≤ ↑↑ν a ∧ a ⊆ U} h : FineSubfamilyOn v f s this : Encodable ↑(FineSubfamilyOn.index h) ⊢ ∑' (x : ↑(FineSubfamilyOn.index h)), ↑↑ν (FineSubfamilyOn.covering h ↑x) = ↑↑ν (⋃ x, FineSubfamilyOn.covering h ↑x) rw [measure_iUnion h.covering_disjoint_subtype fun i => h.measurableSet_u i.2] ** Qed
VitaliFamily.ae_eventually_measure_zero_of_singular α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 0) have A : ∀ ε > (0 : ℝ≥0), ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, ρ a < ε * μ a := by intro ε εpos set s := {x \| ¬∀ᶠ a in v.filterAt x, ρ a < ε * μ a} with hs change μ s = 0 obtain ⟨o, _, ρo, μo⟩ : ∃ o : Set α, MeasurableSet o ∧ ρ o = 0 ∧ μ oᶜ = 0 := hρ apply le_antisymm _ bot_le calc μ s ≤ μ (s ∩ o ∪ oᶜ) := by conv_lhs => rw [← inter_union_compl s o] exact measure_mono (union_subset_union_right _ (inter_subset_right _ _)) _ ≤ μ (s ∩ o) + μ oᶜ := (measure_union_le _ _) _ = μ (s ∩ o) := by rw [μo, add_zero] _ = (ε : ℝ≥0∞)⁻¹ * (ε • μ) (s ∩ o) := by simp only [coe_nnreal_smul_apply, ← mul_assoc, mul_comm _ (ε : ℝ≥0∞)] rw [ENNReal.mul_inv_cancel (ENNReal.coe_pos.2 εpos).ne' ENNReal.coe_ne_top, one_mul] _ ≤ (ε : ℝ≥0∞)⁻¹ * ρ (s ∩ o) := by refine' mul_le_mul_left' _ _ refine' v.measure_le_of_frequently_le ρ ((Measure.AbsolutelyContinuous.refl μ).smul ε) _ _ intro x hx rw [hs] at hx simp only [mem_inter_iff, not_lt, not_eventually, mem_setOf_eq] at hx exact hx.1 _ ≤ (ε : ℝ≥0∞)⁻¹ * ρ o := (mul_le_mul_left' (measure_mono (inter_subset_right _ _)) _) _ = 0 := by rw [ρo, mul_zero] ** α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ A : ∀ (ε : ℝ≥0), ε > 0 → ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 0) obtain ⟨u, _, u_pos, u_lim⟩ : ∃ u : ℕ → ℝ≥0, StrictAnti u ∧ (∀ n : ℕ, 0 < u n) ∧ Tendsto u atTop (𝓝 0) := exists_seq_strictAnti_tendsto (0 : ℝ≥0) case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ A : ∀ (ε : ℝ≥0), ε > 0 → ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a u : ℕ → ℝ≥0 left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 0) ** have B : ∀ᵐ x ∂μ, ∀ n, ∀ᶠ a in v.filterAt x, ρ a < u n * μ a := ae_all_iff.2 fun n => A (u n) (u_pos n) ** case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ A : ∀ (ε : ℝ≥0), ε > 0 → ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a u : ℕ → ℝ≥0 left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) B : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 0) filter_upwards [B, v.ae_eventually_measure_pos] case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ A : ∀ (ε : ℝ≥0), ε > 0 → ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a u : ℕ → ℝ≥0 left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) B : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a ⊢ ∀ (a : α), (∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v a, ↑↑ρ a < ↑(u n) * ↑↑μ a) → (∀ᶠ (a : Set α) in filterAt v a, 0 < ↑↑μ a) → Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v a) (𝓝 0) intro x hx h'x case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ A : ∀ (ε : ℝ≥0), ε > 0 → ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a u : ℕ → ℝ≥0 left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) B : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a x : α hx : ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a ⊢ Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 0) refine' tendsto_order.2 ⟨fun z hz => (ENNReal.not_lt_zero hz).elim, fun z hz => _⟩ case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ A : ∀ (ε : ℝ≥0), ε > 0 → ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a u : ℕ → ℝ≥0 left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) B : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a x : α hx : ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a z : ℝ≥0∞ hz : z > 0 ⊢ ∀ᶠ (b : Set α) in filterAt v x, ↑↑ρ b / ↑↑μ b < z obtain ⟨w, w_pos, w_lt⟩ : ∃ w : ℝ≥0, (0 : ℝ≥0∞) < w ∧ (w : ℝ≥0∞) < z := ENNReal.lt_iff_exists_nnreal_btwn.1 hz case h.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ A : ∀ (ε : ℝ≥0), ε > 0 → ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a u : ℕ → ℝ≥0 left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) B : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a x : α hx : ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a z : ℝ≥0∞ hz : z > 0 w : ℝ≥0 w_pos : 0 < ↑w w_lt : ↑w < z ⊢ ∀ᶠ (b : Set α) in filterAt v x, ↑↑ρ b / ↑↑μ b < z obtain ⟨n, hn⟩ : ∃ n, u n < w := ((tendsto_order.1 u_lim).2 w (ENNReal.coe_pos.1 w_pos)).exists case h.intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ A : ∀ (ε : ℝ≥0), ε > 0 → ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a u : ℕ → ℝ≥0 left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) B : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a x : α hx : ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a z : ℝ≥0∞ hz : z > 0 w : ℝ≥0 w_pos : 0 < ↑w w_lt : ↑w < z n : ℕ hn : u n < w ⊢ ∀ᶠ (b : Set α) in filterAt v x, ↑↑ρ b / ↑↑μ b < z filter_upwards [hx n, h'x, v.eventually_measure_lt_top x] case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ A : ∀ (ε : ℝ≥0), ε > 0 → ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a u : ℕ → ℝ≥0 left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) B : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a x : α hx : ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a z : ℝ≥0∞ hz : z > 0 w : ℝ≥0 w_pos : 0 < ↑w w_lt : ↑w < z n : ℕ hn : u n < w ⊢ ∀ (a : Set α), ↑↑ρ a < ↑(u n) * ↑↑μ a → 0 < ↑↑μ a → ↑↑μ a < ⊤ → ↑↑ρ a / ↑↑μ a < z intro a ha μa_pos μa_lt_top case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ A : ∀ (ε : ℝ≥0), ε > 0 → ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a u : ℕ → ℝ≥0 left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) B : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a x : α hx : ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a z : ℝ≥0∞ hz : z > 0 w : ℝ≥0 w_pos : 0 < ↑w w_lt : ↑w < z n : ℕ hn : u n < w a : Set α ha : ↑↑ρ a < ↑(u n) * ↑↑μ a μa_pos : 0 < ↑↑μ a μa_lt_top : ↑↑μ a < ⊤ ⊢ ↑↑ρ a / ↑↑μ a < z rw [ENNReal.div_lt_iff (Or.inl μa_pos.ne') (Or.inl μa_lt_top.ne)] case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ A : ∀ (ε : ℝ≥0), ε > 0 → ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a u : ℕ → ℝ≥0 left✝ : StrictAnti u u_pos : ∀ (n : ℕ), 0 < u n u_lim : Tendsto u atTop (𝓝 0) B : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a x : α hx : ∀ (n : ℕ), ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑(u n) * ↑↑μ a h'x : ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a z : ℝ≥0∞ hz : z > 0 w : ℝ≥0 w_pos : 0 < ↑w w_lt : ↑w < z n : ℕ hn : u n < w a : Set α ha : ↑↑ρ a < ↑(u n) * ↑↑μ a μa_pos : 0 < ↑↑μ a μa_lt_top : ↑↑μ a < ⊤ ⊢ ↑↑ρ a < z * ↑↑μ a exact ha.trans_le (mul_le_mul_right' ((ENNReal.coe_le_coe.2 hn.le).trans w_lt.le) _) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ ⊢ ∀ (ε : ℝ≥0), ε > 0 → ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a intro ε εpos α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ ε : ℝ≥0 εpos : ε > 0 ⊢ ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a ** set s := {x \| ¬∀ᶠ a in v.filterAt x, ρ a < ε * μ a} with hs ** α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} ⊢ ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a change μ s = 0 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ⟂ₘ μ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} ⊢ ↑↑μ s = 0 obtain ⟨o, _, ρo, μo⟩ : ∃ o : Set α, MeasurableSet o ∧ ρ o = 0 ∧ μ oᶜ = 0 := hρ case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} o : Set α left✝ : MeasurableSet o ρo : ↑↑ρ o = 0 μo : ↑↑μ oᶜ = 0 ⊢ ↑↑μ s = 0 apply le_antisymm _ bot_le α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} o : Set α left✝ : MeasurableSet o ρo : ↑↑ρ o = 0 μo : ↑↑μ oᶜ = 0 ⊢ ↑↑μ s ≤ ⊥ ** calc μ s ≤ μ (s ∩ o ∪ oᶜ) := by conv_lhs => rw [← inter_union_compl s o] exact measure_mono (union_subset_union_right _ (inter_subset_right _ _)) _ ≤ μ (s ∩ o) + μ oᶜ := (measure_union_le _ _) _ = μ (s ∩ o) := by rw [μo, add_zero] _ = (ε : ℝ≥0∞)⁻¹ * (ε • μ) (s ∩ o) := by simp only [coe_nnreal_smul_apply, ← mul_assoc, mul_comm _ (ε : ℝ≥0∞)] rw [ENNReal.mul_inv_cancel (ENNReal.coe_pos.2 εpos).ne' ENNReal.coe_ne_top, one_mul] _ ≤ (ε : ℝ≥0∞)⁻¹ * ρ (s ∩ o) := by refine' mul_le_mul_left' _ _ refine' v.measure_le_of_frequently_le ρ ((Measure.AbsolutelyContinuous.refl μ).smul ε) _ _ intro x hx rw [hs] at hx simp only [mem_inter_iff, not_lt, not_eventually, mem_setOf_eq] at hx exact hx.1 _ ≤ (ε : ℝ≥0∞)⁻¹ * ρ o := (mul_le_mul_left' (measure_mono (inter_subset_right _ _)) _) _ = 0 := by rw [ρo, mul_zero] ** α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} o : Set α left✝ : MeasurableSet o ρo : ↑↑ρ o = 0 μo : ↑↑μ oᶜ = 0 ⊢ ↑↑μ s ≤ ↑↑μ (s ∩ o ∪ oᶜ) conv_lhs => rw [← inter_union_compl s o] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} o : Set α left✝ : MeasurableSet o ρo : ↑↑ρ o = 0 μo : ↑↑μ oᶜ = 0 ⊢ ↑↑μ (s ∩ o ∪ s ∩ oᶜ) ≤ ↑↑μ (s ∩ o ∪ oᶜ) exact measure_mono (union_subset_union_right _ (inter_subset_right _ _)) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} o : Set α left✝ : MeasurableSet o ρo : ↑↑ρ o = 0 μo : ↑↑μ oᶜ = 0 ⊢ ↑↑μ (s ∩ o) + ↑↑μ oᶜ = ↑↑μ (s ∩ o) rw [μo, add_zero] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} o : Set α left✝ : MeasurableSet o ρo : ↑↑ρ o = 0 μo : ↑↑μ oᶜ = 0 ⊢ ↑↑μ (s ∩ o) = (↑ε)⁻¹ * ↑↑(ε • μ) (s ∩ o) simp only [coe_nnreal_smul_apply, ← mul_assoc, mul_comm _ (ε : ℝ≥0∞)] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} o : Set α left✝ : MeasurableSet o ρo : ↑↑ρ o = 0 μo : ↑↑μ oᶜ = 0 ⊢ ↑↑μ ({x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} ∩ o) = ↑ε * (↑ε)⁻¹ * ↑↑μ ({x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} ∩ o) rw [ENNReal.mul_inv_cancel (ENNReal.coe_pos.2 εpos).ne' ENNReal.coe_ne_top, one_mul] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} o : Set α left✝ : MeasurableSet o ρo : ↑↑ρ o = 0 μo : ↑↑μ oᶜ = 0 ⊢ (↑ε)⁻¹ * ↑↑(ε • μ) (s ∩ o) ≤ (↑ε)⁻¹ * ↑↑ρ (s ∩ o) refine' mul_le_mul_left' _ _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} o : Set α left✝ : MeasurableSet o ρo : ↑↑ρ o = 0 μo : ↑↑μ oᶜ = 0 ⊢ ↑↑(ε • μ) (s ∩ o) ≤ ↑↑ρ (s ∩ o) refine' v.measure_le_of_frequently_le ρ ((Measure.AbsolutelyContinuous.refl μ).smul ε) _ _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} o : Set α left✝ : MeasurableSet o ρo : ↑↑ρ o = 0 μo : ↑↑μ oᶜ = 0 ⊢ ∀ (x : α), x ∈ s ∩ o → ∃ᶠ (a : Set α) in filterAt v x, ↑↑(ε • μ) a ≤ ↑↑ρ a intro x hx α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} o : Set α left✝ : MeasurableSet o ρo : ↑↑ρ o = 0 μo : ↑↑μ oᶜ = 0 x : α hx : x ∈ s ∩ o ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑(ε • μ) a ≤ ↑↑ρ a rw [hs] at hx α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} o : Set α left✝ : MeasurableSet o ρo : ↑↑ρ o = 0 μo : ↑↑μ oᶜ = 0 x : α hx : x ∈ {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} ∩ o ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑(ε • μ) a ≤ ↑↑ρ a simp only [mem_inter_iff, not_lt, not_eventually, mem_setOf_eq] at hx α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} o : Set α left✝ : MeasurableSet o ρo : ↑↑ρ o = 0 μo : ↑↑μ oᶜ = 0 x : α hx : (∃ᶠ (x : Set α) in filterAt v x, ↑ε * ↑↑μ x ≤ ↑↑ρ x) ∧ x ∈ o ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑(ε • μ) a ≤ ↑↑ρ a exact hx.1 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ε : ℝ≥0 εpos : ε > 0 s : Set α := {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} hs : s = {x \| ¬∀ᶠ (a : Set α) in filterAt v x, ↑↑ρ a < ↑ε * ↑↑μ a} o : Set α left✝ : MeasurableSet o ρo : ↑↑ρ o = 0 μo : ↑↑μ oᶜ = 0 ⊢ (↑ε)⁻¹ * ↑↑ρ o = 0 rw [ρo, mul_zero] Qed
VitaliFamily.null_of_frequently_le_of_frequently_ge ** α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ c d : ℝ≥0 hcd : c < d s : Set α hc : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑c * ↑↑μ a hd : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑d * ↑↑μ a ≤ ↑↑ρ a ⊢ ↑↑μ s = 0 apply null_of_locally_null s fun x _ => ?_ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ c d : ℝ≥0 hcd : c < d s : Set α hc : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑c * ↑↑μ a hd : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑d * ↑↑μ a ≤ ↑↑ρ a x : α x✝ : x ∈ s ⊢ ∃ u, u ∈ 𝓝[s] x ∧ ↑↑μ u = 0 obtain ⟨o, xo, o_open, μo⟩ : ∃ o : Set α, x ∈ o ∧ IsOpen o ∧ μ o < ∞ := Measure.exists_isOpen_measure_lt_top μ x case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ c d : ℝ≥0 hcd : c < d s : Set α hc : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑c * ↑↑μ a hd : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑d * ↑↑μ a ≤ ↑↑ρ a x : α x✝ : x ∈ s o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ ⊢ ∃ u, u ∈ 𝓝[s] x ∧ ↑↑μ u = 0 refine' ⟨s ∩ o, inter_mem_nhdsWithin _ (o_open.mem_nhds xo), _⟩ case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ c d : ℝ≥0 hcd : c < d s : Set α hc : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑c * ↑↑μ a hd : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑d * ↑↑μ a ≤ ↑↑ρ a x : α x✝ : x ∈ s o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ ⊢ ↑↑μ (s ∩ o) = 0 let s' := s ∩ o case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ c d : ℝ≥0 hcd : c < d s : Set α hc : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑c * ↑↑μ a hd : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑d * ↑↑μ a ≤ ↑↑ρ a x : α x✝ : x ∈ s o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s' : Set α := s ∩ o ⊢ ↑↑μ (s ∩ o) = 0 by_contra h case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ c d : ℝ≥0 hcd : c < d s : Set α hc : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑c * ↑↑μ a hd : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑d * ↑↑μ a ≤ ↑↑ρ a x : α x✝ : x ∈ s o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s' : Set α := s ∩ o h : ¬↑↑μ (s ∩ o) = 0 ⊢ False apply lt_irrefl (ρ s') case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ c d : ℝ≥0 hcd : c < d s : Set α hc : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑c * ↑↑μ a hd : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑d * ↑↑μ a ≤ ↑↑ρ a x : α x✝ : x ∈ s o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s' : Set α := s ∩ o h : ¬↑↑μ (s ∩ o) = 0 ⊢ ↑↑ρ s' < ↑↑ρ s' ** calc ρ s' ≤ c * μ s' := v.measure_le_of_frequently_le (c • μ) hρ s' fun x hx => hc x hx.1 _ < d * μ s' := by apply (ENNReal.mul_lt_mul_right h _).2 (ENNReal.coe_lt_coe.2 hcd) exact (lt_of_le_of_lt (measure_mono (inter_subset_right _ _)) μo).ne _ ≤ ρ s' := v.measure_le_of_frequently_le ρ ((Measure.AbsolutelyContinuous.refl μ).smul d) s' fun x hx => hd x hx.1 ** α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ c d : ℝ≥0 hcd : c < d s : Set α hc : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑c * ↑↑μ a hd : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑d * ↑↑μ a ≤ ↑↑ρ a x : α x✝ : x ∈ s o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s' : Set α := s ∩ o h : ¬↑↑μ (s ∩ o) = 0 ⊢ ↑c * ↑↑μ s' < ↑d * ↑↑μ s' apply (ENNReal.mul_lt_mul_right h _).2 (ENNReal.coe_lt_coe.2 hcd) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ c d : ℝ≥0 hcd : c < d s : Set α hc : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑c * ↑↑μ a hd : ∀ (x : α), x ∈ s → ∃ᶠ (a : Set α) in filterAt v x, ↑d * ↑↑μ a ≤ ↑↑ρ a x : α x✝ : x ∈ s o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s' : Set α := s ∩ o h : ¬↑↑μ (s ∩ o) = 0 ⊢ ↑↑μ (s ∩ o) ≠ ⊤ exact (lt_of_le_of_lt (measure_mono (inter_subset_right _ _)) μo).ne Qed
VitaliFamily.ae_tendsto_div α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ ⊢ ∀ᵐ (x : α) ∂μ, ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c) obtain ⟨w, w_count, w_dense, _, w_top⟩ : ∃ w : Set ℝ≥0∞, w.Countable ∧ Dense w ∧ 0 ∉ w ∧ ∞ ∉ w := ENNReal.exists_countable_dense_no_zero_top case intro.intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w ⊢ ∀ᵐ (x : α) ∂μ, ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c) have I : ∀ x ∈ w, x ≠ ∞ := fun x xs hx => w_top (hx ▸ xs) case intro.intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ A : ∀ (c : ℝ≥0∞), c ∈ w → ∀ (d : ℝ≥0∞), d ∈ w → c < d → ∀ᵐ (x : α) ∂μ, ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a) ⊢ ∀ᵐ (x : α) ∂μ, ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c) have B : ∀ᵐ x ∂μ, ∀ c ∈ w, ∀ d ∈ w, c < d → ¬((∃ᶠ a in v.filterAt x, ρ a / μ a < c) ∧ ∃ᶠ a in v.filterAt x, d < ρ a / μ a) := by simpa only [ae_ball_iff w_count, ae_all_iff] case intro.intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ A : ∀ (c : ℝ≥0∞), c ∈ w → ∀ (d : ℝ≥0∞), d ∈ w → c < d → ∀ᵐ (x : α) ∂μ, ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a) B : ∀ᵐ (x : α) ∂μ, ∀ (c : ℝ≥0∞), c ∈ w → ∀ (d : ℝ≥0∞), d ∈ w → c < d → ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a) ⊢ ∀ᵐ (x : α) ∂μ, ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c) filter_upwards [B] case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ A : ∀ (c : ℝ≥0∞), c ∈ w → ∀ (d : ℝ≥0∞), d ∈ w → c < d → ∀ᵐ (x : α) ∂μ, ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a) B : ∀ᵐ (x : α) ∂μ, ∀ (c : ℝ≥0∞), c ∈ w → ∀ (d : ℝ≥0∞), d ∈ w → c < d → ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a) ⊢ ∀ (a : α), (∀ (c : ℝ≥0∞), c ∈ w → ∀ (d : ℝ≥0∞), d ∈ w → c < d → ¬((∃ᶠ (a : Set α) in filterAt v a, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v a, d < ↑↑ρ a / ↑↑μ a)) → ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v a) (𝓝 c) intro x hx case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ A : ∀ (c : ℝ≥0∞), c ∈ w → ∀ (d : ℝ≥0∞), d ∈ w → c < d → ∀ᵐ (x : α) ∂μ, ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a) B : ∀ᵐ (x : α) ∂μ, ∀ (c : ℝ≥0∞), c ∈ w → ∀ (d : ℝ≥0∞), d ∈ w → c < d → ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a) x : α hx : ∀ (c : ℝ≥0∞), c ∈ w → ∀ (d : ℝ≥0∞), d ∈ w → c < d → ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a) ⊢ ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c) exact tendsto_of_no_upcrossings w_dense hx α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ ⊢ ∀ (c : ℝ≥0∞), c ∈ w → ∀ (d : ℝ≥0∞), d ∈ w → c < d → ∀ᵐ (x : α) ∂μ, ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a) intro c hc d hd hcd α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ c : ℝ≥0∞ hc : c ∈ w d : ℝ≥0∞ hd : d ∈ w hcd : c < d ⊢ ∀ᵐ (x : α) ∂μ, ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a) lift c to ℝ≥0 using I c hc case intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ d : ℝ≥0∞ hd : d ∈ w c : ℝ≥0 hc : ↑c ∈ w hcd : ↑c < d ⊢ ∀ᵐ (x : α) ∂μ, ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a) lift d to ℝ≥0 using I d hd case intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ c : ℝ≥0 hc : ↑c ∈ w d : ℝ≥0 hd : ↑d ∈ w hcd : ↑c < ↑d ⊢ ∀ᵐ (x : α) ∂μ, ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c) ∧ ∃ᶠ (a : Set α) in filterAt v x, ↑d < ↑↑ρ a / ↑↑μ a) apply v.null_of_frequently_le_of_frequently_ge hρ (ENNReal.coe_lt_coe.1 hcd) ** case intro.intro.hc α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ c : ℝ≥0 hc : ↑c ∈ w d : ℝ≥0 hd : ↑d ∈ w hcd : ↑c < ↑d ⊢ ∀ (x : α), x ∈ {x \| (fun x => ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c) ∧ ∃ᶠ (a : Set α) in filterAt v x, ↑d < ↑↑ρ a / ↑↑μ a)) x}ᶜ → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑c * ↑↑μ a simp only [and_imp, exists_prop, not_frequently, not_and, not_lt, not_le, not_eventually, mem_setOf_eq, mem_compl_iff, not_forall] case intro.intro.hc α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ c : ℝ≥0 hc : ↑c ∈ w d : ℝ≥0 hd : ↑d ∈ w hcd : ↑c < ↑d ⊢ ∀ (x : α), (∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c) → (∃ᶠ (x : Set α) in filterAt v x, ↑d < ↑↑ρ x / ↑↑μ x) → ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑c * ↑↑μ a intro x h1x _ case intro.intro.hc α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ c : ℝ≥0 hc : ↑c ∈ w d : ℝ≥0 hd : ↑d ∈ w hcd : ↑c < ↑d x : α h1x : ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c a✝ : ∃ᶠ (x : Set α) in filterAt v x, ↑d < ↑↑ρ x / ↑↑μ x ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑c * ↑↑μ a apply h1x.mono fun a ha => ?_ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ c : ℝ≥0 hc : ↑c ∈ w d : ℝ≥0 hd : ↑d ∈ w hcd : ↑c < ↑d x : α h1x : ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c a✝ : ∃ᶠ (x : Set α) in filterAt v x, ↑d < ↑↑ρ x / ↑↑μ x a : Set α ha : ↑↑ρ a / ↑↑μ a < ↑c ⊢ ↑↑ρ a ≤ ↑c * ↑↑μ a refine' (ENNReal.div_le_iff_le_mul _ (Or.inr (bot_le.trans_lt ha).ne')).1 ha.le α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ c : ℝ≥0 hc : ↑c ∈ w d : ℝ≥0 hd : ↑d ∈ w hcd : ↑c < ↑d x : α h1x : ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c a✝ : ∃ᶠ (x : Set α) in filterAt v x, ↑d < ↑↑ρ x / ↑↑μ x a : Set α ha : ↑↑ρ a / ↑↑μ a < ↑c ⊢ ↑↑μ a ≠ 0 ∨ ↑c ≠ ⊤ simp only [ENNReal.coe_ne_top, Ne.def, or_true_iff, not_false_iff] case intro.intro.hd α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ c : ℝ≥0 hc : ↑c ∈ w d : ℝ≥0 hd : ↑d ∈ w hcd : ↑c < ↑d ⊢ ∀ (x : α), x ∈ {x \| (fun x => ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c) ∧ ∃ᶠ (a : Set α) in filterAt v x, ↑d < ↑↑ρ a / ↑↑μ a)) x}ᶜ → ∃ᶠ (a : Set α) in filterAt v x, ↑d * ↑↑μ a ≤ ↑↑ρ a simp only [and_imp, exists_prop, not_frequently, not_and, not_lt, not_le, not_eventually, mem_setOf_eq, mem_compl_iff, not_forall] case intro.intro.hd α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ c : ℝ≥0 hc : ↑c ∈ w d : ℝ≥0 hd : ↑d ∈ w hcd : ↑c < ↑d ⊢ ∀ (x : α), (∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c) → (∃ᶠ (x : Set α) in filterAt v x, ↑d < ↑↑ρ x / ↑↑μ x) → ∃ᶠ (a : Set α) in filterAt v x, ↑d * ↑↑μ a ≤ ↑↑ρ a intro x _ h2x case intro.intro.hd α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ c : ℝ≥0 hc : ↑c ∈ w d : ℝ≥0 hd : ↑d ∈ w hcd : ↑c < ↑d x : α a✝ : ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c h2x : ∃ᶠ (x : Set α) in filterAt v x, ↑d < ↑↑ρ x / ↑↑μ x ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑d * ↑↑μ a ≤ ↑↑ρ a apply h2x.mono fun a ha => ?_ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ c : ℝ≥0 hc : ↑c ∈ w d : ℝ≥0 hd : ↑d ∈ w hcd : ↑c < ↑d x : α a✝ : ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < ↑c h2x : ∃ᶠ (x : Set α) in filterAt v x, ↑d < ↑↑ρ x / ↑↑μ x a : Set α ha : ↑d < ↑↑ρ a / ↑↑μ a ⊢ ↑d * ↑↑μ a ≤ ↑↑ρ a exact ENNReal.mul_le_of_le_div ha.le α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ w : Set ℝ≥0∞ w_count : Set.Countable w w_dense : Dense w left✝ : ¬0 ∈ w w_top : ¬⊤ ∈ w I : ∀ (x : ℝ≥0∞), x ∈ w → x ≠ ⊤ A : ∀ (c : ℝ≥0∞), c ∈ w → ∀ (d : ℝ≥0∞), d ∈ w → c < d → ∀ᵐ (x : α) ∂μ, ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a) ⊢ ∀ᵐ (x : α) ∂μ, ∀ (c : ℝ≥0∞), c ∈ w → ∀ (d : ℝ≥0∞), d ∈ w → c < d → ¬((∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a / ↑↑μ a < c) ∧ ∃ᶠ (a : Set α) in filterAt v x, d < ↑↑ρ a / ↑↑μ a) simpa only [ae_ball_iff w_count, ae_all_iff] Qed
VitaliFamily.ae_tendsto_limRatio α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatio v ρ x)) filter_upwards [v.ae_tendsto_div hρ] case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ ⊢ ∀ (a : α), (∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v a) (𝓝 c)) → Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v a) (𝓝 (limRatio v ρ a)) intro x hx case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α hx : ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c) ⊢ Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatio v ρ x)) exact tendsto_nhds_limUnder hx ** Qed
VitaliFamily.exists_measurable_supersets_limRatio α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q ⊢ ∃ a b, MeasurableSet a ∧ MeasurableSet b ∧ {x \| limRatio v ρ x < ↑p} ⊆ a ∧ {x \| ↑q < limRatio v ρ x} ⊆ b ∧ ↑↑μ (a ∩ b) = 0 let s := {x \| ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)} α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ⊢ ∃ a b, MeasurableSet a ∧ MeasurableSet b ∧ {x \| limRatio v ρ x < ↑p} ⊆ a ∧ {x \| ↑q < limRatio v ρ x} ⊆ b ∧ ↑↑μ (a ∩ b) = 0 let o : ℕ → Set α := spanningSets (ρ + μ) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) ⊢ ∃ a b, MeasurableSet a ∧ MeasurableSet b ∧ {x \| limRatio v ρ x < ↑p} ⊆ a ∧ {x \| ↑q < limRatio v ρ x} ⊆ b ∧ ↑↑μ (a ∩ b) = 0 let u n := s ∩ {x \| v.limRatio ρ x < p} ∩ o n α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n ⊢ ∃ a b, MeasurableSet a ∧ MeasurableSet b ∧ {x \| limRatio v ρ x < ↑p} ⊆ a ∧ {x \| ↑q < limRatio v ρ x} ⊆ b ∧ ↑↑μ (a ∩ b) = 0 let w n := s ∩ {x \| (q : ℝ≥0∞) < v.limRatio ρ x} ∩ o n α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n ⊢ ∃ a b, MeasurableSet a ∧ MeasurableSet b ∧ {x \| limRatio v ρ x < ↑p} ⊆ a ∧ {x \| ↑q < limRatio v ρ x} ⊆ b ∧ ↑↑μ (a ∩ b) = 0 refine' ⟨toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n), toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n), _, _, _, _, _⟩ case refine'_5 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n ⊢ ↑↑μ ((toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n)) ∩ (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n))) = 0 suffices H : ∀ m n : ℕ, μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n ⊢ ∀ (m n : ℕ), ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 intro m n case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ ⊢ ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 have I : (ρ + μ) (u m) ≠ ∞ := by apply (lt_of_le_of_lt (measure_mono _) (measure_spanningSets_lt_top (ρ + μ) m)).ne exact inter_subset_right _ _ case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ ⊢ ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 have J : (ρ + μ) (w n) ≠ ∞ := by apply (lt_of_le_of_lt (measure_mono _) (measure_spanningSets_lt_top (ρ + μ) n)).ne exact inter_subset_right _ _ case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ ⊢ ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 have A : ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ p * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) := calc ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = ρ (u m ∩ toMeasurable (ρ + μ) (w n)) := measure_toMeasurable_add_inter_left (measurableSet_toMeasurable _ _) I _ ≤ (p • μ) (u m ∩ toMeasurable (ρ + μ) (w n)) := by refine' v.measure_le_of_frequently_le (p • μ) hρ _ fun x hx => _ have L : Tendsto (fun a : Set α => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatio ρ x)) := tendsto_nhds_limUnder hx.1.1.1 have I : ∀ᶠ b : Set α in v.filterAt x, ρ b / μ b < p := (tendsto_order.1 L).2 _ hx.1.1.2 apply I.frequently.mono fun a ha => ?_ rw [coe_nnreal_smul_apply] refine' (ENNReal.div_le_iff_le_mul _ (Or.inr (bot_le.trans_lt ha).ne')).1 ha.le simp only [ENNReal.coe_ne_top, Ne.def, or_true_iff, not_false_iff] _ = p * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) := by simp only [coe_nnreal_smul_apply, measure_toMeasurable_add_inter_right (measurableSet_toMeasurable _ _) I] ** case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ⊢ ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 ** have B : (q : ℝ≥0∞) * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) := calc (q : ℝ≥0∞) * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = (q : ℝ≥0∞) * μ (toMeasurable (ρ + μ) (u m) ∩ w n) := by conv_rhs => rw [inter_comm] rw [inter_comm, measure_toMeasurable_add_inter_right (measurableSet_toMeasurable _ _) J] _ ≤ ρ (toMeasurable (ρ + μ) (u m) ∩ w n) := by rw [← coe_nnreal_smul_apply] refine' v.measure_le_of_frequently_le _ (AbsolutelyContinuous.rfl.smul _) _ _ intro x hx have L : Tendsto (fun a : Set α => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatio ρ x)) := tendsto_nhds_limUnder hx.2.1.1 have I : ∀ᶠ b : Set α in v.filterAt x, (q : ℝ≥0∞) < ρ b / μ b := (tendsto_order.1 L).1 _ hx.2.1.2 apply I.frequently.mono fun a ha => ?_ rw [coe_nnreal_smul_apply] exact ENNReal.mul_le_of_le_div ha.le _ = ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) := by conv_rhs => rw [inter_comm] rw [inter_comm] exact (measure_toMeasurable_add_inter_left (measurableSet_toMeasurable _ _) J).symm ** case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) B : ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ⊢ ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 by_contra h case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) B : ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) h : ¬↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 ⊢ False apply lt_irrefl (ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))) case refine'_1 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n ⊢ MeasurableSet (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n)) exact (measurableSet_toMeasurable _ _).union (MeasurableSet.iUnion fun n => measurableSet_toMeasurable _ _) case refine'_2 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n ⊢ MeasurableSet (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n)) exact (measurableSet_toMeasurable _ _).union (MeasurableSet.iUnion fun n => measurableSet_toMeasurable _ _) case refine'_3 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n ⊢ {x \| limRatio v ρ x < ↑p} ⊆ toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n) intro x hx case refine'_3 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n x : α hx : x ∈ {x \| limRatio v ρ x < ↑p} ⊢ x ∈ toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n) by_cases h : x ∈ s case pos α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n x : α hx : x ∈ {x \| limRatio v ρ x < ↑p} h : x ∈ s ⊢ x ∈ toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n) refine' Or.inr (mem_iUnion.2 ⟨spanningSetsIndex (ρ + μ) x, _⟩) case pos α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n x : α hx : x ∈ {x \| limRatio v ρ x < ↑p} h : x ∈ s ⊢ x ∈ toMeasurable (ρ + μ) (u (spanningSetsIndex (ρ + μ) x)) exact subset_toMeasurable _ _ ⟨⟨h, hx⟩, mem_spanningSetsIndex _ _⟩ case neg α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n x : α hx : x ∈ {x \| limRatio v ρ x < ↑p} h : ¬x ∈ s ⊢ x ∈ toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n) exact Or.inl (subset_toMeasurable μ sᶜ h) case refine'_4 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n ⊢ {x \| ↑q < limRatio v ρ x} ⊆ toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n) intro x hx case refine'_4 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n x : α hx : x ∈ {x \| ↑q < limRatio v ρ x} ⊢ x ∈ toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n) by_cases h : x ∈ s case pos α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n x : α hx : x ∈ {x \| ↑q < limRatio v ρ x} h : x ∈ s ⊢ x ∈ toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n) refine' Or.inr (mem_iUnion.2 ⟨spanningSetsIndex (ρ + μ) x, _⟩) case pos α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n x : α hx : x ∈ {x \| ↑q < limRatio v ρ x} h : x ∈ s ⊢ x ∈ toMeasurable (ρ + μ) (w (spanningSetsIndex (ρ + μ) x)) exact subset_toMeasurable _ _ ⟨⟨h, hx⟩, mem_spanningSetsIndex _ _⟩ case neg α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n x : α hx : x ∈ {x \| ↑q < limRatio v ρ x} h : ¬x ∈ s ⊢ x ∈ toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n) exact Or.inl (subset_toMeasurable μ sᶜ h) case refine'_5 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n H : ∀ (m n : ℕ), ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 A : (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n)) ∩ (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n)) ⊆ toMeasurable μ sᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n) ⊢ ↑↑μ ((toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n)) ∩ (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n))) = 0 refine' le_antisymm ((measure_mono A).trans _) bot_le case refine'_5 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n H : ∀ (m n : ℕ), ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 A : (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n)) ∩ (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n)) ⊆ toMeasurable μ sᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n) ⊢ ↑↑μ (toMeasurable μ sᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ 0 calc μ (toMeasurable μ sᶜ ∪ ⋃ (m) (n), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ μ (toMeasurable μ sᶜ) + μ (⋃ (m) (n), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) := measure_union_le _ _ _ = μ (⋃ (m) (n), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) := by have : μ sᶜ = 0 := v.ae_tendsto_div hρ; rw [measure_toMeasurable, this, zero_add] _ ≤ ∑' (m) (n), μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) := ((measure_iUnion_le _).trans (ENNReal.tsum_le_tsum fun m => measure_iUnion_le _)) _ = 0 := by simp only [H, tsum_zero] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n H : ∀ (m n : ℕ), ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 ⊢ (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n)) ∩ (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n)) ⊆ toMeasurable μ sᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n) simp only [inter_distrib_left, inter_distrib_right, true_and_iff, subset_union_left, union_subset_iff, inter_self] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n H : ∀ (m n : ℕ), ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 ⊢ (⋃ n, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| limRatio v ρ x < ↑p} ∩ spanningSets (ρ + μ) n)) ∩ toMeasurable μ {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)}ᶜ ⊆ toMeasurable μ {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)}ᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| limRatio v ρ x < ↑p} ∩ spanningSets (ρ + μ) m) ∩ toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| ↑q < limRatio v ρ x} ∩ spanningSets (ρ + μ) n) ∧ toMeasurable μ {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)}ᶜ ∩ ⋃ n, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| ↑q < limRatio v ρ x} ∩ spanningSets (ρ + μ) n) ⊆ toMeasurable μ {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)}ᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| limRatio v ρ x < ↑p} ∩ spanningSets (ρ + μ) m) ∩ toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| ↑q < limRatio v ρ x} ∩ spanningSets (ρ + μ) n) ∧ (⋃ n, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| limRatio v ρ x < ↑p} ∩ spanningSets (ρ + μ) n)) ∩ ⋃ n, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| ↑q < limRatio v ρ x} ∩ spanningSets (ρ + μ) n) ⊆ toMeasurable μ {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)}ᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| limRatio v ρ x < ↑p} ∩ spanningSets (ρ + μ) m) ∩ toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| ↑q < limRatio v ρ x} ∩ spanningSets (ρ + μ) n) refine' ⟨_, _, _⟩ case refine'_1 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n H : ∀ (m n : ℕ), ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 ⊢ (⋃ n, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| limRatio v ρ x < ↑p} ∩ spanningSets (ρ + μ) n)) ∩ toMeasurable μ {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)}ᶜ ⊆ toMeasurable μ {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)}ᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| limRatio v ρ x < ↑p} ∩ spanningSets (ρ + μ) m) ∩ toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| ↑q < limRatio v ρ x} ∩ spanningSets (ρ + μ) n) exact (inter_subset_right _ _).trans (subset_union_left _ _) case refine'_2 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n H : ∀ (m n : ℕ), ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 ⊢ toMeasurable μ {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)}ᶜ ∩ ⋃ n, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| ↑q < limRatio v ρ x} ∩ spanningSets (ρ + μ) n) ⊆ toMeasurable μ {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)}ᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| limRatio v ρ x < ↑p} ∩ spanningSets (ρ + μ) m) ∩ toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| ↑q < limRatio v ρ x} ∩ spanningSets (ρ + μ) n) exact (inter_subset_left _ _).trans (subset_union_left _ _) case refine'_3 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n H : ∀ (m n : ℕ), ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 ⊢ (⋃ n, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| limRatio v ρ x < ↑p} ∩ spanningSets (ρ + μ) n)) ∩ ⋃ n, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| ↑q < limRatio v ρ x} ∩ spanningSets (ρ + μ) n) ⊆ toMeasurable μ {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)}ᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| limRatio v ρ x < ↑p} ∩ spanningSets (ρ + μ) m) ∩ toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| ↑q < limRatio v ρ x} ∩ spanningSets (ρ + μ) n) simp_rw [iUnion_inter, inter_iUnion] case refine'_3 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n H : ∀ (m n : ℕ), ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 ⊢ ⋃ i, ⋃ i_1, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| limRatio v ρ x < ↑p} ∩ spanningSets (ρ + μ) i) ∩ toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| ↑q < limRatio v ρ x} ∩ spanningSets (ρ + μ) i_1) ⊆ toMeasurable μ {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)}ᶜ ∪ ⋃ i, ⋃ i_1, toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| limRatio v ρ x < ↑p} ∩ spanningSets (ρ + μ) i) ∩ toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| ↑q < limRatio v ρ x} ∩ spanningSets (ρ + μ) i_1) exact subset_union_right _ _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n H : ∀ (m n : ℕ), ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 A : (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n)) ∩ (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n)) ⊆ toMeasurable μ sᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n) ⊢ ↑↑μ (toMeasurable μ sᶜ) + ↑↑μ (⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = ↑↑μ (⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) have : μ sᶜ = 0 := v.ae_tendsto_div hρ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n H : ∀ (m n : ℕ), ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 A : (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n)) ∩ (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n)) ⊆ toMeasurable μ sᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n) this : ↑↑μ sᶜ = 0 ⊢ ↑↑μ (toMeasurable μ sᶜ) + ↑↑μ (⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = ↑↑μ (⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) rw [measure_toMeasurable, this, zero_add] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n H : ∀ (m n : ℕ), ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 A : (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n)) ∩ (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n)) ⊆ toMeasurable μ sᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n) ⊢ ∑' (m : ℕ) (n : ℕ), ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 simp only [H, tsum_zero] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ ⊢ ↑↑(ρ + μ) (u m) ≠ ⊤ apply (lt_of_le_of_lt (measure_mono _) (measure_spanningSets_lt_top (ρ + μ) m)).ne α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ ⊢ u m ⊆ spanningSets (ρ + μ) m exact inter_subset_right _ _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ ⊢ ↑↑(ρ + μ) (w n) ≠ ⊤ apply (lt_of_le_of_lt (measure_mono _) (measure_spanningSets_lt_top (ρ + μ) n)).ne α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ ⊢ w n ⊆ spanningSets (ρ + μ) n exact inter_subset_right _ _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ ⊢ ↑↑ρ (u m ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑↑(p • μ) (u m ∩ toMeasurable (ρ + μ) (w n)) refine' v.measure_le_of_frequently_le (p • μ) hρ _ fun x hx => _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ x : α hx : x ∈ u m ∩ toMeasurable (ρ + μ) (w n) ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑(p • μ) a have L : Tendsto (fun a : Set α => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatio ρ x)) := tendsto_nhds_limUnder hx.1.1.1 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ x : α hx : x ∈ u m ∩ toMeasurable (ρ + μ) (w n) L : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatio v ρ x)) ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑(p • μ) a have I : ∀ᶠ b : Set α in v.filterAt x, ρ b / μ b < p := (tendsto_order.1 L).2 _ hx.1.1.2 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I✝ : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ x : α hx : x ∈ u m ∩ toMeasurable (ρ + μ) (w n) L : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatio v ρ x)) I : ∀ᶠ (b : Set α) in filterAt v x, ↑↑ρ b / ↑↑μ b < ↑p ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑(p • μ) a apply I.frequently.mono fun a ha => ?_ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I✝ : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ x : α hx : x ∈ u m ∩ toMeasurable (ρ + μ) (w n) L : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatio v ρ x)) I : ∀ᶠ (b : Set α) in filterAt v x, ↑↑ρ b / ↑↑μ b < ↑p a : Set α ha : ↑↑ρ a / ↑↑μ a < ↑p ⊢ ↑↑ρ a ≤ ↑↑(p • μ) a rw [coe_nnreal_smul_apply] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I✝ : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ x : α hx : x ∈ u m ∩ toMeasurable (ρ + μ) (w n) L : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatio v ρ x)) I : ∀ᶠ (b : Set α) in filterAt v x, ↑↑ρ b / ↑↑μ b < ↑p a : Set α ha : ↑↑ρ a / ↑↑μ a < ↑p ⊢ ↑↑ρ a ≤ ↑p * ↑↑μ a refine' (ENNReal.div_le_iff_le_mul _ (Or.inr (bot_le.trans_lt ha).ne')).1 ha.le α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I✝ : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ x : α hx : x ∈ u m ∩ toMeasurable (ρ + μ) (w n) L : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatio v ρ x)) I : ∀ᶠ (b : Set α) in filterAt v x, ↑↑ρ b / ↑↑μ b < ↑p a : Set α ha : ↑↑ρ a / ↑↑μ a < ↑p ⊢ ↑↑μ a ≠ 0 ∨ ↑p ≠ ⊤ simp only [ENNReal.coe_ne_top, Ne.def, or_true_iff, not_false_iff] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ ⊢ ↑↑(p • μ) (u m ∩ toMeasurable (ρ + μ) (w n)) = ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) simp only [coe_nnreal_smul_apply, measure_toMeasurable_add_inter_right (measurableSet_toMeasurable _ _) I] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ⊢ ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ w n) conv_rhs => rw [inter_comm] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ⊢ ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = ↑q * ↑↑μ (w n ∩ toMeasurable (ρ + μ) (u m)) rw [inter_comm, measure_toMeasurable_add_inter_right (measurableSet_toMeasurable _ _) J] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ⊢ ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ w n) ≤ ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ w n) rw [← coe_nnreal_smul_apply] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ⊢ ↑↑(q • μ) (toMeasurable (ρ + μ) (u m) ∩ w n) ≤ ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ w n) refine' v.measure_le_of_frequently_le _ (AbsolutelyContinuous.rfl.smul _) _ _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ⊢ ∀ (x : α), x ∈ toMeasurable (ρ + μ) (u m) ∩ w n → ∃ᶠ (a : Set α) in filterAt v x, ↑↑(q • μ) a ≤ ↑↑ρ a intro x hx α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) x : α hx : x ∈ toMeasurable (ρ + μ) (u m) ∩ w n ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑(q • μ) a ≤ ↑↑ρ a have L : Tendsto (fun a : Set α => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatio ρ x)) := tendsto_nhds_limUnder hx.2.1.1 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) x : α hx : x ∈ toMeasurable (ρ + μ) (u m) ∩ w n L : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatio v ρ x)) ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑(q • μ) a ≤ ↑↑ρ a have I : ∀ᶠ b : Set α in v.filterAt x, (q : ℝ≥0∞) < ρ b / μ b := (tendsto_order.1 L).1 _ hx.2.1.2 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I✝ : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) x : α hx : x ∈ toMeasurable (ρ + μ) (u m) ∩ w n L : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatio v ρ x)) I : ∀ᶠ (b : Set α) in filterAt v x, ↑q < ↑↑ρ b / ↑↑μ b ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑(q • μ) a ≤ ↑↑ρ a apply I.frequently.mono fun a ha => ?_ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I✝ : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) x : α hx : x ∈ toMeasurable (ρ + μ) (u m) ∩ w n L : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatio v ρ x)) I : ∀ᶠ (b : Set α) in filterAt v x, ↑q < ↑↑ρ b / ↑↑μ b a : Set α ha : ↑q < ↑↑ρ a / ↑↑μ a ⊢ ↑↑(q • μ) a ≤ ↑↑ρ a rw [coe_nnreal_smul_apply] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I✝ : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) x : α hx : x ∈ toMeasurable (ρ + μ) (u m) ∩ w n L : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatio v ρ x)) I : ∀ᶠ (b : Set α) in filterAt v x, ↑q < ↑↑ρ b / ↑↑μ b a : Set α ha : ↑q < ↑↑ρ a / ↑↑μ a ⊢ ↑q * ↑↑μ a ≤ ↑↑ρ a exact ENNReal.mul_le_of_le_div ha.le α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ⊢ ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ w n) = ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) conv_rhs => rw [inter_comm] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ⊢ ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ w n) = ↑↑ρ (toMeasurable (ρ + μ) (w n) ∩ toMeasurable (ρ + μ) (u m)) rw [inter_comm] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ⊢ ↑↑ρ (w n ∩ toMeasurable (ρ + μ) (u m)) = ↑↑ρ (toMeasurable (ρ + μ) (w n) ∩ toMeasurable (ρ + μ) (u m)) exact (measure_toMeasurable_add_inter_left (measurableSet_toMeasurable _ _) J).symm α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) B : ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) h : ¬↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 ⊢ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) < ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) apply (ENNReal.mul_lt_mul_right h _).2 (ENNReal.coe_lt_coe.2 hpq) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) B : ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) h : ¬↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 ⊢ ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≠ ⊤ suffices H : (ρ + μ) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≠ ∞ case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) B : ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) h : ¬↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 ⊢ ↑↑(ρ + μ) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≠ ⊤ apply (lt_of_le_of_lt (measure_mono (inter_subset_left _ _)) _).ne α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) B : ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) h : ¬↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 ⊢ ↑↑(ρ + μ) (toMeasurable (ρ + μ) (u m)) < ⊤ rw [measure_toMeasurable] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) B : ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) h : ¬↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 ⊢ ↑↑(ρ + μ) (u m) < ⊤ apply lt_of_le_of_lt (measure_mono _) (measure_spanningSets_lt_top (ρ + μ) m) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) B : ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) h : ¬↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 ⊢ u m ⊆ spanningSets (ρ + μ) m exact inter_subset_right _ _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) B : ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) h : ¬↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 H : ↑↑(ρ + μ) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≠ ⊤ ⊢ ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≠ ⊤ simp only [not_or, ENNReal.add_eq_top, Pi.add_apply, Ne.def, coe_add] at H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p q : ℝ≥0 hpq : p < q s : Set α := {x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} o : ℕ → Set α := spanningSets (ρ + μ) u : ℕ → Set α := fun n => s ∩ {x \| limRatio v ρ x < ↑p} ∩ o n w : ℕ → Set α := fun n => s ∩ {x \| ↑q < limRatio v ρ x} ∩ o n m n : ℕ I : ↑↑(ρ + μ) (u m) ≠ ⊤ J : ↑↑(ρ + μ) (w n) ≠ ⊤ A : ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑p * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) B : ↑q * ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ ↑↑ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) h : ¬↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 H : ¬↑↑ρ (toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| limRatio v ρ x < ↑p} ∩ spanningSets (ρ + μ) m) ∩ toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| ↑q < limRatio v ρ x} ∩ spanningSets (ρ + μ) n)) = ⊤ ∧ ¬↑↑μ (toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| limRatio v ρ x < ↑p} ∩ spanningSets (ρ + μ) m) ∩ toMeasurable (ρ + μ) ({x \| ∃ c, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 c)} ∩ {x \| ↑q < limRatio v ρ x} ∩ spanningSets (ρ + μ) n)) = ⊤ ⊢ ↑↑μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≠ ⊤ exact H.2 Qed
VitaliFamily.ae_tendsto_limRatioMeas α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x)) filter_upwards [v.ae_tendsto_limRatio hρ, AEMeasurable.ae_eq_mk (v.aemeasurable_limRatio hρ)] case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ ⊢ ∀ (a : α), Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v a) (𝓝 (limRatio v ρ a)) → limRatio v ρ a = AEMeasurable.mk (limRatio v ρ) (_ : AEMeasurable (limRatio v ρ)) a → Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v a) (𝓝 (limRatioMeas v hρ a)) intro x hx h'x case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α hx : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatio v ρ x)) h'x : limRatio v ρ x = AEMeasurable.mk (limRatio v ρ) (_ : AEMeasurable (limRatio v ρ)) x ⊢ Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x)) rwa [h'x] at hx ** Qed
VitaliFamily.measure_le_mul_of_subset_limRatioMeas_lt ** α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p : ℝ≥0 s : Set α h : s ⊆ {x \| limRatioMeas v hρ x < ↑p} ⊢ ↑↑ρ s ≤ ↑p * ↑↑μ s let t := {x : α \| Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))} α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p : ℝ≥0 s : Set α h : s ⊆ {x \| limRatioMeas v hρ x < ↑p} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} ⊢ ↑↑ρ s ≤ ↑p * ↑↑μ s have A : μ tᶜ = 0 := v.ae_tendsto_limRatioMeas hρ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p : ℝ≥0 s : Set α h : s ⊆ {x \| limRatioMeas v hρ x < ↑p} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 ⊢ ↑↑ρ s ≤ ↑p * ↑↑μ s suffices H : ρ (s ∩ t) ≤ (p • μ) (s ∩ t) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p : ℝ≥0 s : Set α h : s ⊆ {x \| limRatioMeas v hρ x < ↑p} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 H : ↑↑ρ (s ∩ t) ≤ ↑↑(p • μ) (s ∩ t) ⊢ ↑↑ρ s ≤ ↑p * ↑↑μ s case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p : ℝ≥0 s : Set α h : s ⊆ {x \| limRatioMeas v hρ x < ↑p} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 ⊢ ↑↑ρ (s ∩ t) ≤ ↑↑(p • μ) (s ∩ t) ** exact calc ρ s = ρ (s ∩ t ∪ s ∩ tᶜ) := by rw [inter_union_compl] _ ≤ ρ (s ∩ t) + ρ (s ∩ tᶜ) := (measure_union_le _ _) _ ≤ p * μ (s ∩ t) + 0 := (add_le_add H ((measure_mono (inter_subset_right _ _)).trans (hρ A).le)) _ ≤ p * μ s := by rw [add_zero]; exact mul_le_mul_left' (measure_mono (inter_subset_left _ _)) _ case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p : ℝ≥0 s : Set α h : s ⊆ {x \| limRatioMeas v hρ x < ↑p} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 ⊢ ↑↑ρ (s ∩ t) ≤ ↑↑(p • μ) (s ∩ t) refine' v.measure_le_of_frequently_le (p • μ) hρ _ fun x hx => _ case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p : ℝ≥0 s : Set α h : s ⊆ {x \| limRatioMeas v hρ x < ↑p} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 x : α hx : x ∈ s ∩ t ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑(p • μ) a have I : ∀ᶠ b : Set α in v.filterAt x, ρ b / μ b < p := (tendsto_order.1 hx.2).2 _ (h hx.1) case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p : ℝ≥0 s : Set α h : s ⊆ {x \| limRatioMeas v hρ x < ↑p} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 x : α hx : x ∈ s ∩ t I : ∀ᶠ (b : Set α) in filterAt v x, ↑↑ρ b / ↑↑μ b < ↑p ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑ρ a ≤ ↑↑(p • μ) a apply I.frequently.mono fun a ha => ?_ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p : ℝ≥0 s : Set α h : s ⊆ {x \| limRatioMeas v hρ x < ↑p} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 x : α hx : x ∈ s ∩ t I : ∀ᶠ (b : Set α) in filterAt v x, ↑↑ρ b / ↑↑μ b < ↑p a : Set α ha : ↑↑ρ a / ↑↑μ a < ↑p ⊢ ↑↑ρ a ≤ ↑↑(p • μ) a rw [coe_nnreal_smul_apply] ** α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p : ℝ≥0 s : Set α h : s ⊆ {x \| limRatioMeas v hρ x < ↑p} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 x : α hx : x ∈ s ∩ t I : ∀ᶠ (b : Set α) in filterAt v x, ↑↑ρ b / ↑↑μ b < ↑p a : Set α ha : ↑↑ρ a / ↑↑μ a < ↑p ⊢ ↑↑ρ a ≤ ↑p * ↑↑μ a refine' (ENNReal.div_le_iff_le_mul _ (Or.inr (bot_le.trans_lt ha).ne')).1 ha.le α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p : ℝ≥0 s : Set α h : s ⊆ {x \| limRatioMeas v hρ x < ↑p} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 x : α hx : x ∈ s ∩ t I : ∀ᶠ (b : Set α) in filterAt v x, ↑↑ρ b / ↑↑μ b < ↑p a : Set α ha : ↑↑ρ a / ↑↑μ a < ↑p ⊢ ↑↑μ a ≠ 0 ∨ ↑p ≠ ⊤ simp only [ENNReal.coe_ne_top, Ne.def, or_true_iff, not_false_iff] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p : ℝ≥0 s : Set α h : s ⊆ {x \| limRatioMeas v hρ x < ↑p} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 H : ↑↑ρ (s ∩ t) ≤ ↑↑(p • μ) (s ∩ t) ⊢ ↑↑ρ s = ↑↑ρ (s ∩ t ∪ s ∩ tᶜ) rw [inter_union_compl] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p : ℝ≥0 s : Set α h : s ⊆ {x \| limRatioMeas v hρ x < ↑p} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 H : ↑↑ρ (s ∩ t) ≤ ↑↑(p • μ) (s ∩ t) ⊢ ↑p * ↑↑μ (s ∩ t) + 0 ≤ ↑p * ↑↑μ s rw [add_zero] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ p : ℝ≥0 s : Set α h : s ⊆ {x \| limRatioMeas v hρ x < ↑p} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 H : ↑↑ρ (s ∩ t) ≤ ↑↑(p • μ) (s ∩ t) ⊢ ↑p * ↑↑μ (s ∩ t) ≤ ↑p * ↑↑μ s exact mul_le_mul_left' (measure_mono (inter_subset_left _ _)) _ Qed
VitaliFamily.mul_measure_le_of_subset_lt_limRatioMeas ** α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} ⊢ ↑q * ↑↑μ s ≤ ↑↑ρ s let t := {x : α \| Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))} α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} ⊢ ↑q * ↑↑μ s ≤ ↑↑ρ s have A : μ tᶜ = 0 := v.ae_tendsto_limRatioMeas hρ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 ⊢ ↑q * ↑↑μ s ≤ ↑↑ρ s suffices H : (q • μ) (s ∩ t) ≤ ρ (s ∩ t) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 H : ↑↑(q • μ) (s ∩ t) ≤ ↑↑ρ (s ∩ t) ⊢ ↑q * ↑↑μ s ≤ ↑↑ρ s case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 ⊢ ↑↑(q • μ) (s ∩ t) ≤ ↑↑ρ (s ∩ t) ** exact calc (q • μ) s = (q • μ) (s ∩ t ∪ s ∩ tᶜ) := by rw [inter_union_compl] _ ≤ (q • μ) (s ∩ t) + (q • μ) (s ∩ tᶜ) := (measure_union_le _ _) _ ≤ ρ (s ∩ t) + q * μ tᶜ := by apply add_le_add H rw [coe_nnreal_smul_apply] exact mul_le_mul_left' (measure_mono (inter_subset_right _ _)) _ _ ≤ ρ s := by rw [A, mul_zero, add_zero]; exact measure_mono (inter_subset_left _ _) case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 ⊢ ↑↑(q • μ) (s ∩ t) ≤ ↑↑ρ (s ∩ t) refine' v.measure_le_of_frequently_le _ (AbsolutelyContinuous.rfl.smul _) _ _ case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 ⊢ ∀ (x : α), x ∈ s ∩ t → ∃ᶠ (a : Set α) in filterAt v x, ↑↑(q • μ) a ≤ ↑↑ρ a intro x hx case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 x : α hx : x ∈ s ∩ t ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑(q • μ) a ≤ ↑↑ρ a have I : ∀ᶠ a in v.filterAt x, (q : ℝ≥0∞) < ρ a / μ a := (tendsto_order.1 hx.2).1 _ (h hx.1) case H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 x : α hx : x ∈ s ∩ t I : ∀ᶠ (a : Set α) in filterAt v x, ↑q < ↑↑ρ a / ↑↑μ a ⊢ ∃ᶠ (a : Set α) in filterAt v x, ↑↑(q • μ) a ≤ ↑↑ρ a apply I.frequently.mono fun a ha => ?_ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 x : α hx : x ∈ s ∩ t I : ∀ᶠ (a : Set α) in filterAt v x, ↑q < ↑↑ρ a / ↑↑μ a a : Set α ha : ↑q < ↑↑ρ a / ↑↑μ a ⊢ ↑↑(q • μ) a ≤ ↑↑ρ a rw [coe_nnreal_smul_apply] ** α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 x : α hx : x ∈ s ∩ t I : ∀ᶠ (a : Set α) in filterAt v x, ↑q < ↑↑ρ a / ↑↑μ a a : Set α ha : ↑q < ↑↑ρ a / ↑↑μ a ⊢ ↑q * ↑↑μ a ≤ ↑↑ρ a exact ENNReal.mul_le_of_le_div ha.le α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 H : ↑↑(q • μ) (s ∩ t) ≤ ↑↑ρ (s ∩ t) ⊢ ↑↑(q • μ) s = ↑↑(q • μ) (s ∩ t ∪ s ∩ tᶜ) rw [inter_union_compl] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 H : ↑↑(q • μ) (s ∩ t) ≤ ↑↑ρ (s ∩ t) ⊢ ↑↑(q • μ) (s ∩ t) + ↑↑(q • μ) (s ∩ tᶜ) ≤ ↑↑ρ (s ∩ t) + ↑q * ↑↑μ tᶜ apply add_le_add H α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 H : ↑↑(q • μ) (s ∩ t) ≤ ↑↑ρ (s ∩ t) ⊢ ↑↑(q • μ) (s ∩ tᶜ) ≤ ↑q * ↑↑μ tᶜ rw [coe_nnreal_smul_apply] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 H : ↑↑(q • μ) (s ∩ t) ≤ ↑↑ρ (s ∩ t) ⊢ ↑q * ↑↑μ (s ∩ tᶜ) ≤ ↑q * ↑↑μ tᶜ exact mul_le_mul_left' (measure_mono (inter_subset_right _ _)) _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 H : ↑↑(q • μ) (s ∩ t) ≤ ↑↑ρ (s ∩ t) ⊢ ↑↑ρ (s ∩ t) + ↑q * ↑↑μ tᶜ ≤ ↑↑ρ s rw [A, mul_zero, add_zero] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ q : ℝ≥0 s : Set α h : s ⊆ {x \| ↑q < limRatioMeas v hρ x} t : Set α := {x \| Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (limRatioMeas v hρ x))} A : ↑↑μ tᶜ = 0 H : ↑↑(q • μ) (s ∩ t) ≤ ↑↑ρ (s ∩ t) ⊢ ↑↑ρ (s ∩ t) ≤ ↑↑ρ s exact measure_mono (inter_subset_left _ _) Qed
VitaliFamily.measure_limRatioMeas_top α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ ⊢ ↑↑μ {x \| limRatioMeas v hρ x = ⊤} = 0 refine' null_of_locally_null _ fun x _ => _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} ⊢ ∃ u, u ∈ 𝓝[{x \| limRatioMeas v hρ x = ⊤}] x ∧ ↑↑μ u = 0 obtain ⟨o, xo, o_open, μo⟩ : ∃ o : Set α, x ∈ o ∧ IsOpen o ∧ ρ o < ∞ := Measure.exists_isOpen_measure_lt_top ρ x case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ ⊢ ∃ u, u ∈ 𝓝[{x \| limRatioMeas v hρ x = ⊤}] x ∧ ↑↑μ u = 0 let s := {x : α \| v.limRatioMeas hρ x = ∞} ∩ o case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ⊢ ∃ u, u ∈ 𝓝[{x \| limRatioMeas v hρ x = ⊤}] x ∧ ↑↑μ u = 0 refine' ⟨s, inter_mem_nhdsWithin _ (o_open.mem_nhds xo), le_antisymm _ bot_le⟩ case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ⊢ ↑↑μ s ≤ 0 have ρs : ρ s ≠ ∞ := ((measure_mono (inter_subset_right _ _)).trans_lt μo).ne ** case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ρs : ↑↑ρ s ≠ ⊤ A : ∀ (q : ℝ≥0), 1 ≤ q → ↑↑μ s ≤ (↑q)⁻¹ * ↑↑ρ s ⊢ ↑↑μ s ≤ 0 ** have B : Tendsto (fun q : ℝ≥0 => (q : ℝ≥0∞)⁻¹ * ρ s) atTop (𝓝 (∞⁻¹ * ρ s)) := by apply ENNReal.Tendsto.mul_const _ (Or.inr ρs) exact ENNReal.tendsto_inv_iff.2 (ENNReal.tendsto_coe_nhds_top.2 tendsto_id) ** case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ρs : ↑↑ρ s ≠ ⊤ A : ∀ (q : ℝ≥0), 1 ≤ q → ↑↑μ s ≤ (↑q)⁻¹ * ↑↑ρ s B : Tendsto (fun q => (↑q)⁻¹ * ↑↑ρ s) atTop (𝓝 (⊤⁻¹ * ↑↑ρ s)) ⊢ ↑↑μ s ≤ 0 simp only [zero_mul, ENNReal.inv_top] at B case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ρs : ↑↑ρ s ≠ ⊤ A : ∀ (q : ℝ≥0), 1 ≤ q → ↑↑μ s ≤ (↑q)⁻¹ * ↑↑ρ s B : Tendsto (fun q => (↑q)⁻¹ * ↑↑ρ ({x \| limRatioMeas v hρ x = ⊤} ∩ o)) atTop (𝓝 0) ⊢ ↑↑μ s ≤ 0 apply ge_of_tendsto B case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ρs : ↑↑ρ s ≠ ⊤ A : ∀ (q : ℝ≥0), 1 ≤ q → ↑↑μ s ≤ (↑q)⁻¹ * ↑↑ρ s B : Tendsto (fun q => (↑q)⁻¹ * ↑↑ρ ({x \| limRatioMeas v hρ x = ⊤} ∩ o)) atTop (𝓝 0) ⊢ ∀ᶠ (c : ℝ≥0) in atTop, ↑↑μ s ≤ (↑c)⁻¹ * ↑↑ρ ({x \| limRatioMeas v hρ x = ⊤} ∩ o) exact eventually_atTop.2 ⟨1, A⟩ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ρs : ↑↑ρ s ≠ ⊤ ⊢ ∀ (q : ℝ≥0), 1 ≤ q → ↑↑μ s ≤ (↑q)⁻¹ * ↑↑ρ s intro q hq α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ρs : ↑↑ρ s ≠ ⊤ q : ℝ≥0 hq : 1 ≤ q ⊢ ↑↑μ s ≤ (↑q)⁻¹ * ↑↑ρ s rw [mul_comm, ← div_eq_mul_inv, ENNReal.le_div_iff_mul_le _ (Or.inr ρs), mul_comm] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ρs : ↑↑ρ s ≠ ⊤ q : ℝ≥0 hq : 1 ≤ q ⊢ ↑q * ↑↑μ s ≤ ↑↑ρ s apply v.mul_measure_le_of_subset_lt_limRatioMeas hρ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ρs : ↑↑ρ s ≠ ⊤ q : ℝ≥0 hq : 1 ≤ q ⊢ s ⊆ {x \| ↑q < limRatioMeas v hρ x} intro y hy α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ρs : ↑↑ρ s ≠ ⊤ q : ℝ≥0 hq : 1 ≤ q y : α hy : y ∈ s ⊢ y ∈ {x \| ↑q < limRatioMeas v hρ x} have : v.limRatioMeas hρ y = ∞ := hy.1 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ρs : ↑↑ρ s ≠ ⊤ q : ℝ≥0 hq : 1 ≤ q y : α hy : y ∈ s this : limRatioMeas v hρ y = ⊤ ⊢ y ∈ {x \| ↑q < limRatioMeas v hρ x} simp only [this, ENNReal.coe_lt_top, mem_setOf_eq] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ρs : ↑↑ρ s ≠ ⊤ q : ℝ≥0 hq : 1 ≤ q ⊢ ↑q ≠ 0 ∨ ↑↑ρ s ≠ 0 simp only [(zero_lt_one.trans_le hq).ne', true_or_iff, ENNReal.coe_eq_zero, Ne.def, not_false_iff] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ρs : ↑↑ρ s ≠ ⊤ A : ∀ (q : ℝ≥0), 1 ≤ q → ↑↑μ s ≤ (↑q)⁻¹ * ↑↑ρ s ⊢ Tendsto (fun q => (↑q)⁻¹ * ↑↑ρ s) atTop (𝓝 (⊤⁻¹ * ↑↑ρ s)) apply ENNReal.Tendsto.mul_const _ (Or.inr ρs) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = ⊤} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑ρ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = ⊤} ∩ o ρs : ↑↑ρ s ≠ ⊤ A : ∀ (q : ℝ≥0), 1 ≤ q → ↑↑μ s ≤ (↑q)⁻¹ * ↑↑ρ s ⊢ Tendsto (fun x => (↑x)⁻¹) atTop (𝓝 ⊤⁻¹) exact ENNReal.tendsto_inv_iff.2 (ENNReal.tendsto_coe_nhds_top.2 tendsto_id) Qed
VitaliFamily.measure_limRatioMeas_zero α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ ⊢ ↑↑ρ {x \| limRatioMeas v hρ x = 0} = 0 refine' null_of_locally_null _ fun x _ => _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} ⊢ ∃ u, u ∈ 𝓝[{x \| limRatioMeas v hρ x = 0}] x ∧ ↑↑ρ u = 0 obtain ⟨o, xo, o_open, μo⟩ : ∃ o : Set α, x ∈ o ∧ IsOpen o ∧ μ o < ∞ := Measure.exists_isOpen_measure_lt_top μ x case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ ⊢ ∃ u, u ∈ 𝓝[{x \| limRatioMeas v hρ x = 0}] x ∧ ↑↑ρ u = 0 let s := {x : α \| v.limRatioMeas hρ x = 0} ∩ o case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o ⊢ ∃ u, u ∈ 𝓝[{x \| limRatioMeas v hρ x = 0}] x ∧ ↑↑ρ u = 0 refine' ⟨s, inter_mem_nhdsWithin _ (o_open.mem_nhds xo), le_antisymm _ bot_le⟩ case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o ⊢ ↑↑ρ s ≤ 0 have μs : μ s ≠ ∞ := ((measure_mono (inter_subset_right _ _)).trans_lt μo).ne case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o μs : ↑↑μ s ≠ ⊤ ⊢ ↑↑ρ s ≤ 0 have A : ∀ q : ℝ≥0, 0 < q → ρ s ≤ q * μ s := by intro q hq apply v.measure_le_mul_of_subset_limRatioMeas_lt hρ intro y hy have : v.limRatioMeas hρ y = 0 := hy.1 simp only [this, mem_setOf_eq, hq, ENNReal.coe_pos] ** case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o μs : ↑↑μ s ≠ ⊤ A : ∀ (q : ℝ≥0), 0 < q → ↑↑ρ s ≤ ↑q * ↑↑μ s ⊢ ↑↑ρ s ≤ 0 ** have B : Tendsto (fun q : ℝ≥0 => (q : ℝ≥0∞) * μ s) (𝓝[>] (0 : ℝ≥0)) (𝓝 ((0 : ℝ≥0) * μ s)) := by apply ENNReal.Tendsto.mul_const _ (Or.inr μs) rw [ENNReal.tendsto_coe] exact nhdsWithin_le_nhds ** case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o μs : ↑↑μ s ≠ ⊤ A : ∀ (q : ℝ≥0), 0 < q → ↑↑ρ s ≤ ↑q * ↑↑μ s B : Tendsto (fun q => ↑q * ↑↑μ s) (𝓝[Ioi 0] 0) (𝓝 (↑0 * ↑↑μ s)) ⊢ ↑↑ρ s ≤ 0 simp only [zero_mul, ENNReal.coe_zero] at B case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o μs : ↑↑μ s ≠ ⊤ A : ∀ (q : ℝ≥0), 0 < q → ↑↑ρ s ≤ ↑q * ↑↑μ s B : Tendsto (fun q => ↑q * ↑↑μ ({x \| limRatioMeas v hρ x = 0} ∩ o)) (𝓝[Ioi 0] 0) (𝓝 0) ⊢ ↑↑ρ s ≤ 0 apply ge_of_tendsto B case intro.intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o μs : ↑↑μ s ≠ ⊤ A : ∀ (q : ℝ≥0), 0 < q → ↑↑ρ s ≤ ↑q * ↑↑μ s B : Tendsto (fun q => ↑q * ↑↑μ ({x \| limRatioMeas v hρ x = 0} ∩ o)) (𝓝[Ioi 0] 0) (𝓝 0) ⊢ ∀ᶠ (c : ℝ≥0) in 𝓝[Ioi 0] 0, ↑↑ρ s ≤ ↑c * ↑↑μ ({x \| limRatioMeas v hρ x = 0} ∩ o) filter_upwards [self_mem_nhdsWithin] using A α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o μs : ↑↑μ s ≠ ⊤ ⊢ ∀ (q : ℝ≥0), 0 < q → ↑↑ρ s ≤ ↑q * ↑↑μ s intro q hq α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o μs : ↑↑μ s ≠ ⊤ q : ℝ≥0 hq : 0 < q ⊢ ↑↑ρ s ≤ ↑q * ↑↑μ s apply v.measure_le_mul_of_subset_limRatioMeas_lt hρ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o μs : ↑↑μ s ≠ ⊤ q : ℝ≥0 hq : 0 < q ⊢ s ⊆ {x \| limRatioMeas v hρ x < ↑q} intro y hy α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o μs : ↑↑μ s ≠ ⊤ q : ℝ≥0 hq : 0 < q y : α hy : y ∈ s ⊢ y ∈ {x \| limRatioMeas v hρ x < ↑q} have : v.limRatioMeas hρ y = 0 := hy.1 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o μs : ↑↑μ s ≠ ⊤ q : ℝ≥0 hq : 0 < q y : α hy : y ∈ s this : limRatioMeas v hρ y = 0 ⊢ y ∈ {x \| limRatioMeas v hρ x < ↑q} simp only [this, mem_setOf_eq, hq, ENNReal.coe_pos] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o μs : ↑↑μ s ≠ ⊤ A : ∀ (q : ℝ≥0), 0 < q → ↑↑ρ s ≤ ↑q * ↑↑μ s ⊢ Tendsto (fun q => ↑q * ↑↑μ s) (𝓝[Ioi 0] 0) (𝓝 (↑0 * ↑↑μ s)) apply ENNReal.Tendsto.mul_const _ (Or.inr μs) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o μs : ↑↑μ s ≠ ⊤ A : ∀ (q : ℝ≥0), 0 < q → ↑↑ρ s ≤ ↑q * ↑↑μ s ⊢ Tendsto (fun x => ↑x) (𝓝[Ioi 0] 0) (𝓝 ↑0) rw [ENNReal.tendsto_coe] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ x : α x✝ : x ∈ {x \| limRatioMeas v hρ x = 0} o : Set α xo : x ∈ o o_open : IsOpen o μo : ↑↑μ o < ⊤ s : Set α := {x \| limRatioMeas v hρ x = 0} ∩ o μs : ↑↑μ s ≠ ⊤ A : ∀ (q : ℝ≥0), 0 < q → ↑↑ρ s ≤ ↑q * ↑↑μ s ⊢ Tendsto (fun x => x) (𝓝[Ioi 0] 0) (𝓝 0) exact nhdsWithin_le_nhds Qed
VitaliFamily.le_mul_withDensity ** α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t ⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s have t_ne_zero' : t ≠ 0 := (zero_lt_one.trans ht).ne' α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 ⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s have t_ne_zero : (t : ℝ≥0∞) ≠ 0 := by simpa only [ENNReal.coe_eq_zero, Ne.def] using t_ne_zero' α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s let ν := μ.withDensity (v.limRatioMeas hρ) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) ⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s let f := v.limRatioMeas hρ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ ⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s have f_meas : Measurable f := v.limRatioMeas_measurable hρ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f ⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s have A : ρ (s ∩ f ⁻¹' {0}) ≤ (t • ν) (s ∩ f ⁻¹' {0}) := by refine' le_trans (measure_mono (inter_subset_right _ _)) (le_trans (le_of_eq _) (zero_le _)) exact v.measure_limRatioMeas_zero hρ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) ⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s have B : ρ (s ∩ f ⁻¹' {∞}) ≤ (t • ν) (s ∩ f ⁻¹' {∞}) := by apply le_trans (le_of_eq _) (zero_le _) apply hρ rw [← nonpos_iff_eq_zero] exact (measure_mono (inter_subset_right _ _)).trans (v.measure_limRatioMeas_top hρ).le α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) ⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s ** have C : ∀ n : ℤ, ρ (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) ≤ (t • ν) (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) := by intro n let I := Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1)) have M : MeasurableSet (s ∩ f ⁻¹' I) := hs.inter (f_meas measurableSet_Ico) simp only [M, withDensity_apply, coe_nnreal_smul_apply] calc ρ (s ∩ f ⁻¹' I) ≤ (t : ℝ≥0∞) ^ (n + 1) * μ (s ∩ f ⁻¹' I) := by rw [← ENNReal.coe_zpow t_ne_zero'] apply v.measure_le_mul_of_subset_limRatioMeas_lt hρ intro x hx apply hx.2.2.trans_le (le_of_eq _) rw [ENNReal.coe_zpow t_ne_zero'] _ = ∫⁻ x in s ∩ f ⁻¹' I, (t : ℝ≥0∞) ^ (n + 1) ∂μ := by simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter] _ ≤ ∫⁻ x in s ∩ f ⁻¹' I, t * f x ∂μ := by apply lintegral_mono_ae ((ae_restrict_iff' M).2 (eventually_of_forall fun x hx => ?_)) rw [add_comm, ENNReal.zpow_add t_ne_zero ENNReal.coe_ne_top, zpow_one] exact mul_le_mul_left' hx.2.1 _ _ = t * ∫⁻ x in s ∩ f ⁻¹' I, f x ∂μ := lintegral_const_mul _ f_meas ** α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) C : ∀ (n : ℤ), ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ⊢ ↑↑ρ s ≤ ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s calc ρ s = ρ (s ∩ f ⁻¹' {0}) + ρ (s ∩ f ⁻¹' {∞}) + ∑' n : ℤ, ρ (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) := measure_eq_measure_preimage_add_measure_tsum_Ico_zpow ρ f_meas hs ht _ ≤ (t • ν) (s ∩ f ⁻¹' {0}) + (t • ν) (s ∩ f ⁻¹' {∞}) + ∑' n : ℤ, (t • ν) (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) := (add_le_add (add_le_add A B) (ENNReal.tsum_le_tsum C)) _ = (t • ν) s := (measure_eq_measure_preimage_add_measure_tsum_Ico_zpow (t • ν) f_meas hs ht).symm α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 ⊢ ↑t ≠ 0 simpa only [ENNReal.coe_eq_zero, Ne.def] using t_ne_zero' α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f ⊢ ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) refine' le_trans (measure_mono (inter_subset_right _ _)) (le_trans (le_of_eq _) (zero_le _)) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f ⊢ ↑↑ρ (f ⁻¹' {0}) = 0 exact v.measure_limRatioMeas_zero hρ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) ⊢ ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) apply le_trans (le_of_eq _) (zero_le _) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) ⊢ ↑↑ρ (s ∩ f ⁻¹' {⊤}) = 0 apply hρ case a α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) ⊢ ↑↑μ (s ∩ f ⁻¹' {⊤}) = 0 rw [← nonpos_iff_eq_zero] case a α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) ⊢ ↑↑μ (s ∩ f ⁻¹' {⊤}) ≤ 0 exact (measure_mono (inter_subset_right _ _)).trans (v.measure_limRatioMeas_top hρ).le α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) ⊢ ∀ (n : ℤ), ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) intro n α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) n : ℤ ⊢ ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) let I := Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1)) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) n : ℤ I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1)) ⊢ ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) have M : MeasurableSet (s ∩ f ⁻¹' I) := hs.inter (f_meas measurableSet_Ico) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) n : ℤ I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1)) M : MeasurableSet (s ∩ f ⁻¹' I) ⊢ ↑↑ρ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) simp only [M, withDensity_apply, coe_nnreal_smul_apply] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) n : ℤ I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1)) M : MeasurableSet (s ∩ f ⁻¹' I) ⊢ ↑↑ρ (s ∩ limRatioMeas v hρ ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑t * ∫⁻ (a : α) in s ∩ limRatioMeas v hρ ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)), limRatioMeas v hρ a ∂μ ** calc ρ (s ∩ f ⁻¹' I) ≤ (t : ℝ≥0∞) ^ (n + 1) * μ (s ∩ f ⁻¹' I) := by rw [← ENNReal.coe_zpow t_ne_zero'] apply v.measure_le_mul_of_subset_limRatioMeas_lt hρ intro x hx apply hx.2.2.trans_le (le_of_eq _) rw [ENNReal.coe_zpow t_ne_zero'] _ = ∫⁻ x in s ∩ f ⁻¹' I, (t : ℝ≥0∞) ^ (n + 1) ∂μ := by simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter] _ ≤ ∫⁻ x in s ∩ f ⁻¹' I, t * f x ∂μ := by apply lintegral_mono_ae ((ae_restrict_iff' M).2 (eventually_of_forall fun x hx => ?_)) rw [add_comm, ENNReal.zpow_add t_ne_zero ENNReal.coe_ne_top, zpow_one] exact mul_le_mul_left' hx.2.1 _ _ = t * ∫⁻ x in s ∩ f ⁻¹' I, f x ∂μ := lintegral_const_mul _ f_meas ** α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) n : ℤ I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1)) M : MeasurableSet (s ∩ f ⁻¹' I) ⊢ ↑↑ρ (s ∩ f ⁻¹' I) ≤ ↑t ^ (n + 1) * ↑↑μ (s ∩ f ⁻¹' I) rw [← ENNReal.coe_zpow t_ne_zero'] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) n : ℤ I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1)) M : MeasurableSet (s ∩ f ⁻¹' I) ⊢ ↑↑ρ (s ∩ f ⁻¹' I) ≤ ↑(t ^ (n + 1)) * ↑↑μ (s ∩ f ⁻¹' I) apply v.measure_le_mul_of_subset_limRatioMeas_lt hρ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) n : ℤ I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1)) M : MeasurableSet (s ∩ f ⁻¹' I) ⊢ s ∩ f ⁻¹' I ⊆ {x \| limRatioMeas v hρ x < ↑(t ^ (n + 1))} intro x hx α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) n : ℤ I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1)) M : MeasurableSet (s ∩ f ⁻¹' I) x : α hx : x ∈ s ∩ f ⁻¹' I ⊢ x ∈ {x \| limRatioMeas v hρ x < ↑(t ^ (n + 1))} apply hx.2.2.trans_le (le_of_eq _) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) n : ℤ I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1)) M : MeasurableSet (s ∩ f ⁻¹' I) x : α hx : x ∈ s ∩ f ⁻¹' I ⊢ ↑t ^ (n + 1) = ↑(t ^ (n + 1)) rw [ENNReal.coe_zpow t_ne_zero'] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) n : ℤ I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1)) M : MeasurableSet (s ∩ f ⁻¹' I) ⊢ ↑t ^ (n + 1) * ↑↑μ (s ∩ f ⁻¹' I) = ∫⁻ (x : α) in s ∩ f ⁻¹' I, ↑t ^ (n + 1) ∂μ simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) n : ℤ I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1)) M : MeasurableSet (s ∩ f ⁻¹' I) ⊢ ∫⁻ (x : α) in s ∩ f ⁻¹' I, ↑t ^ (n + 1) ∂μ ≤ ∫⁻ (x : α) in s ∩ f ⁻¹' I, ↑t * f x ∂μ apply lintegral_mono_ae ((ae_restrict_iff' M).2 (eventually_of_forall fun x hx => ?_)) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) n : ℤ I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1)) M : MeasurableSet (s ∩ f ⁻¹' I) x : α hx : x ∈ s ∩ f ⁻¹' I ⊢ ↑t ^ (n + 1) ≤ ↑t * f x rw [add_comm, ENNReal.zpow_add t_ne_zero ENNReal.coe_ne_top, zpow_one] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ s : Set α hs : MeasurableSet s t : ℝ≥0 ht : 1 < t t_ne_zero' : t ≠ 0 t_ne_zero : ↑t ≠ 0 ν : Measure α := withDensity μ (limRatioMeas v hρ) f : α → ℝ≥0∞ := limRatioMeas v hρ f_meas : Measurable f A : ↑↑ρ (s ∩ f ⁻¹' {0}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {0}) B : ↑↑ρ (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(t • ν) (s ∩ f ⁻¹' {⊤}) n : ℤ I : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1)) M : MeasurableSet (s ∩ f ⁻¹' I) x : α hx : x ∈ s ∩ f ⁻¹' I ⊢ ↑t * ↑t ^ n ≤ ↑t * f x exact mul_le_mul_left' hx.2.1 _ Qed
VitaliFamily.ae_tendsto_rnDeriv_of_absolutelyContinuous α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv ρ μ x)) have A : (μ.withDensity (v.limRatioMeas hρ)).rnDeriv μ =ᵐ[μ] v.limRatioMeas hρ := rnDeriv_withDensity μ (v.limRatioMeas_measurable hρ) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ A : rnDeriv (withDensity μ (limRatioMeas v hρ)) μ =ᶠ[ae μ] limRatioMeas v hρ ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv ρ μ x)) rw [v.withDensity_limRatioMeas_eq hρ] at A α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ A : rnDeriv ρ μ =ᶠ[ae μ] limRatioMeas v hρ ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv ρ μ x)) filter_upwards [v.ae_tendsto_limRatioMeas hρ, A] with _ _ h'x case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ hρ : ρ ≪ μ A : rnDeriv ρ μ =ᶠ[ae μ] limRatioMeas v hρ a✝¹ : α a✝ : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v a✝¹) (𝓝 (limRatioMeas v hρ a✝¹)) h'x : rnDeriv ρ μ a✝¹ = limRatioMeas v hρ a✝¹ ⊢ Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v a✝¹) (𝓝 (rnDeriv ρ μ a✝¹)) rwa [h'x] ** Qed
VitaliFamily.ae_tendsto_rnDeriv α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv ρ μ x)) let t := μ.withDensity (ρ.rnDeriv μ) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ t : Measure α := withDensity μ (rnDeriv ρ μ) ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv ρ μ x)) have eq_add : ρ = ρ.singularPart μ + t := haveLebesgueDecomposition_add _ _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ t : Measure α := withDensity μ (rnDeriv ρ μ) eq_add : ρ = singularPart ρ μ + t ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv ρ μ x)) have A : ∀ᵐ x ∂μ, Tendsto (fun a => ρ.singularPart μ a / μ a) (v.filterAt x) (𝓝 0) := v.ae_eventually_measure_zero_of_singular (mutuallySingular_singularPart ρ μ) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ t : Measure α := withDensity μ (rnDeriv ρ μ) eq_add : ρ = singularPart ρ μ + t A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v x) (𝓝 0) ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv ρ μ x)) have B : ∀ᵐ x ∂μ, t.rnDeriv μ x = ρ.rnDeriv μ x := rnDeriv_withDensity μ (measurable_rnDeriv ρ μ) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ t : Measure α := withDensity μ (rnDeriv ρ μ) eq_add : ρ = singularPart ρ μ + t A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v x) (𝓝 0) B : ∀ᵐ (x : α) ∂μ, rnDeriv t μ x = rnDeriv ρ μ x ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv ρ μ x)) have C : ∀ᵐ x ∂μ, Tendsto (fun a => t a / μ a) (v.filterAt x) (𝓝 (t.rnDeriv μ x)) := v.ae_tendsto_rnDeriv_of_absolutelyContinuous (withDensity_absolutelyContinuous _ _) α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ t : Measure α := withDensity μ (rnDeriv ρ μ) eq_add : ρ = singularPart ρ μ + t A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v x) (𝓝 0) B : ∀ᵐ (x : α) ∂μ, rnDeriv t μ x = rnDeriv ρ μ x C : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv t μ x)) ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv ρ μ x)) filter_upwards [A, B, C] with _ Ax Bx Cx case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ t : Measure α := withDensity μ (rnDeriv ρ μ) eq_add : ρ = singularPart ρ μ + t A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v x) (𝓝 0) B : ∀ᵐ (x : α) ∂μ, rnDeriv t μ x = rnDeriv ρ μ x C : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv t μ x)) a✝ : α Ax : Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v a✝) (𝓝 0) Bx : rnDeriv t μ a✝ = rnDeriv ρ μ a✝ Cx : Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v a✝) (𝓝 (rnDeriv t μ a✝)) ⊢ Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v a✝) (𝓝 (rnDeriv ρ μ a✝)) convert Ax.add Cx using 1 case h.e'_3 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ t : Measure α := withDensity μ (rnDeriv ρ μ) eq_add : ρ = singularPart ρ μ + t A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v x) (𝓝 0) B : ∀ᵐ (x : α) ∂μ, rnDeriv t μ x = rnDeriv ρ μ x C : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv t μ x)) a✝ : α Ax : Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v a✝) (𝓝 0) Bx : rnDeriv t μ a✝ = rnDeriv ρ μ a✝ Cx : Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v a✝) (𝓝 (rnDeriv t μ a✝)) ⊢ (fun a => ↑↑ρ a / ↑↑μ a) = fun x => ↑↑(singularPart ρ μ) x / ↑↑μ x + ↑↑t x / ↑↑μ x ext1 a case h.e'_3.h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ t : Measure α := withDensity μ (rnDeriv ρ μ) eq_add : ρ = singularPart ρ μ + t A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v x) (𝓝 0) B : ∀ᵐ (x : α) ∂μ, rnDeriv t μ x = rnDeriv ρ μ x C : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv t μ x)) a✝ : α Ax : Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v a✝) (𝓝 0) Bx : rnDeriv t μ a✝ = rnDeriv ρ μ a✝ Cx : Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v a✝) (𝓝 (rnDeriv t μ a✝)) a : Set α ⊢ ↑↑ρ a / ↑↑μ a = ↑↑(singularPart ρ μ) a / ↑↑μ a + ↑↑t a / ↑↑μ a conv_lhs => rw [eq_add] case h.e'_3.h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ t : Measure α := withDensity μ (rnDeriv ρ μ) eq_add : ρ = singularPart ρ μ + t A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v x) (𝓝 0) B : ∀ᵐ (x : α) ∂μ, rnDeriv t μ x = rnDeriv ρ μ x C : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv t μ x)) a✝ : α Ax : Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v a✝) (𝓝 0) Bx : rnDeriv t μ a✝ = rnDeriv ρ μ a✝ Cx : Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v a✝) (𝓝 (rnDeriv t μ a✝)) a : Set α ⊢ ↑↑(singularPart ρ μ + t) a / ↑↑μ a = ↑↑(singularPart ρ μ) a / ↑↑μ a + ↑↑t a / ↑↑μ a simp only [Pi.add_apply, coe_add, ENNReal.add_div] case h.e'_5 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ t : Measure α := withDensity μ (rnDeriv ρ μ) eq_add : ρ = singularPart ρ μ + t A : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v x) (𝓝 0) B : ∀ᵐ (x : α) ∂μ, rnDeriv t μ x = rnDeriv ρ μ x C : ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv t μ x)) a✝ : α Ax : Tendsto (fun a => ↑↑(singularPart ρ μ) a / ↑↑μ a) (filterAt v a✝) (𝓝 0) Bx : rnDeriv t μ a✝ = rnDeriv ρ μ a✝ Cx : Tendsto (fun a => ↑↑t a / ↑↑μ a) (filterAt v a✝) (𝓝 (rnDeriv t μ a✝)) ⊢ 𝓝 (rnDeriv ρ μ a✝) = 𝓝 (0 + rnDeriv t μ a✝) simp only [Bx, zero_add] ** Qed
VitaliFamily.ae_tendsto_measure_inter_div α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α ⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1) let t := toMeasurable μ s α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α t : Set α := toMeasurable μ s ⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1) have A : ∀ᵐ x ∂μ.restrict s, Tendsto (fun a => μ (t ∩ a) / μ a) (v.filterAt x) (𝓝 (t.indicator 1 x)) := by apply ae_mono restrict_le_self apply ae_tendsto_measure_inter_div_of_measurableSet exact measurableSet_toMeasurable _ _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α t : Set α := toMeasurable μ s A : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x)) ⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1) have B : ∀ᵐ x ∂μ.restrict s, t.indicator 1 x = (1 : ℝ≥0∞) := by refine' ae_restrict_of_ae_restrict_of_subset (subset_toMeasurable μ s) _ filter_upwards [ae_restrict_mem (measurableSet_toMeasurable μ s)] with _ hx simp only [hx, Pi.one_apply, indicator_of_mem] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α t : Set α := toMeasurable μ s A : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x)) B : ∀ᵐ (x : α) ∂Measure.restrict μ s, indicator t 1 x = 1 ⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1) filter_upwards [A, B] with x hx h'x case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α t : Set α := toMeasurable μ s A : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x)) B : ∀ᵐ (x : α) ∂Measure.restrict μ s, indicator t 1 x = 1 x : α hx : Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x)) h'x : indicator t 1 x = 1 ⊢ Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1) rw [h'x] at hx case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α t : Set α := toMeasurable μ s A : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x)) B : ∀ᵐ (x : α) ∂Measure.restrict μ s, indicator t 1 x = 1 x : α hx : Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1) h'x : indicator t 1 x = 1 ⊢ Tendsto (fun a => ↑↑μ (s ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1) apply hx.congr' _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α t : Set α := toMeasurable μ s A : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x)) B : ∀ᵐ (x : α) ∂Measure.restrict μ s, indicator t 1 x = 1 x : α hx : Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1) h'x : indicator t 1 x = 1 ⊢ (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) =ᶠ[filterAt v x] fun a => ↑↑μ (s ∩ a) / ↑↑μ a filter_upwards [v.eventually_filterAt_measurableSet x] with _ ha case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α t : Set α := toMeasurable μ s A : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x)) B : ∀ᵐ (x : α) ∂Measure.restrict μ s, indicator t 1 x = 1 x : α hx : Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1) h'x : indicator t 1 x = 1 a✝ : Set α ha : MeasurableSet a✝ ⊢ ↑↑μ (t ∩ a✝) / ↑↑μ a✝ = ↑↑μ (s ∩ a✝) / ↑↑μ a✝ congr 1 case h.e_a α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α t : Set α := toMeasurable μ s A : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x)) B : ∀ᵐ (x : α) ∂Measure.restrict μ s, indicator t 1 x = 1 x : α hx : Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 1) h'x : indicator t 1 x = 1 a✝ : Set α ha : MeasurableSet a✝ ⊢ ↑↑μ (t ∩ a✝) = ↑↑μ (s ∩ a✝) exact measure_toMeasurable_inter_of_sigmaFinite ha _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α t : Set α := toMeasurable μ s ⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x)) apply ae_mono restrict_le_self case a α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α t : Set α := toMeasurable μ s ⊢ {x \| (fun x => Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x))) x} ∈ ae μ apply ae_tendsto_measure_inter_div_of_measurableSet case a.hs α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α t : Set α := toMeasurable μ s ⊢ MeasurableSet t exact measurableSet_toMeasurable _ _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α t : Set α := toMeasurable μ s A : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x)) ⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, indicator t 1 x = 1 refine' ae_restrict_of_ae_restrict_of_subset (subset_toMeasurable μ s) _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α t : Set α := toMeasurable μ s A : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x)) ⊢ ∀ᵐ (x : α) ∂Measure.restrict μ (toMeasurable μ s), indicator t 1 x = 1 filter_upwards [ae_restrict_mem (measurableSet_toMeasurable μ s)] with _ hx case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ s : Set α t : Set α := toMeasurable μ s A : ∀ᵐ (x : α) ∂Measure.restrict μ s, Tendsto (fun a => ↑↑μ (t ∩ a) / ↑↑μ a) (filterAt v x) (𝓝 (indicator t 1 x)) a✝ : α hx : a✝ ∈ toMeasurable μ s ⊢ indicator t 1 a✝ = 1 simp only [hx, Pi.one_apply, indicator_of_mem] ** Qed
VitaliFamily.ae_tendsto_lintegral_div' α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → ℝ≥0∞ hf : Measurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => (∫⁻ (y : α) in a, f y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (f x)) let ρ := μ.withDensity f α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ✝ : Measure α inst✝ : IsLocallyFiniteMeasure ρ✝ f : α → ℝ≥0∞ hf : Measurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ ρ : Measure α := withDensity μ f ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => (∫⁻ (y : α) in a, f y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (f x)) have : IsFiniteMeasure ρ := isFiniteMeasure_withDensity h'f α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ✝ : Measure α inst✝ : IsLocallyFiniteMeasure ρ✝ f : α → ℝ≥0∞ hf : Measurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ ρ : Measure α := withDensity μ f this : IsFiniteMeasure ρ ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => (∫⁻ (y : α) in a, f y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (f x)) filter_upwards [ae_tendsto_rnDeriv v ρ, rnDeriv_withDensity μ hf] with x hx h'x case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ✝ : Measure α inst✝ : IsLocallyFiniteMeasure ρ✝ f : α → ℝ≥0∞ hf : Measurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ ρ : Measure α := withDensity μ f this : IsFiniteMeasure ρ x : α hx : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv ρ μ x)) h'x : rnDeriv (withDensity μ f) μ x = f x ⊢ Tendsto (fun a => (∫⁻ (y : α) in a, f y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (f x)) rw [← h'x] case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ✝ : Measure α inst✝ : IsLocallyFiniteMeasure ρ✝ f : α → ℝ≥0∞ hf : Measurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ ρ : Measure α := withDensity μ f this : IsFiniteMeasure ρ x : α hx : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv ρ μ x)) h'x : rnDeriv (withDensity μ f) μ x = f x ⊢ Tendsto (fun a => (∫⁻ (y : α) in a, f y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv (withDensity μ f) μ x)) apply hx.congr' _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ✝ : Measure α inst✝ : IsLocallyFiniteMeasure ρ✝ f : α → ℝ≥0∞ hf : Measurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ ρ : Measure α := withDensity μ f this : IsFiniteMeasure ρ x : α hx : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv ρ μ x)) h'x : rnDeriv (withDensity μ f) μ x = f x ⊢ (fun a => ↑↑ρ a / ↑↑μ a) =ᶠ[filterAt v x] fun a => (∫⁻ (y : α) in a, f y ∂μ) / ↑↑μ a filter_upwards [v.eventually_filterAt_measurableSet x] with a ha case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ✝ : Measure α inst✝ : IsLocallyFiniteMeasure ρ✝ f : α → ℝ≥0∞ hf : Measurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ ρ : Measure α := withDensity μ f this : IsFiniteMeasure ρ x : α hx : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (filterAt v x) (𝓝 (rnDeriv ρ μ x)) h'x : rnDeriv (withDensity μ f) μ x = f x a : Set α ha : MeasurableSet a ⊢ ↑↑ρ a / ↑↑μ a = (∫⁻ (y : α) in a, f y ∂μ) / ↑↑μ a rw [← withDensity_apply f ha] ** Qed
VitaliFamily.ae_tendsto_lintegral_div α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => (∫⁻ (y : α) in a, f y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (f x)) have A : (∫⁻ y, hf.mk f y ∂μ) ≠ ∞ := by convert h'f using 1 apply lintegral_congr_ae exact hf.ae_eq_mk.symm α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ A : ∫⁻ (y : α), AEMeasurable.mk f hf y ∂μ ≠ ⊤ ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => (∫⁻ (y : α) in a, f y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (f x)) filter_upwards [v.ae_tendsto_lintegral_div' hf.measurable_mk A, hf.ae_eq_mk] with x hx h'x case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ A : ∫⁻ (y : α), AEMeasurable.mk f hf y ∂μ ≠ ⊤ x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, AEMeasurable.mk f hf y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (AEMeasurable.mk f hf x)) h'x : f x = AEMeasurable.mk f hf x ⊢ Tendsto (fun a => (∫⁻ (y : α) in a, f y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (f x)) rw [h'x] case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ A : ∫⁻ (y : α), AEMeasurable.mk f hf y ∂μ ≠ ⊤ x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, AEMeasurable.mk f hf y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (AEMeasurable.mk f hf x)) h'x : f x = AEMeasurable.mk f hf x ⊢ Tendsto (fun a => (∫⁻ (y : α) in a, f y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (AEMeasurable.mk f hf x)) convert hx using 1 case h.e'_3 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ A : ∫⁻ (y : α), AEMeasurable.mk f hf y ∂μ ≠ ⊤ x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, AEMeasurable.mk f hf y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (AEMeasurable.mk f hf x)) h'x : f x = AEMeasurable.mk f hf x ⊢ (fun a => (∫⁻ (y : α) in a, f y ∂μ) / ↑↑μ a) = fun a => (∫⁻ (y : α) in a, AEMeasurable.mk f hf y ∂μ) / ↑↑μ a ext1 a case h.e'_3.h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ A : ∫⁻ (y : α), AEMeasurable.mk f hf y ∂μ ≠ ⊤ x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, AEMeasurable.mk f hf y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (AEMeasurable.mk f hf x)) h'x : f x = AEMeasurable.mk f hf x a : Set α ⊢ (∫⁻ (y : α) in a, f y ∂μ) / ↑↑μ a = (∫⁻ (y : α) in a, AEMeasurable.mk f hf y ∂μ) / ↑↑μ a congr 1 case h.e'_3.h.e_a α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ A : ∫⁻ (y : α), AEMeasurable.mk f hf y ∂μ ≠ ⊤ x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, AEMeasurable.mk f hf y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (AEMeasurable.mk f hf x)) h'x : f x = AEMeasurable.mk f hf x a : Set α ⊢ ∫⁻ (y : α) in a, f y ∂μ = ∫⁻ (y : α) in a, AEMeasurable.mk f hf y ∂μ apply lintegral_congr_ae case h.e'_3.h.e_a.h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ A : ∫⁻ (y : α), AEMeasurable.mk f hf y ∂μ ≠ ⊤ x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, AEMeasurable.mk f hf y ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (AEMeasurable.mk f hf x)) h'x : f x = AEMeasurable.mk f hf x a : Set α ⊢ (fun a => f a) =ᶠ[ae (Measure.restrict μ a)] fun a => AEMeasurable.mk f hf a exact ae_restrict_of_ae hf.ae_eq_mk α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ ⊢ ∫⁻ (y : α), AEMeasurable.mk f hf y ∂μ ≠ ⊤ convert h'f using 1 case h.e'_2 α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ ⊢ ∫⁻ (y : α), AEMeasurable.mk f hf y ∂μ = ∫⁻ (y : α), f y ∂μ apply lintegral_congr_ae case h.e'_2.h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → ℝ≥0∞ hf : AEMeasurable f h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤ ⊢ (fun a => AEMeasurable.mk f hf a) =ᶠ[ae μ] fun a => f a exact hf.ae_eq_mk.symm ** Qed
VitaliFamily.ae_tendsto_lintegral_nnnorm_sub_div_of_integrable α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : Integrable f ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) have I : Integrable (hf.1.mk f) μ := hf.congr hf.1.ae_eq_mk α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : Integrable f I : Integrable (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)) ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) filter_upwards [v.ae_tendsto_lintegral_nnnorm_sub_div'_of_integrable I hf.1.stronglyMeasurable_mk, hf.1.ae_eq_mk] with x hx h'x case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : Integrable f I : Integrable (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)) x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y - AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) h'x : f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x ⊢ Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) apply hx.congr _ α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : Integrable f I : Integrable (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)) x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y - AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) h'x : f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x ⊢ ∀ (x_1 : Set α), (∫⁻ (y : α) in x_1, ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y - AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ) / ↑↑μ x_1 = (∫⁻ (y : α) in x_1, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ x_1 intro a α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : Integrable f I : Integrable (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)) x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y - AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) h'x : f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x a : Set α ⊢ (∫⁻ (y : α) in a, ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y - AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ) / ↑↑μ a = (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a congr 1 case e_a α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : Integrable f I : Integrable (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)) x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y - AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) h'x : f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x a : Set α ⊢ ∫⁻ (y : α) in a, ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y - AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ = ∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ apply lintegral_congr_ae case e_a.h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : Integrable f I : Integrable (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)) x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y - AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) h'x : f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x a : Set α ⊢ (fun a => ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) a - AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊) =ᶠ[ae (Measure.restrict μ a)] fun a => ↑‖f a - f x‖₊ apply ae_restrict_of_ae case e_a.h.h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : Integrable f I : Integrable (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)) x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y - AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) h'x : f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x a : Set α ⊢ ∀ᵐ (x_1 : α) ∂μ, (fun a => ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) a - AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊) x_1 = (fun a => ↑‖f a - f x‖₊) x_1 filter_upwards [hf.1.ae_eq_mk] with y hy case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : Integrable f I : Integrable (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ)) x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y - AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) h'x : f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x a : Set α y : α hy : f y = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y ⊢ ↑‖AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) y - AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f μ) x‖₊ = ↑‖f y - f x‖₊ rw [hy, h'x] ** Qed
VitaliFamily.eventually_filterAt_integrableOn α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ x : α f : α → E hf : LocallyIntegrable f ⊢ ∀ᶠ (a : Set α) in filterAt v x, IntegrableOn f a rcases hf x with ⟨w, w_nhds, hw⟩ case intro.intro α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ x : α f : α → E hf : LocallyIntegrable f w : Set α w_nhds : w ∈ 𝓝 x hw : IntegrableOn f w ⊢ ∀ᶠ (a : Set α) in filterAt v x, IntegrableOn f a filter_upwards [v.eventually_filterAt_subset_of_nhds w_nhds] with a ha case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ x : α f : α → E hf : LocallyIntegrable f w : Set α w_nhds : w ∈ 𝓝 x hw : IntegrableOn f w a : Set α ha : a ⊆ w ⊢ IntegrableOn f a exact hw.mono_set ha ** Qed
VitaliFamily.ae_tendsto_average_norm_sub α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f ⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0) filter_upwards [v.ae_tendsto_lintegral_nnnorm_sub_div hf] with x hx case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) ⊢ Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0) have := (ENNReal.tendsto_toReal ENNReal.zero_ne_top).comp hx case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) this : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 (ENNReal.toReal 0)) ⊢ Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0) simp only [ENNReal.zero_toReal] at this case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) this : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) ⊢ Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (filterAt v x) (𝓝 0) apply Tendsto.congr' _ this α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) this : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) ⊢ (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) =ᶠ[filterAt v x] fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ filter_upwards [v.eventually_measure_lt_top x, v.eventually_filterAt_integrableOn x hf] with a h'a h''a case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) this : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) a : Set α h'a : ↑↑μ a < ⊤ h''a : IntegrableOn f a ⊢ (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) a = ⨍ (y : α) in a, ‖f y - f x‖ ∂μ simp only [Function.comp_apply, ENNReal.toReal_div, setAverage_eq, div_eq_inv_mul] ** case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) this : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) a : Set α h'a : ↑↑μ a < ⊤ h''a : IntegrableOn f a ⊢ (ENNReal.toReal (↑↑μ a))⁻¹ * ENNReal.toReal (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) = (ENNReal.toReal (↑↑μ a))⁻¹ • ∫ (y : α) in a, ‖f y - f x‖ ∂μ have A : IntegrableOn (fun y => (‖f y - f x‖₊ : ℝ)) a μ := by simp_rw [coe_nnnorm] exact (h''a.sub (integrableOn_const.2 (Or.inr h'a))).norm case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) this : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) a : Set α h'a : ↑↑μ a < ⊤ h''a : IntegrableOn f a A : IntegrableOn (fun y => ↑‖f y - f x‖₊) a ⊢ (ENNReal.toReal (↑↑μ a))⁻¹ * ENNReal.toReal (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) = (ENNReal.toReal (↑↑μ a))⁻¹ • ∫ (y : α) in a, ‖f y - f x‖ ∂μ rw [lintegral_coe_eq_integral _ A, ENNReal.toReal_ofReal] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) this : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) a : Set α h'a : ↑↑μ a < ⊤ h''a : IntegrableOn f a ⊢ IntegrableOn (fun y => ↑‖f y - f x‖₊) a simp_rw [coe_nnnorm] α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) this : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) a : Set α h'a : ↑↑μ a < ⊤ h''a : IntegrableOn f a ⊢ IntegrableOn (fun y => ‖f y - f x‖) a exact (h''a.sub (integrableOn_const.2 (Or.inr h'a))).norm case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) this : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) a : Set α h'a : ↑↑μ a < ⊤ h''a : IntegrableOn f a A : IntegrableOn (fun y => ↑‖f y - f x‖₊) a ⊢ (ENNReal.toReal (↑↑μ a))⁻¹ * ∫ (a : α) in a, ↑‖f a - f x‖₊ ∂μ = (ENNReal.toReal (↑↑μ a))⁻¹ • ∫ (y : α) in a, ‖f y - f x‖ ∂μ simp_rw [coe_nnnorm] case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) this : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) a : Set α h'a : ↑↑μ a < ⊤ h''a : IntegrableOn f a A : IntegrableOn (fun y => ↑‖f y - f x‖₊) a ⊢ (ENNReal.toReal (↑↑μ a))⁻¹ * ∫ (a : α) in a, ‖f a - f x‖ ∂μ = (ENNReal.toReal (↑↑μ a))⁻¹ • ∫ (y : α) in a, ‖f y - f x‖ ∂μ rfl case h α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) this : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) a : Set α h'a : ↑↑μ a < ⊤ h''a : IntegrableOn f a A : IntegrableOn (fun y => ↑‖f y - f x‖₊) a ⊢ 0 ≤ ∫ (a : α) in a, ↑‖f a - f x‖₊ ∂μ apply integral_nonneg case h.hf α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f x : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) this : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x‖₊ ∂μ) / ↑↑μ a) (filterAt v x) (𝓝 0) a : Set α h'a : ↑↑μ a < ⊤ h''a : IntegrableOn f a A : IntegrableOn (fun y => ↑‖f y - f x‖₊) a ⊢ 0 ≤ fun a => ↑‖f a - f x‖₊ intro x case h.hf α : Type u_1 inst✝⁵ : MetricSpace α m0 : MeasurableSpace α μ : Measure α v : VitaliFamily μ E : Type u_2 inst✝⁴ : NormedAddCommGroup E inst✝³ : SecondCountableTopology α inst✝² : BorelSpace α inst✝¹ : IsLocallyFiniteMeasure μ ρ : Measure α inst✝ : IsLocallyFiniteMeasure ρ f : α → E hf : LocallyIntegrable f x✝ : α hx : Tendsto (fun a => (∫⁻ (y : α) in a, ↑‖f y - f x✝‖₊ ∂μ) / ↑↑μ a) (filterAt v x✝) (𝓝 0) this : Tendsto (ENNReal.toReal ∘ fun a => (∫⁻ (y : α) in a, ↑‖f y - f x✝‖₊ ∂μ) / ↑↑μ a) (filterAt v x✝) (𝓝 0) a : Set α h'a : ↑↑μ a < ⊤ h''a : IntegrableOn f a A : IntegrableOn (fun y => ↑‖f y - f x✝‖₊) a x : α ⊢ OfNat.ofNat 0 x ≤ (fun a => ↑‖f a - f x✝‖₊) x exact NNReal.coe_nonneg _ Qed
Turing.BlankExtends.refl Γ : Type u_1 inst✝ : Inhabited Γ l : List Γ ⊢ l = l ++ List.replicate 0 default simp ** Qed
Turing.BlankExtends.trans Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ l₃ : List Γ ⊢ BlankExtends l₁ l₂ → BlankExtends l₂ l₃ → BlankExtends l₁ l₃ rintro ⟨i, rfl⟩ ⟨j, rfl⟩ case intro.intro Γ : Type u_1 inst✝ : Inhabited Γ l₁ : List Γ i j : ℕ ⊢ BlankExtends l₁ (l₁ ++ List.replicate i default ++ List.replicate j default) exact ⟨i + j, by simp [List.replicate_add]⟩ Γ : Type u_1 inst✝ : Inhabited Γ l₁ : List Γ i j : ℕ ⊢ l₁ ++ List.replicate i default ++ List.replicate j default = l₁ ++ List.replicate (i + j) default simp [List.replicate_add] ** Qed
Turing.BlankExtends.below_of_le Γ : Type u_1 inst✝ : Inhabited Γ l l₁ l₂ : List Γ ⊢ BlankExtends l l₁ → BlankExtends l l₂ → List.length l₁ ≤ List.length l₂ → BlankExtends l₁ l₂ rintro ⟨i, rfl⟩ ⟨j, rfl⟩ h case intro.intro Γ : Type u_1 inst✝ : Inhabited Γ l : List Γ i j : ℕ h : List.length (l ++ List.replicate i default) ≤ List.length (l ++ List.replicate j default) ⊢ BlankExtends (l ++ List.replicate i default) (l ++ List.replicate j default) use j - i case h Γ : Type u_1 inst✝ : Inhabited Γ l : List Γ i j : ℕ h : List.length (l ++ List.replicate i default) ≤ List.length (l ++ List.replicate j default) ⊢ l ++ List.replicate j default = l ++ List.replicate i default ++ List.replicate (j - i) default simp only [List.length_append, add_le_add_iff_left, List.length_replicate] at h case h Γ : Type u_1 inst✝ : Inhabited Γ l : List Γ i j : ℕ h : i ≤ j ⊢ l ++ List.replicate j default = l ++ List.replicate i default ++ List.replicate (j - i) default simp only [← List.replicate_add, add_tsub_cancel_of_le h, List.append_assoc] ** Qed
Turing.BlankExtends.above_of_le Γ : Type u_1 inst✝ : Inhabited Γ l l₁ l₂ : List Γ ⊢ BlankExtends l₁ l → BlankExtends l₂ l → List.length l₁ ≤ List.length l₂ → BlankExtends l₁ l₂ rintro ⟨i, rfl⟩ ⟨j, e⟩ h case intro.intro Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ : List Γ i j : ℕ e : l₁ ++ List.replicate i default = l₂ ++ List.replicate j default h : List.length l₁ ≤ List.length l₂ ⊢ BlankExtends l₁ l₂ use i - j case h Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ : List Γ i j : ℕ e : l₁ ++ List.replicate i default = l₂ ++ List.replicate j default h : List.length l₁ ≤ List.length l₂ ⊢ l₂ = l₁ ++ List.replicate (i - j) default refine' List.append_right_cancel (e.symm.trans _) case h Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ : List Γ i j : ℕ e : l₁ ++ List.replicate i default = l₂ ++ List.replicate j default h : List.length l₁ ≤ List.length l₂ ⊢ l₁ ++ List.replicate i default = l₁ ++ List.replicate (i - j) default ++ List.replicate j default rw [List.append_assoc, ← List.replicate_add, tsub_add_cancel_of_le] case h Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ : List Γ i j : ℕ e : l₁ ++ List.replicate i default = l₂ ++ List.replicate j default h : List.length l₁ ≤ List.length l₂ ⊢ j ≤ i apply_fun List.length at e case h Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ : List Γ i j : ℕ h : List.length l₁ ≤ List.length l₂ e : List.length (l₁ ++ List.replicate i default) = List.length (l₂ ++ List.replicate j default) ⊢ j ≤ i simp only [List.length_append, List.length_replicate] at e case h Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ : List Γ i j : ℕ h : List.length l₁ ≤ List.length l₂ e : List.length l₁ + i = List.length l₂ + j ⊢ j ≤ i rwa [← add_le_add_iff_left, e, add_le_add_iff_right] ** Qed
Turing.BlankRel.trans Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ l₃ : List Γ ⊢ BlankRel l₁ l₂ → BlankRel l₂ l₃ → BlankRel l₁ l₃ rintro (h₁ \| h₁) (h₂ \| h₂) case inl.inl Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ l₃ : List Γ h₁ : BlankExtends l₁ l₂ h₂ : BlankExtends l₂ l₃ ⊢ BlankRel l₁ l₃ exact Or.inl (h₁.trans h₂) case inl.inr Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ l₃ : List Γ h₁ : BlankExtends l₁ l₂ h₂ : BlankExtends l₃ l₂ ⊢ BlankRel l₁ l₃ cases' le_total l₁.length l₃.length with h h case inl.inr.inl Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ l₃ : List Γ h₁ : BlankExtends l₁ l₂ h₂ : BlankExtends l₃ l₂ h : List.length l₁ ≤ List.length l₃ ⊢ BlankRel l₁ l₃ exact Or.inl (h₁.above_of_le h₂ h) case inl.inr.inr Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ l₃ : List Γ h₁ : BlankExtends l₁ l₂ h₂ : BlankExtends l₃ l₂ h : List.length l₃ ≤ List.length l₁ ⊢ BlankRel l₁ l₃ exact Or.inr (h₂.above_of_le h₁ h) case inr.inl Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ l₃ : List Γ h₁ : BlankExtends l₂ l₁ h₂ : BlankExtends l₂ l₃ ⊢ BlankRel l₁ l₃ cases' le_total l₁.length l₃.length with h h case inr.inl.inl Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ l₃ : List Γ h₁ : BlankExtends l₂ l₁ h₂ : BlankExtends l₂ l₃ h : List.length l₁ ≤ List.length l₃ ⊢ BlankRel l₁ l₃ exact Or.inl (h₁.below_of_le h₂ h) case inr.inl.inr Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ l₃ : List Γ h₁ : BlankExtends l₂ l₁ h₂ : BlankExtends l₂ l₃ h : List.length l₃ ≤ List.length l₁ ⊢ BlankRel l₁ l₃ exact Or.inr (h₂.below_of_le h₁ h) case inr.inr Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ l₃ : List Γ h₁ : BlankExtends l₂ l₁ h₂ : BlankExtends l₃ l₂ ⊢ BlankRel l₁ l₃ exact Or.inr (h₂.trans h₁) ** Qed
Turing.ListBlank.cons_head_tail Γ : Type u_1 inst✝ : Inhabited Γ ⊢ ∀ (l : ListBlank Γ), cons (head l) (tail l) = l apply Quotient.ind' case h Γ : Type u_1 inst✝ : Inhabited Γ ⊢ ∀ (a : List Γ), cons (head (Quotient.mk'' a)) (tail (Quotient.mk'' a)) = Quotient.mk'' a refine' fun l ↦ Quotient.sound' (Or.inr _) case h Γ : Type u_1 inst✝ : Inhabited Γ l : List Γ ⊢ BlankExtends l (head (Quotient.mk'' l) :: List.tail l) cases l case h.nil Γ : Type u_1 inst✝ : Inhabited Γ ⊢ BlankExtends [] (head (Quotient.mk'' []) :: List.tail []) exact ⟨1, rfl⟩ case h.cons Γ : Type u_1 inst✝ : Inhabited Γ head✝ : Γ tail✝ : List Γ ⊢ BlankExtends (head✝ :: tail✝) (head (Quotient.mk'' (head✝ :: tail✝)) :: List.tail (head✝ :: tail✝)) rfl ** Qed
Turing.ListBlank.nth_zero Γ : Type u_1 inst✝ : Inhabited Γ l : ListBlank Γ ⊢ nth l 0 = head l conv => lhs; rw [← ListBlank.cons_head_tail l] Γ : Type u_1 inst✝ : Inhabited Γ l : ListBlank Γ ⊢ nth (cons (head l) (tail l)) 0 = head l exact Quotient.inductionOn' l.tail fun l ↦ rfl ** Qed
Turing.ListBlank.nth_succ Γ : Type u_1 inst✝ : Inhabited Γ l : ListBlank Γ n : ℕ ⊢ nth l (n + 1) = nth (tail l) n conv => lhs; rw [← ListBlank.cons_head_tail l] Γ : Type u_1 inst✝ : Inhabited Γ l : ListBlank Γ n : ℕ ⊢ nth (cons (head l) (tail l)) (n + 1) = nth (tail l) n exact Quotient.inductionOn' l.tail fun l ↦ rfl ** Qed
Turing.ListBlank.ext Γ : Type u_1 i : Inhabited Γ L₁ L₂ : ListBlank Γ ⊢ (∀ (i_1 : ℕ), nth L₁ i_1 = nth L₂ i_1) → L₁ = L₂ refine' ListBlank.induction_on L₁ fun l₁ ↦ ListBlank.induction_on L₂ fun l₂ H ↦ _ Γ : Type u_1 i : Inhabited Γ L₁ L₂ : ListBlank Γ l₁ l₂ : List Γ H : ∀ (i_1 : ℕ), nth (mk l₁) i_1 = nth (mk l₂) i_1 ⊢ mk l₁ = mk l₂ wlog h : l₁.length ≤ l₂.length Γ : Type u_1 i : Inhabited Γ L₁ L₂ : ListBlank Γ l₁ l₂ : List Γ H : ∀ (i_1 : ℕ), nth (mk l₁) i_1 = nth (mk l₂) i_1 h : List.length l₁ ≤ List.length l₂ ⊢ mk l₁ = mk l₂ refine' Quotient.sound' (Or.inl ⟨l₂.length - l₁.length, _⟩) Γ : Type u_1 i : Inhabited Γ L₁ L₂ : ListBlank Γ l₁ l₂ : List Γ H : ∀ (i_1 : ℕ), nth (mk l₁) i_1 = nth (mk l₂) i_1 h : List.length l₁ ≤ List.length l₂ ⊢ l₂ = l₁ ++ List.replicate (List.length l₂ - List.length l₁) default refine' List.ext_get _ fun i h h₂ ↦ Eq.symm _ case refine'_2 Γ : Type u_1 i✝ : Inhabited Γ L₁ L₂ : ListBlank Γ l₁ l₂ : List Γ H : ∀ (i : ℕ), nth (mk l₁) i = nth (mk l₂) i h✝ : List.length l₁ ≤ List.length l₂ i : ℕ h : i < List.length l₂ h₂ : i < List.length (l₁ ++ List.replicate (List.length l₂ - List.length l₁) default) ⊢ List.get (l₁ ++ List.replicate (List.length l₂ - List.length l₁) default) { val := i, isLt := h₂ } = List.get l₂ { val := i, isLt := h } simp only [ListBlank.nth_mk] at H case refine'_2 Γ : Type u_1 i✝ : Inhabited Γ L₁ L₂ : ListBlank Γ l₁ l₂ : List Γ H : ∀ (i : ℕ), List.getI l₁ i = List.getI l₂ i h✝ : List.length l₁ ≤ List.length l₂ i : ℕ h : i < List.length l₂ h₂ : i < List.length (l₁ ++ List.replicate (List.length l₂ - List.length l₁) default) ⊢ List.get (l₁ ++ List.replicate (List.length l₂ - List.length l₁) default) { val := i, isLt := h₂ } = List.get l₂ { val := i, isLt := h } cases' lt_or_le i l₁.length with h' h' case inr Γ : Type u_1 i : Inhabited Γ L₁ L₂ : ListBlank Γ l₁ l₂ : List Γ H : ∀ (i_1 : ℕ), nth (mk l₁) i_1 = nth (mk l₂) i_1 this : ∀ {Γ : Type u_1} [i : Inhabited Γ] {L₁ L₂ : ListBlank Γ} (l₁ l₂ : List Γ), (∀ (i_1 : ℕ), nth (mk l₁) i_1 = nth (mk l₂) i_1) → List.length l₁ ≤ List.length l₂ → mk l₁ = mk l₂ h : ¬List.length l₁ ≤ List.length l₂ ⊢ mk l₁ = mk l₂ cases le_total l₁.length l₂.length <;> [skip; symm] <;> apply this <;> try assumption case inr.inr.H Γ : Type u_1 i : Inhabited Γ L₁ L₂ : ListBlank Γ l₁ l₂ : List Γ H : ∀ (i_1 : ℕ), nth (mk l₁) i_1 = nth (mk l₂) i_1 this : ∀ {Γ : Type u_1} [i : Inhabited Γ] {L₁ L₂ : ListBlank Γ} (l₁ l₂ : List Γ), (∀ (i_1 : ℕ), nth (mk l₁) i_1 = nth (mk l₂) i_1) → List.length l₁ ≤ List.length l₂ → mk l₁ = mk l₂ h : ¬List.length l₁ ≤ List.length l₂ h✝ : List.length l₂ ≤ List.length l₁ ⊢ ∀ (i_1 : ℕ), nth (mk l₂) i_1 = nth (mk l₁) i_1 intro case inr.inr.H Γ : Type u_1 i : Inhabited Γ L₁ L₂ : ListBlank Γ l₁ l₂ : List Γ H : ∀ (i_1 : ℕ), nth (mk l₁) i_1 = nth (mk l₂) i_1 this : ∀ {Γ : Type u_1} [i : Inhabited Γ] {L₁ L₂ : ListBlank Γ} (l₁ l₂ : List Γ), (∀ (i_1 : ℕ), nth (mk l₁) i_1 = nth (mk l₂) i_1) → List.length l₁ ≤ List.length l₂ → mk l₁ = mk l₂ h : ¬List.length l₁ ≤ List.length l₂ h✝ : List.length l₂ ≤ List.length l₁ i✝ : ℕ ⊢ nth (mk l₂) i✝ = nth (mk l₁) i✝ rw [H] case inr.inr.h Γ : Type u_1 i : Inhabited Γ L₁ L₂ : ListBlank Γ l₁ l₂ : List Γ H : ∀ (i_1 : ℕ), nth (mk l₁) i_1 = nth (mk l₂) i_1 this : ∀ {Γ : Type u_1} [i : Inhabited Γ] {L₁ L₂ : ListBlank Γ} (l₁ l₂ : List Γ), (∀ (i_1 : ℕ), nth (mk l₁) i_1 = nth (mk l₂) i_1) → List.length l₁ ≤ List.length l₂ → mk l₁ = mk l₂ h : ¬List.length l₁ ≤ List.length l₂ h✝ : List.length l₂ ≤ List.length l₁ ⊢ List.length l₂ ≤ List.length l₁ assumption case refine'_1 Γ : Type u_1 i : Inhabited Γ L₁ L₂ : ListBlank Γ l₁ l₂ : List Γ H : ∀ (i_1 : ℕ), nth (mk l₁) i_1 = nth (mk l₂) i_1 h : List.length l₁ ≤ List.length l₂ ⊢ List.length l₂ = List.length (l₁ ++ List.replicate (List.length l₂ - List.length l₁) default) simp only [add_tsub_cancel_of_le h, List.length_append, List.length_replicate] case refine'_2.inl Γ : Type u_1 i✝ : Inhabited Γ L₁ L₂ : ListBlank Γ l₁ l₂ : List Γ H : ∀ (i : ℕ), List.getI l₁ i = List.getI l₂ i h✝ : List.length l₁ ≤ List.length l₂ i : ℕ h : i < List.length l₂ h₂ : i < List.length (l₁ ++ List.replicate (List.length l₂ - List.length l₁) default) h' : i < List.length l₁ ⊢ List.get (l₁ ++ List.replicate (List.length l₂ - List.length l₁) default) { val := i, isLt := h₂ } = List.get l₂ { val := i, isLt := h } simp only [List.get_append _ h', List.get?_eq_get h, List.get?_eq_get h', ← List.getI_eq_get _ h, ← List.getI_eq_get _ h', H] case refine'_2.inr Γ : Type u_1 i✝ : Inhabited Γ L₁ L₂ : ListBlank Γ l₁ l₂ : List Γ H : ∀ (i : ℕ), List.getI l₁ i = List.getI l₂ i h✝ : List.length l₁ ≤ List.length l₂ i : ℕ h : i < List.length l₂ h₂ : i < List.length (l₁ ++ List.replicate (List.length l₂ - List.length l₁) default) h' : List.length l₁ ≤ i ⊢ List.get (l₁ ++ List.replicate (List.length l₂ - List.length l₁) default) { val := i, isLt := h₂ } = List.get l₂ { val := i, isLt := h } simp only [List.get_append_right' h', List.get_replicate, List.get?_eq_get h, List.get?_len_le h', ← List.getI_eq_default _ h', H, List.getI_eq_get _ h] ** Qed
Turing.ListBlank.nth_modifyNth Γ : Type u_1 inst✝ : Inhabited Γ f : Γ → Γ n i : ℕ L : ListBlank Γ ⊢ nth (modifyNth f n L) i = if i = n then f (nth L i) else nth L i induction' n with n IH generalizing i L case zero Γ : Type u_1 inst✝ : Inhabited Γ f : Γ → Γ i✝ : ℕ L✝ : ListBlank Γ i : ℕ L : ListBlank Γ ⊢ nth (modifyNth f Nat.zero L) i = if i = Nat.zero then f (nth L i) else nth L i cases i <;> simp only [ListBlank.nth_zero, if_true, ListBlank.head_cons, ListBlank.modifyNth, ListBlank.nth_succ, if_false, ListBlank.tail_cons, Nat.zero_eq] case succ Γ : Type u_1 inst✝ : Inhabited Γ f : Γ → Γ i✝ : ℕ L✝ : ListBlank Γ n : ℕ IH : ∀ (i : ℕ) (L : ListBlank Γ), nth (modifyNth f n L) i = if i = n then f (nth L i) else nth L i i : ℕ L : ListBlank Γ ⊢ nth (modifyNth f (Nat.succ n) L) i = if i = Nat.succ n then f (nth L i) else nth L i cases i case succ.zero Γ : Type u_1 inst✝ : Inhabited Γ f : Γ → Γ i : ℕ L✝ : ListBlank Γ n : ℕ IH : ∀ (i : ℕ) (L : ListBlank Γ), nth (modifyNth f n L) i = if i = n then f (nth L i) else nth L i L : ListBlank Γ ⊢ nth (modifyNth f (Nat.succ n) L) Nat.zero = if Nat.zero = Nat.succ n then f (nth L Nat.zero) else nth L Nat.zero rw [if_neg (Nat.succ_ne_zero _).symm] case succ.zero Γ : Type u_1 inst✝ : Inhabited Γ f : Γ → Γ i : ℕ L✝ : ListBlank Γ n : ℕ IH : ∀ (i : ℕ) (L : ListBlank Γ), nth (modifyNth f n L) i = if i = n then f (nth L i) else nth L i L : ListBlank Γ ⊢ nth (modifyNth f (Nat.succ n) L) Nat.zero = nth L Nat.zero simp only [ListBlank.nth_zero, ListBlank.head_cons, ListBlank.modifyNth, Nat.zero_eq] case succ.succ Γ : Type u_1 inst✝ : Inhabited Γ f : Γ → Γ i : ℕ L✝ : ListBlank Γ n : ℕ IH : ∀ (i : ℕ) (L : ListBlank Γ), nth (modifyNth f n L) i = if i = n then f (nth L i) else nth L i L : ListBlank Γ n✝ : ℕ ⊢ nth (modifyNth f (Nat.succ n) L) (Nat.succ n✝) = if Nat.succ n✝ = Nat.succ n then f (nth L (Nat.succ n✝)) else nth L (Nat.succ n✝) simp only [IH, ListBlank.modifyNth, ListBlank.nth_succ, ListBlank.tail_cons, Nat.succ.injEq] ** Qed
Turing.ListBlank.head_map Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' l : ListBlank Γ ⊢ head (map f l) = PointedMap.f f (head l) conv => lhs; rw [← ListBlank.cons_head_tail l] Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' l : ListBlank Γ ⊢ head (map f (cons (head l) (tail l))) = PointedMap.f f (head l) exact Quotient.inductionOn' l fun a ↦ rfl ** Qed
Turing.ListBlank.tail_map Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' l : ListBlank Γ ⊢ tail (map f l) = map f (tail l) conv => lhs; rw [← ListBlank.cons_head_tail l] Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' l : ListBlank Γ ⊢ tail (map f (cons (head l) (tail l))) = map f (tail l) exact Quotient.inductionOn' l fun a ↦ rfl ** Qed
Turing.ListBlank.map_cons Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' l : ListBlank Γ a : Γ ⊢ map f (cons a l) = cons (PointedMap.f f a) (map f l) refine' (ListBlank.cons_head_tail _).symm.trans _ Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' l : ListBlank Γ a : Γ ⊢ cons (head (map f (cons a l))) (tail (map f (cons a l))) = cons (PointedMap.f f a) (map f l) simp only [ListBlank.head_map, ListBlank.head_cons, ListBlank.tail_map, ListBlank.tail_cons] ** Qed
Turing.ListBlank.nth_map Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' l : ListBlank Γ n : ℕ ⊢ nth (map f l) n = PointedMap.f f (nth l n) refine' l.inductionOn fun l ↦ _ Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' l✝ : ListBlank Γ n : ℕ l : List Γ ⊢ nth (map f (Quotient.mk (BlankRel.setoid Γ) l)) n = PointedMap.f f (nth (Quotient.mk (BlankRel.setoid Γ) l) n) suffices ((mk l).map f).nth n = f ((mk l).nth n) by exact this Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' l✝ : ListBlank Γ n : ℕ l : List Γ ⊢ nth (map f (mk l)) n = PointedMap.f f (nth (mk l) n) simp only [List.get?_map, ListBlank.map_mk, ListBlank.nth_mk, List.getI_eq_iget_get?] Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' l✝ : ListBlank Γ n : ℕ l : List Γ ⊢ Option.iget (Option.map f.f (List.get? l n)) = PointedMap.f f (Option.iget (List.get? l n)) cases l.get? n Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' l✝ : ListBlank Γ n : ℕ l : List Γ this : nth (map f (mk l)) n = PointedMap.f f (nth (mk l) n) ⊢ nth (map f (Quotient.mk (BlankRel.setoid Γ) l)) n = PointedMap.f f (nth (Quotient.mk (BlankRel.setoid Γ) l) n) exact this case none Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' l✝ : ListBlank Γ n : ℕ l : List Γ ⊢ Option.iget (Option.map f.f none) = PointedMap.f f (Option.iget none) exact f.2.symm case some Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' l✝ : ListBlank Γ n : ℕ l : List Γ val✝ : Γ ⊢ Option.iget (Option.map f.f (some val✝)) = PointedMap.f f (Option.iget (some val✝)) rfl ** Qed
Turing.proj_map_nth ι : Type u_1 Γ : ι → Type u_2 inst✝ : (i : ι) → Inhabited (Γ i) i : ι L : ListBlank ((i : ι) → Γ i) n : ℕ ⊢ ListBlank.nth (ListBlank.map (proj i) L) n = ListBlank.nth L n i rw [ListBlank.nth_map] ι : Type u_1 Γ : ι → Type u_2 inst✝ : (i : ι) → Inhabited (Γ i) i : ι L : ListBlank ((i : ι) → Γ i) n : ℕ ⊢ PointedMap.f (proj i) (ListBlank.nth L n) = ListBlank.nth L n i rfl ** Qed
Turing.ListBlank.map_modifyNth Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' F : PointedMap Γ Γ' f : Γ → Γ f' : Γ' → Γ' H : ∀ (x : Γ), PointedMap.f F (f x) = f' (PointedMap.f F x) n : ℕ L : ListBlank Γ ⊢ map F (modifyNth f n L) = modifyNth f' n (map F L) induction' n with n IH generalizing L <;> simp only [, ListBlank.head_map, ListBlank.modifyNth, ListBlank.map_cons, ListBlank.tail_map] * Qed
Turing.ListBlank.append_mk Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ : List Γ ⊢ append l₁ (mk l₂) = mk (l₁ ++ l₂) induction l₁ <;> simp only [, ListBlank.append, List.nil_append, List.cons_append, ListBlank.cons_mk] * Qed
Turing.ListBlank.append_assoc Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ : List Γ l₃ : ListBlank Γ ⊢ append (l₁ ++ l₂) l₃ = append l₁ (append l₂ l₃) refine' l₃.inductionOn fun l ↦ _ Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ : List Γ l₃ : ListBlank Γ l : List Γ ⊢ append (l₁ ++ l₂) (Quotient.mk (BlankRel.setoid Γ) l) = append l₁ (append l₂ (Quotient.mk (BlankRel.setoid Γ) l)) suffices append (l₁ ++ l₂) (mk l) = append l₁ (append l₂ (mk l)) by exact this Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ : List Γ l₃ : ListBlank Γ l : List Γ ⊢ append (l₁ ++ l₂) (mk l) = append l₁ (append l₂ (mk l)) simp only [ListBlank.append_mk, List.append_assoc] Γ : Type u_1 inst✝ : Inhabited Γ l₁ l₂ : List Γ l₃ : ListBlank Γ l : List Γ this : append (l₁ ++ l₂) (mk l) = append l₁ (append l₂ (mk l)) ⊢ append (l₁ ++ l₂) (Quotient.mk (BlankRel.setoid Γ) l) = append l₁ (append l₂ (Quotient.mk (BlankRel.setoid Γ) l)) exact this ** Qed
Turing.Tape.move_left_right Γ : Type u_1 inst✝ : Inhabited Γ T : Tape Γ ⊢ move Dir.right (move Dir.left T) = T cases T case mk Γ : Type u_1 inst✝ : Inhabited Γ head✝ : Γ left✝ right✝ : ListBlank Γ ⊢ move Dir.right (move Dir.left { head := head✝, left := left✝, right := right✝ }) = { head := head✝, left := left✝, right := right✝ } simp [Tape.move] ** Qed
Turing.Tape.move_right_left Γ : Type u_1 inst✝ : Inhabited Γ T : Tape Γ ⊢ move Dir.left (move Dir.right T) = T cases T case mk Γ : Type u_1 inst✝ : Inhabited Γ head✝ : Γ left✝ right✝ : ListBlank Γ ⊢ move Dir.left (move Dir.right { head := head✝, left := left✝, right := right✝ }) = { head := head✝, left := left✝, right := right✝ } simp [Tape.move] ** Qed
Turing.Tape.mk'_left_right₀ Γ : Type u_1 inst✝ : Inhabited Γ T : Tape Γ ⊢ mk' T.left (right₀ T) = T cases T case mk Γ : Type u_1 inst✝ : Inhabited Γ head✝ : Γ left✝ right✝ : ListBlank Γ ⊢ mk' { head := head✝, left := left✝, right := right✝ }.left (right₀ { head := head✝, left := left✝, right := right✝ }) = { head := head✝, left := left✝, right := right✝ } simp only [Tape.right₀, Tape.mk', ListBlank.head_cons, ListBlank.tail_cons, eq_self_iff_true, and_self_iff] ** Qed
Turing.Tape.move_left_mk' Γ : Type u_1 inst✝ : Inhabited Γ L R : ListBlank Γ ⊢ move Dir.left (mk' L R) = mk' (ListBlank.tail L) (ListBlank.cons (ListBlank.head L) R) simp only [Tape.move, Tape.mk', ListBlank.head_cons, eq_self_iff_true, ListBlank.cons_head_tail, and_self_iff, ListBlank.tail_cons] ** Qed
Turing.Tape.move_right_mk' Γ : Type u_1 inst✝ : Inhabited Γ L R : ListBlank Γ ⊢ move Dir.right (mk' L R) = mk' (ListBlank.cons (ListBlank.head R) L) (ListBlank.tail R) simp only [Tape.move, Tape.mk', ListBlank.head_cons, eq_self_iff_true, ListBlank.cons_head_tail, and_self_iff, ListBlank.tail_cons] ** Qed
Turing.Tape.right₀_nth Γ : Type u_1 inst✝ : Inhabited Γ T : Tape Γ n : ℕ ⊢ ListBlank.nth (right₀ T) n = nth T ↑n cases n <;> simp only [Tape.nth, Tape.right₀, Int.ofNat_zero, ListBlank.nth_zero, ListBlank.nth_succ, ListBlank.head_cons, ListBlank.tail_cons, Nat.zero_eq] ** Qed
Turing.Tape.move_left_nth Γ : Type u_1 inst✝ : Inhabited Γ a : Γ L R : ListBlank Γ n : ℕ ⊢ nth (move Dir.left { head := a, left := L, right := R }) (↑(n + 1) + 1) = nth { head := a, left := L, right := R } (↑(n + 1) + 1 - 1) rw [add_sub_cancel] Γ : Type u_1 inst✝ : Inhabited Γ a : Γ L R : ListBlank Γ n : ℕ ⊢ nth (move Dir.left { head := a, left := L, right := R }) (↑(n + 1) + 1) = nth { head := a, left := L, right := R } ↑(n + 1) change (R.cons a).nth (n + 1) = R.nth n Γ : Type u_1 inst✝ : Inhabited Γ a : Γ L R : ListBlank Γ n : ℕ ⊢ ListBlank.nth (ListBlank.cons a R) (n + 1) = ListBlank.nth R n rw [ListBlank.nth_succ, ListBlank.tail_cons] ** Qed
Turing.Tape.write_self Γ : Type u_1 inst✝ : Inhabited Γ ⊢ ∀ (T : Tape Γ), write T.head T = T rintro ⟨⟩ case mk Γ : Type u_1 inst✝ : Inhabited Γ head✝ : Γ left✝ right✝ : ListBlank Γ ⊢ write { head := head✝, left := left✝, right := right✝ }.head { head := head✝, left := left✝, right := right✝ } = { head := head✝, left := left✝, right := right✝ } rfl ** Qed
Turing.Tape.write_mk' Γ : Type u_1 inst✝ : Inhabited Γ a b : Γ L R : ListBlank Γ ⊢ write b (mk' L (ListBlank.cons a R)) = mk' L (ListBlank.cons b R) simp only [Tape.write, Tape.mk', ListBlank.head_cons, ListBlank.tail_cons, eq_self_iff_true, and_self_iff] ** Qed
Turing.Tape.map_fst Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' ⊢ ∀ (T : Tape Γ), (map f T).head = PointedMap.f f T.head rintro ⟨⟩ case mk Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' head✝ : Γ left✝ right✝ : ListBlank Γ ⊢ (map f { head := head✝, left := left✝, right := right✝ }).head = PointedMap.f f { head := head✝, left := left✝, right := right✝ }.head rfl ** Qed
Turing.Tape.map_write Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' b : Γ ⊢ ∀ (T : Tape Γ), map f (write b T) = write (PointedMap.f f b) (map f T) rintro ⟨⟩ case mk Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' b head✝ : Γ left✝ right✝ : ListBlank Γ ⊢ map f (write b { head := head✝, left := left✝, right := right✝ }) = write (PointedMap.f f b) (map f { head := head✝, left := left✝, right := right✝ }) rfl ** Qed
Turing.Tape.write_move_right_n Γ : Type u_1 inst✝ : Inhabited Γ f : Γ → Γ L R : ListBlank Γ n : ℕ ⊢ write (f (ListBlank.nth R n)) ((move Dir.right)^[n] (mk' L R)) = (move Dir.right)^[n] (mk' L (ListBlank.modifyNth f n R)) induction' n with n IH generalizing L R case succ Γ : Type u_1 inst✝ : Inhabited Γ f : Γ → Γ L✝ R✝ : ListBlank Γ n : ℕ IH : ∀ (L R : ListBlank Γ), write (f (ListBlank.nth R n)) ((move Dir.right)^[n] (mk' L R)) = (move Dir.right)^[n] (mk' L (ListBlank.modifyNth f n R)) L R : ListBlank Γ ⊢ write (f (ListBlank.nth R (Nat.succ n))) ((move Dir.right)^[Nat.succ n] (mk' L R)) = (move Dir.right)^[Nat.succ n] (mk' L (ListBlank.modifyNth f (Nat.succ n) R)) simp only [ListBlank.head_cons, ListBlank.nth_succ, ListBlank.modifyNth, Tape.move_right_mk', ListBlank.tail_cons, iterate_succ_apply, IH] case zero Γ : Type u_1 inst✝ : Inhabited Γ f : Γ → Γ L✝ R✝ L R : ListBlank Γ ⊢ write (f (ListBlank.nth R Nat.zero)) ((move Dir.right)^[Nat.zero] (mk' L R)) = (move Dir.right)^[Nat.zero] (mk' L (ListBlank.modifyNth f Nat.zero R)) simp only [ListBlank.nth_zero, ListBlank.modifyNth, iterate_zero_apply, Nat.zero_eq] case zero Γ : Type u_1 inst✝ : Inhabited Γ f : Γ → Γ L✝ R✝ L R : ListBlank Γ ⊢ write (f (ListBlank.head R)) (mk' L R) = mk' L (ListBlank.cons (f (ListBlank.head R)) (ListBlank.tail R)) rw [← Tape.write_mk', ListBlank.cons_head_tail] ** Qed
Turing.Tape.map_move Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' T : Tape Γ d : Dir ⊢ map f (move d T) = move d (map f T) cases T case mk Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' d : Dir head✝ : Γ left✝ right✝ : ListBlank Γ ⊢ map f (move d { head := head✝, left := left✝, right := right✝ }) = move d (map f { head := head✝, left := left✝, right := right✝ }) cases d <;> simp only [Tape.move, Tape.map, ListBlank.head_map, eq_self_iff_true, ListBlank.map_cons, and_self_iff, ListBlank.tail_map] ** Qed
Turing.Tape.map_mk' Γ : Type u_1 Γ' : Type u_2 inst✝¹ : Inhabited Γ inst✝ : Inhabited Γ' f : PointedMap Γ Γ' L R : ListBlank Γ ⊢ map f (mk' L R) = mk' (ListBlank.map f L) (ListBlank.map f R) simp only [Tape.mk', Tape.map, ListBlank.head_map, eq_self_iff_true, and_self_iff, ListBlank.tail_map] ** Qed
Turing.reaches₁_eq σ : Type u_1 f : σ → Option σ a b c : σ h : f a = f b ⊢ (∃ b_1, b_1 ∈ f b ∧ ReflTransGen (fun a b => b ∈ f a) b_1 c) ↔ ∃ b, b ∈ f a ∧ ReflTransGen (fun a b => b ∈ f a) b c rw [h] ** Qed
Turing.reaches₁_fwd σ : Type u_1 f : σ → Option σ a b c : σ h₁ : Reaches₁ f a c h₂ : b ∈ f a ⊢ Reaches f b c rcases TransGen.head'_iff.1 h₁ with ⟨b', hab, hbc⟩ case intro.intro σ : Type u_1 f : σ → Option σ a b c : σ h₁ : Reaches₁ f a c h₂ : b ∈ f a b' : σ hab : b' ∈ f a hbc : ReflTransGen (fun a b => b ∈ f a) b' c ⊢ Reaches f b c cases Option.mem_unique hab h₂ case intro.intro.refl σ : Type u_1 f : σ → Option σ a b c : σ h₁ : Reaches₁ f a c h₂ hab : b ∈ f a hbc : ReflTransGen (fun a b => b ∈ f a) b c ⊢ Reaches f b c exact hbc ** Qed
Turing.mem_eval σ : Type u_1 f : σ → Option σ a b : σ ⊢ b ∈ eval f a ↔ Reaches f a b ∧ f b = none refine' ⟨fun h ↦ _, fun ⟨h₁, h₂⟩ ↦ _⟩ case refine'_1 σ : Type u_1 f : σ → Option σ a b : σ h : b ∈ eval f a ⊢ Reaches f a b ∧ f b = none refine' @evalInduction _ _ _ (fun a ↦ Reaches f a b ∧ f b = none) _ h fun a h IH ↦ _ case refine'_1 σ : Type u_1 f : σ → Option σ a✝ b : σ h✝ : b ∈ eval f a✝ a : σ h : b ∈ eval f a IH : ∀ (a' : σ), f a = some a' → (fun a => Reaches f a b ∧ f b = none) a' ⊢ (fun a => Reaches f a b ∧ f b = none) a cases' e : f a with a' case refine'_1.none σ : Type u_1 f : σ → Option σ a✝ b : σ h✝ : b ∈ eval f a✝ a : σ h : b ∈ eval f a IH : ∀ (a' : σ), f a = some a' → (fun a => Reaches f a b ∧ f b = none) a' e : f a = none ⊢ Reaches f a b ∧ f b = none rw [Part.mem_unique h (PFun.mem_fix_iff.2 <\| Or.inl <\| Part.mem_some_iff.2 <\| by rw [e] <;> rfl)] case refine'_1.none σ : Type u_1 f : σ → Option σ a✝ b : σ h✝ : b ∈ eval f a✝ a : σ h : b ∈ eval f a IH : ∀ (a' : σ), f a = some a' → (fun a => Reaches f a b ∧ f b = none) a' e : f a = none ⊢ Reaches f a a ∧ f a = none exact ⟨ReflTransGen.refl, e⟩ σ : Type u_1 f : σ → Option σ a✝ b : σ h✝ : b ∈ eval f a✝ a : σ h : b ∈ eval f a IH : ∀ (a' : σ), f a = some a' → (fun a => Reaches f a b ∧ f b = none) a' e : f a = none ⊢ Sum.inl ?m.63851 = Option.elim (f a) (Sum.inl a) Sum.inr rw [e] <;> rfl case refine'_1.some σ : Type u_1 f : σ → Option σ a✝ b : σ h✝ : b ∈ eval f a✝ a : σ h : b ∈ eval f a IH : ∀ (a' : σ), f a = some a' → (fun a => Reaches f a b ∧ f b = none) a' a' : σ e : f a = some a' ⊢ Reaches f a b ∧ f b = none rcases PFun.mem_fix_iff.1 h with (h \| ⟨_, h, _⟩) <;> rw [e] at h <;> cases Part.mem_some_iff.1 h case refine'_1.some.inr.intro.intro.refl σ : Type u_1 f : σ → Option σ a✝ b : σ h✝¹ : b ∈ eval f a✝ a : σ h✝ : b ∈ eval f a IH : ∀ (a' : σ), f a = some a' → (fun a => Reaches f a b ∧ f b = none) a' a' : σ e : f a = some a' h : Sum.inr a' ∈ Part.some (Option.elim (some a') (Sum.inl a) Sum.inr) right✝ : b ∈ PFun.fix (fun s => Part.some (Option.elim (f s) (Sum.inl s) Sum.inr)) a' ⊢ Reaches f a b ∧ f b = none cases' IH a' e with h₁ h₂ case refine'_1.some.inr.intro.intro.refl.intro σ : Type u_1 f : σ → Option σ a✝ b : σ h✝¹ : b ∈ eval f a✝ a : σ h✝ : b ∈ eval f a IH : ∀ (a' : σ), f a = some a' → (fun a => Reaches f a b ∧ f b = none) a' a' : σ e : f a = some a' h : Sum.inr a' ∈ Part.some (Option.elim (some a') (Sum.inl a) Sum.inr) right✝ : b ∈ PFun.fix (fun s => Part.some (Option.elim (f s) (Sum.inl s) Sum.inr)) a' h₁ : Reaches f a' b h₂ : f b = none ⊢ Reaches f a b ∧ f b = none exact ⟨ReflTransGen.head e h₁, h₂⟩ case refine'_2 σ : Type u_1 f : σ → Option σ a b : σ x✝ : Reaches f a b ∧ f b = none h₁ : Reaches f a b h₂ : f b = none ⊢ b ∈ eval f a refine' ReflTransGen.head_induction_on h₁ _ fun h _ IH ↦ _ case refine'_2.refine'_1 σ : Type u_1 f : σ → Option σ a b : σ x✝ : Reaches f a b ∧ f b = none h₁ : Reaches f a b h₂ : f b = none ⊢ b ∈ eval f b refine' PFun.mem_fix_iff.2 (Or.inl _) case refine'_2.refine'_1 σ : Type u_1 f : σ → Option σ a b : σ x✝ : Reaches f a b ∧ f b = none h₁ : Reaches f a b h₂ : f b = none ⊢ Sum.inl b ∈ Part.some (Option.elim (f b) (Sum.inl b) Sum.inr) rw [h₂] case refine'_2.refine'_1 σ : Type u_1 f : σ → Option σ a b : σ x✝ : Reaches f a b ∧ f b = none h₁ : Reaches f a b h₂ : f b = none ⊢ Sum.inl b ∈ Part.some (Option.elim none (Sum.inl b) Sum.inr) apply Part.mem_some case refine'_2.refine'_2 σ : Type u_1 f : σ → Option σ a b : σ x✝¹ : Reaches f a b ∧ f b = none h₁ : Reaches f a b h₂ : f b = none a✝ c✝ : σ h : c✝ ∈ f a✝ x✝ : ReflTransGen (fun a b => b ∈ f a) c✝ b IH : b ∈ eval f c✝ ⊢ b ∈ eval f a✝ refine' PFun.mem_fix_iff.2 (Or.inr ⟨_, _, IH⟩) case refine'_2.refine'_2 σ : Type u_1 f : σ → Option σ a b : σ x✝¹ : Reaches f a b ∧ f b = none h₁ : Reaches f a b h₂ : f b = none a✝ c✝ : σ h : c✝ ∈ f a✝ x✝ : ReflTransGen (fun a b => b ∈ f a) c✝ b IH : b ∈ eval f c✝ ⊢ Sum.inr c✝ ∈ Part.some (Option.elim (f a✝) (Sum.inl a✝) Sum.inr) rw [h] case refine'_2.refine'_2 σ : Type u_1 f : σ → Option σ a b : σ x✝¹ : Reaches f a b ∧ f b = none h₁ : Reaches f a b h₂ : f b = none a✝ c✝ : σ h : c✝ ∈ f a✝ x✝ : ReflTransGen (fun a b => b ∈ f a) c✝ b IH : b ∈ eval f c✝ ⊢ Sum.inr c✝ ∈ Part.some (Option.elim (some c✝) (Sum.inl a✝) Sum.inr) apply Part.mem_some ** Qed
Turing.eval_maximal₁ σ : Type u_1 f : σ → Option σ a b : σ h : b ∈ eval f a c : σ x✝ : Reaches₁ f b c bc : Reaches₁ f b c := x✝ ⊢ False let ⟨_, b0⟩ := mem_eval.1 h σ : Type u_1 f : σ → Option σ a b : σ h : b ∈ eval f a c : σ x✝ : Reaches₁ f b c bc : Reaches₁ f b c := x✝ left✝ : Reaches f a b b0 : f b = none ⊢ False let ⟨b', h', _⟩ := TransGen.head'_iff.1 bc σ : Type u_1 f : σ → Option σ a b : σ h : b ∈ eval f a c : σ x✝ : Reaches₁ f b c bc : Reaches₁ f b c := x✝ left✝ : Reaches f a b b0 : f b = none b' : σ h' : b' ∈ f b right✝ : ReflTransGen (fun a b => b ∈ f a) b' c ⊢ False cases b0.symm.trans h' ** Qed
Turing.eval_maximal σ : Type u_1 f : σ → Option σ a b : σ h : b ∈ eval f a c : σ left✝ : Reaches f a b b0 : f b = none b' : σ h' : b' ∈ f b ⊢ False cases b0.symm.trans h' ** Qed
Turing.tr_reaches₁ σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₁ : σ₁ ab : Reaches₁ f₁ a₁ b₁ ⊢ ∃ b₂, tr b₁ b₂ ∧ Reaches₁ f₂ a₂ b₂ induction' ab with c₁ ac c₁ d₁ _ cd IH case single σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₁ c₁ : σ₁ ac : c₁ ∈ f₁ a₁ ⊢ ∃ b₂, tr c₁ b₂ ∧ Reaches₁ f₂ a₂ b₂ have := H aa case single σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₁ c₁ : σ₁ ac : c₁ ∈ f₁ a₁ this : match f₁ a₁ with \| some b₁ => ∃ b₂, tr b₁ b₂ ∧ Reaches₁ f₂ a₂ b₂ \| none => f₂ a₂ = none ⊢ ∃ b₂, tr c₁ b₂ ∧ Reaches₁ f₂ a₂ b₂ rwa [show f₁ a₁ = _ from ac] at this case tail σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₁ c₁ d₁ : σ₁ a✝ : TransGen (fun a b => b ∈ f₁ a) a₁ c₁ cd : d₁ ∈ f₁ c₁ IH : ∃ b₂, tr c₁ b₂ ∧ Reaches₁ f₂ a₂ b₂ ⊢ ∃ b₂, tr d₁ b₂ ∧ Reaches₁ f₂ a₂ b₂ rcases IH with ⟨c₂, cc, ac₂⟩ case tail.intro.intro σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₁ c₁ d₁ : σ₁ a✝ : TransGen (fun a b => b ∈ f₁ a) a₁ c₁ cd : d₁ ∈ f₁ c₁ c₂ : σ₂ cc : tr c₁ c₂ ac₂ : Reaches₁ f₂ a₂ c₂ ⊢ ∃ b₂, tr d₁ b₂ ∧ Reaches₁ f₂ a₂ b₂ have := H cc case tail.intro.intro σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₁ c₁ d₁ : σ₁ a✝ : TransGen (fun a b => b ∈ f₁ a) a₁ c₁ cd : d₁ ∈ f₁ c₁ c₂ : σ₂ cc : tr c₁ c₂ ac₂ : Reaches₁ f₂ a₂ c₂ this : match f₁ c₁ with \| some b₁ => ∃ b₂, tr b₁ b₂ ∧ Reaches₁ f₂ c₂ b₂ \| none => f₂ c₂ = none ⊢ ∃ b₂, tr d₁ b₂ ∧ Reaches₁ f₂ a₂ b₂ rw [show f₁ c₁ = _ from cd] at this case tail.intro.intro σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₁ c₁ d₁ : σ₁ a✝ : TransGen (fun a b => b ∈ f₁ a) a₁ c₁ cd : d₁ ∈ f₁ c₁ c₂ : σ₂ cc : tr c₁ c₂ ac₂ : Reaches₁ f₂ a₂ c₂ this : match some d₁ with \| some b₁ => ∃ b₂, tr b₁ b₂ ∧ Reaches₁ f₂ c₂ b₂ \| none => f₂ c₂ = none ⊢ ∃ b₂, tr d₁ b₂ ∧ Reaches₁ f₂ a₂ b₂ rcases this with ⟨d₂, dd, cd₂⟩ case tail.intro.intro.intro.intro σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₁ c₁ d₁ : σ₁ a✝ : TransGen (fun a b => b ∈ f₁ a) a₁ c₁ cd : d₁ ∈ f₁ c₁ c₂ : σ₂ cc : tr c₁ c₂ ac₂ : Reaches₁ f₂ a₂ c₂ d₂ : σ₂ dd : tr d₁ d₂ cd₂ : Reaches₁ f₂ c₂ d₂ ⊢ ∃ b₂, tr d₁ b₂ ∧ Reaches₁ f₂ a₂ b₂ exact ⟨_, dd, ac₂.trans cd₂⟩ ** Qed
Turing.tr_reaches_rev σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ : σ₂ ab : Reaches f₂ a₂ b₂ ⊢ ∃ c₁ c₂, Reaches f₂ b₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ induction' ab with c₂ d₂ _ cd IH case refl σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ : σ₂ ⊢ ∃ c₁ c₂, Reaches f₂ a₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ exact ⟨_, _, ReflTransGen.refl, aa, ReflTransGen.refl⟩ case tail σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ c₂ d₂ : σ₂ a✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂ cd : d₂ ∈ f₂ c₂ IH : ∃ c₁ c₂_1, Reaches f₂ c₂ c₂_1 ∧ tr c₁ c₂_1 ∧ Reaches f₁ a₁ c₁ ⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ rcases IH with ⟨e₁, e₂, ce, ee, ae⟩ case tail.intro.intro.intro.intro σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ c₂ d₂ : σ₂ a✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂ cd : d₂ ∈ f₂ c₂ e₁ : σ₁ e₂ : σ₂ ce : Reaches f₂ c₂ e₂ ee : tr e₁ e₂ ae : Reaches f₁ a₁ e₁ ⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ rcases ReflTransGen.cases_head ce with (rfl \| ⟨d', cd', de⟩) case tail.intro.intro.intro.intro.inl σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ c₂ d₂ : σ₂ a✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂ cd : d₂ ∈ f₂ c₂ e₁ : σ₁ ae : Reaches f₁ a₁ e₁ ce : Reaches f₂ c₂ c₂ ee : tr e₁ c₂ ⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ have := H ee case tail.intro.intro.intro.intro.inl σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ c₂ d₂ : σ₂ a✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂ cd : d₂ ∈ f₂ c₂ e₁ : σ₁ ae : Reaches f₁ a₁ e₁ ce : Reaches f₂ c₂ c₂ ee : tr e₁ c₂ this : match f₁ e₁ with \| some b₁ => ∃ b₂, tr b₁ b₂ ∧ Reaches₁ f₂ c₂ b₂ \| none => f₂ c₂ = none ⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ revert this case tail.intro.intro.intro.intro.inl σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ c₂ d₂ : σ₂ a✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂ cd : d₂ ∈ f₂ c₂ e₁ : σ₁ ae : Reaches f₁ a₁ e₁ ce : Reaches f₂ c₂ c₂ ee : tr e₁ c₂ ⊢ (match f₁ e₁ with \| some b₁ => ∃ b₂, tr b₁ b₂ ∧ Reaches₁ f₂ c₂ b₂ \| none => f₂ c₂ = none) → ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ cases' eg : f₁ e₁ with g₁ <;> simp only [Respects, and_imp, exists_imp] case tail.intro.intro.intro.intro.inl.none σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ c₂ d₂ : σ₂ a✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂ cd : d₂ ∈ f₂ c₂ e₁ : σ₁ ae : Reaches f₁ a₁ e₁ ce : Reaches f₂ c₂ c₂ ee : tr e₁ c₂ eg : f₁ e₁ = none ⊢ f₂ c₂ = none → ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ intro c0 case tail.intro.intro.intro.intro.inl.none σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ c₂ d₂ : σ₂ a✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂ cd : d₂ ∈ f₂ c₂ e₁ : σ₁ ae : Reaches f₁ a₁ e₁ ce : Reaches f₂ c₂ c₂ ee : tr e₁ c₂ eg : f₁ e₁ = none c0 : f₂ c₂ = none ⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ cases cd.symm.trans c0 case tail.intro.intro.intro.intro.inl.some σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ c₂ d₂ : σ₂ a✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂ cd : d₂ ∈ f₂ c₂ e₁ : σ₁ ae : Reaches f₁ a₁ e₁ ce : Reaches f₂ c₂ c₂ ee : tr e₁ c₂ g₁ : σ₁ eg : f₁ e₁ = some g₁ ⊢ ∀ (x : σ₂), tr g₁ x → Reaches₁ f₂ c₂ x → ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ intro g₂ gg cg case tail.intro.intro.intro.intro.inl.some σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ c₂ d₂ : σ₂ a✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂ cd : d₂ ∈ f₂ c₂ e₁ : σ₁ ae : Reaches f₁ a₁ e₁ ce : Reaches f₂ c₂ c₂ ee : tr e₁ c₂ g₁ : σ₁ eg : f₁ e₁ = some g₁ g₂ : σ₂ gg : tr g₁ g₂ cg : Reaches₁ f₂ c₂ g₂ ⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ rcases TransGen.head'_iff.1 cg with ⟨d', cd', dg⟩ case tail.intro.intro.intro.intro.inl.some.intro.intro σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ c₂ d₂ : σ₂ a✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂ cd : d₂ ∈ f₂ c₂ e₁ : σ₁ ae : Reaches f₁ a₁ e₁ ce : Reaches f₂ c₂ c₂ ee : tr e₁ c₂ g₁ : σ₁ eg : f₁ e₁ = some g₁ g₂ : σ₂ gg : tr g₁ g₂ cg : Reaches₁ f₂ c₂ g₂ d' : σ₂ cd' : d' ∈ f₂ c₂ dg : ReflTransGen (fun a b => b ∈ f₂ a) d' g₂ ⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ cases Option.mem_unique cd cd' case tail.intro.intro.intro.intro.inl.some.intro.intro.refl σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ c₂ d₂ : σ₂ a✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂ cd : d₂ ∈ f₂ c₂ e₁ : σ₁ ae : Reaches f₁ a₁ e₁ ce : Reaches f₂ c₂ c₂ ee : tr e₁ c₂ g₁ : σ₁ eg : f₁ e₁ = some g₁ g₂ : σ₂ gg : tr g₁ g₂ cg : Reaches₁ f₂ c₂ g₂ cd' : d₂ ∈ f₂ c₂ dg : ReflTransGen (fun a b => b ∈ f₂ a) d₂ g₂ ⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ exact ⟨_, _, dg, gg, ae.tail eg⟩ case tail.intro.intro.intro.intro.inr.intro.intro σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ c₂ d₂ : σ₂ a✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂ cd : d₂ ∈ f₂ c₂ e₁ : σ₁ e₂ : σ₂ ce : Reaches f₂ c₂ e₂ ee : tr e₁ e₂ ae : Reaches f₁ a₁ e₁ d' : σ₂ cd' : d' ∈ f₂ c₂ de : ReflTransGen (fun a b => b ∈ f₂ a) d' e₂ ⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ cases Option.mem_unique cd cd' case tail.intro.intro.intro.intro.inr.intro.intro.refl σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ → Prop H : Respects f₁ f₂ tr a₁ : σ₁ a₂ : σ₂ aa : tr a₁ a₂ b₂ c₂ d₂ : σ₂ a✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂ cd : d₂ ∈ f₂ c₂ e₁ : σ₁ e₂ : σ₂ ce : Reaches f₂ c₂ e₂ ee : tr e₁ e₂ ae : Reaches f₁ a₁ e₁ cd' : d₂ ∈ f₂ c₂ de : ReflTransGen (fun a b => b ∈ f₂ a) d₂ e₂ ⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁ exact ⟨_, _, de, ee, ae⟩ ** Qed
Turing.frespects_eq σ₁ : Type u_1 σ₂ : Type u_2 f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ a₂ b₂ : σ₂ h : f₂ a₂ = f₂ b₂ ⊢ FRespects f₂ tr a₂ none ↔ FRespects f₂ tr b₂ none unfold FRespects σ₁ : Type u_1 σ₂ : Type u_2 f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ a₂ b₂ : σ₂ h : f₂ a₂ = f₂ b₂ ⊢ (match none with \| some b₁ => Reaches₁ f₂ a₂ (tr b₁) \| none => f₂ a₂ = none) ↔ match none with \| some b₁ => Reaches₁ f₂ b₂ (tr b₁) \| none => f₂ b₂ = none rw [h] ** Qed
Turing.fun_respects σ₁ : Type u_1 σ₂ : Type u_2 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ a₁ : σ₁ ⊢ (∀ ⦃a₂ : σ₂⦄, (fun a b => tr a = b) a₁ a₂ → match f₁ a₁ with \| some b₁ => ∃ b₂, (fun a b => tr a = b) b₁ b₂ ∧ Reaches₁ f₂ a₂ b₂ \| none => f₂ a₂ = none) ↔ FRespects f₂ tr (tr a₁) (f₁ a₁) cases f₁ a₁ <;> simp only [FRespects, Respects, exists_eq_left', forall_eq'] ** Qed
Turing.tr_eval' σ₁ σ₂ : Type u_1 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ H : Respects f₁ f₂ fun a b => tr a = b a₁ : σ₁ b₂ : σ₂ h : b₂ ∈ tr <$> eval f₁ a₁ ⊢ b₂ ∈ eval f₂ (tr a₁) rcases (Part.mem_map_iff _).1 h with ⟨b₁, ab, bb⟩ case intro.intro σ₁ σ₂ : Type u_1 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ H : Respects f₁ f₂ fun a b => tr a = b a₁ : σ₁ b₂ : σ₂ h : b₂ ∈ tr <$> eval f₁ a₁ b₁ : σ₁ ab : b₁ ∈ eval f₁ a₁ bb : tr b₁ = b₂ ⊢ b₂ ∈ eval f₂ (tr a₁) rcases tr_eval H rfl ab with ⟨_, rfl, h⟩ case intro.intro.intro.intro σ₁ σ₂ : Type u_1 f₁ : σ₁ → Option σ₁ f₂ : σ₂ → Option σ₂ tr : σ₁ → σ₂ H : Respects f₁ f₂ fun a b => tr a = b a₁ : σ₁ b₂ : σ₂ h✝ : b₂ ∈ tr <$> eval f₁ a₁ b₁ : σ₁ ab : b₁ ∈ eval f₁ a₁ bb : tr b₁ = b₂ h : tr b₁ ∈ eval f₂ (tr a₁) ⊢ b₂ ∈ eval f₂ (tr a₁) rwa [bb] at h ** Qed
Turing.TM0.step_supports Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 inst✝ : Inhabited Λ M : Machine₀ S : Set Λ ss : Supports M S ⊢ ∀ {c c' : Cfg₀}, c' ∈ step M c → c.q ∈ S → c'.q ∈ S intro ⟨q, T⟩ c' h₁ h₂ Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 inst✝ : Inhabited Λ M : Machine₀ S : Set Λ ss : Supports M S q : Λ T : Tape Γ c' : Cfg₀ h₁ : c' ∈ step M { q := q, Tape := T } h₂ : { q := q, Tape := T }.q ∈ S ⊢ c'.q ∈ S rcases Option.map_eq_some'.1 h₁ with ⟨⟨q', a⟩, h, rfl⟩ case intro.mk.intro Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 inst✝ : Inhabited Λ M : Machine₀ S : Set Λ ss : Supports M S q : Λ T : Tape Γ h₂ : { q := q, Tape := T }.q ∈ S q' : Λ a : Stmt₀ h : M q T.head = some (q', a) h₁ : (fun x => match x with \| (q', a) => { q := q', Tape := match a with \| Stmt.move d => Tape.move d T \| Stmt.write a => Tape.write a T }) (q', a) ∈ step M { q := q, Tape := T } ⊢ ((fun x => match x with \| (q', a) => { q := q', Tape := match a with \| Stmt.move d => Tape.move d T \| Stmt.write a => Tape.write a T }) (q', a)).q ∈ S exact ss.2 h h₂ ** Qed
Turing.TM0.univ_supports Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 inst✝ : Inhabited Λ M : Machine₀ ⊢ Supports M Set.univ constructor <;> intros <;> apply Set.mem_univ ** Qed
Turing.TM1.stmts₁_self Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q : Stmt₁ ⊢ q ∈ stmts₁ q cases q <;> simp only [stmts₁, Finset.mem_insert_self, Finset.mem_singleton_self] ** Qed
Turing.TM1.stmts₁_trans Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁ q₂ : Stmt₁ ⊢ q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ intro h₁₂ q₀ h₀₁ Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁ q₂ : Stmt₁ h₁₂ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₁ h₀₁ : q₀ ∈ stmts₁ q₁ ⊢ q₀ ∈ stmts₁ q₂ induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;> simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂ case goto Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁ q₂ : Stmt₁ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₁ h₀₁ : q₀ ∈ stmts₁ q₁ a✝ : Γ → σ → Λ h₁₂ : q₁ = goto a✝ ⊢ q₀ ∈ {goto a✝} case halt Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁ q₂ : Stmt₁ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₁ h₀₁ : q₀ ∈ stmts₁ q₁ h₁₂ : q₁ = halt ⊢ q₀ ∈ {halt} case goto l => subst h₁₂; exact h₀₁ case halt Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁ q₂ : Stmt₁ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₁ h₀₁ : q₀ ∈ stmts₁ q₁ h₁₂ : q₁ = halt ⊢ q₀ ∈ {halt} case halt => subst h₁₂; exact h₀₁ case load Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁ q₂ : Stmt₁ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₁ h₀₁ : q₀ ∈ stmts₁ q₁ a✝ : Γ → σ → σ q : Stmt₁ IH : q₁ ∈ stmts₁ q → q₀ ∈ stmts₁ q h₁₂ : q₁ = load a✝ q ∨ q₁ ∈ stmts₁ q ⊢ q₀ ∈ insert (load a✝ q) (stmts₁ q) case branch Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁ q₂ : Stmt₁ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₁ h₀₁ : q₀ ∈ stmts₁ q₁ a✝² : Γ → σ → Bool a✝¹ a✝ : Stmt₁ a_ih✝¹ : q₁ ∈ stmts₁ a✝¹ → q₀ ∈ stmts₁ a✝¹ a_ih✝ : q₁ ∈ stmts₁ a✝ → q₀ ∈ stmts₁ a✝ h₁₂ : q₁ = branch a✝² a✝¹ a✝ ∨ q₁ ∈ stmts₁ a✝¹ ∨ q₁ ∈ stmts₁ a✝ ⊢ q₀ ∈ insert (branch a✝² a✝¹ a✝) (stmts₁ a✝¹ ∪ stmts₁ a✝) case goto Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁ q₂ : Stmt₁ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₁ h₀₁ : q₀ ∈ stmts₁ q₁ a✝ : Γ → σ → Λ h₁₂ : q₁ = goto a✝ ⊢ q₀ ∈ {goto a✝} case halt Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁ q₂ : Stmt₁ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₁ h₀₁ : q₀ ∈ stmts₁ q₁ h₁₂ : q₁ = halt ⊢ q₀ ∈ {halt} rcases h₁₂ with (rfl \| h₁₂) case load.inl Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₂ q₀ : Stmt₁ a✝ : Γ → σ → σ q : Stmt₁ h₁₂ : load a✝ q ∈ stmts₁ q₂ h₀₁ : q₀ ∈ stmts₁ (load a✝ q) IH : load a✝ q ∈ stmts₁ q → q₀ ∈ stmts₁ q ⊢ q₀ ∈ insert (load a✝ q) (stmts₁ q) unfold stmts₁ at h₀₁ case load.inl Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₂ q₀ : Stmt₁ a✝ : Γ → σ → σ q : Stmt₁ h₁₂ : load a✝ q ∈ stmts₁ q₂ IH : load a✝ q ∈ stmts₁ q → q₀ ∈ stmts₁ q h₀₁ : q₀ ∈ insert (load a✝ q) (stmts₁ q) ⊢ q₀ ∈ insert (load a✝ q) (stmts₁ q) exact h₀₁ case load.inr Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁ q₂ : Stmt₁ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₁ h₀₁ : q₀ ∈ stmts₁ q₁ a✝ : Γ → σ → σ q : Stmt₁ IH : q₁ ∈ stmts₁ q → q₀ ∈ stmts₁ q h₁₂ : q₁ ∈ stmts₁ q ⊢ q₀ ∈ insert (load a✝ q) (stmts₁ q) exact Finset.mem_insert_of_mem (IH h₁₂) Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁✝ q₂✝ : Stmt₁ h₁₂✝ : q₁✝ ∈ stmts₁ q₂✝ q₀ : Stmt₁ h₀₁ : q₀ ∈ stmts₁ q₁✝ p : Γ → σ → Bool q₁ q₂ : Stmt₁ IH₁ : q₁✝ ∈ stmts₁ q₁ → q₀ ∈ stmts₁ q₁ IH₂ : q₁✝ ∈ stmts₁ q₂ → q₀ ∈ stmts₁ q₂ h₁₂ : q₁✝ = branch p q₁ q₂ ∨ q₁✝ ∈ stmts₁ q₁ ∨ q₁✝ ∈ stmts₁ q₂ ⊢ q₀ ∈ insert (branch p q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂) rcases h₁₂ with (rfl \| h₁₂ \| h₁₂) case inl Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₂✝ q₀ : Stmt₁ p : Γ → σ → Bool q₁ q₂ : Stmt₁ h₁₂ : branch p q₁ q₂ ∈ stmts₁ q₂✝ h₀₁ : q₀ ∈ stmts₁ (branch p q₁ q₂) IH₁ : branch p q₁ q₂ ∈ stmts₁ q₁ → q₀ ∈ stmts₁ q₁ IH₂ : branch p q₁ q₂ ∈ stmts₁ q₂ → q₀ ∈ stmts₁ q₂ ⊢ q₀ ∈ insert (branch p q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂) unfold stmts₁ at h₀₁ case inl Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₂✝ q₀ : Stmt₁ p : Γ → σ → Bool q₁ q₂ : Stmt₁ h₁₂ : branch p q₁ q₂ ∈ stmts₁ q₂✝ IH₁ : branch p q₁ q₂ ∈ stmts₁ q₁ → q₀ ∈ stmts₁ q₁ IH₂ : branch p q₁ q₂ ∈ stmts₁ q₂ → q₀ ∈ stmts₁ q₂ h₀₁ : q₀ ∈ insert (branch p q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂) ⊢ q₀ ∈ insert (branch p q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂) exact h₀₁ case inr.inl Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁✝ q₂✝ : Stmt₁ h₁₂✝ : q₁✝ ∈ stmts₁ q₂✝ q₀ : Stmt₁ h₀₁ : q₀ ∈ stmts₁ q₁✝ p : Γ → σ → Bool q₁ q₂ : Stmt₁ IH₁ : q₁✝ ∈ stmts₁ q₁ → q₀ ∈ stmts₁ q₁ IH₂ : q₁✝ ∈ stmts₁ q₂ → q₀ ∈ stmts₁ q₂ h₁₂ : q₁✝ ∈ stmts₁ q₁ ⊢ q₀ ∈ insert (branch p q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂) exact Finset.mem_insert_of_mem (Finset.mem_union_left _ <\| IH₁ h₁₂) case inr.inr Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁✝ q₂✝ : Stmt₁ h₁₂✝ : q₁✝ ∈ stmts₁ q₂✝ q₀ : Stmt₁ h₀₁ : q₀ ∈ stmts₁ q₁✝ p : Γ → σ → Bool q₁ q₂ : Stmt₁ IH₁ : q₁✝ ∈ stmts₁ q₁ → q₀ ∈ stmts₁ q₁ IH₂ : q₁✝ ∈ stmts₁ q₂ → q₀ ∈ stmts₁ q₂ h₁₂ : q₁✝ ∈ stmts₁ q₂ ⊢ q₀ ∈ insert (branch p q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂) exact Finset.mem_insert_of_mem (Finset.mem_union_right _ <\| IH₂ h₁₂) Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁ q₂ : Stmt₁ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₁ h₀₁ : q₀ ∈ stmts₁ q₁ l : Γ → σ → Λ h₁₂ : q₁ = goto l ⊢ q₀ ∈ {goto l} subst h₁₂ Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₂ q₀ : Stmt₁ l : Γ → σ → Λ h₁₂ : goto l ∈ stmts₁ q₂ h₀₁ : q₀ ∈ stmts₁ (goto l) ⊢ q₀ ∈ {goto l} exact h₀₁ Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₁ q₂ : Stmt₁ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₁ h₀₁ : q₀ ∈ stmts₁ q₁ h₁₂ : q₁ = halt ⊢ q₀ ∈ {halt} subst h₁₂ Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 q₂ q₀ : Stmt₁ h₁₂ : halt ∈ stmts₁ q₂ h₀₁ : q₀ ∈ stmts₁ halt ⊢ q₀ ∈ {halt} exact h₀₁ ** Qed
Turing.TM1.stmts₁_supportsStmt_mono Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h : q₁ ∈ stmts₁ q₂ hs : SupportsStmt S q₂ ⊢ SupportsStmt S q₁ induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h hs case move Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝ : Dir q : Stmt₁ IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁ hs : SupportsStmt S q h : q₁ = move a✝ q ∨ q₁ ∈ stmts₁ q ⊢ SupportsStmt S q₁ case write Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝ : Γ → σ → Γ q : Stmt₁ IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁ hs : SupportsStmt S q h : q₁ = write a✝ q ∨ q₁ ∈ stmts₁ q ⊢ SupportsStmt S q₁ case load Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝ : Γ → σ → σ q : Stmt₁ IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁ hs : SupportsStmt S q h : q₁ = load a✝ q ∨ q₁ ∈ stmts₁ q ⊢ SupportsStmt S q₁ case branch Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝² : Γ → σ → Bool a✝¹ a✝ : Stmt₁ a_ih✝¹ : q₁ ∈ stmts₁ a✝¹ → SupportsStmt S a✝¹ → SupportsStmt S q₁ a_ih✝ : q₁ ∈ stmts₁ a✝ → SupportsStmt S a✝ → SupportsStmt S q₁ hs : SupportsStmt S a✝¹ ∧ SupportsStmt S a✝ h : q₁ = branch a✝² a✝¹ a✝ ∨ q₁ ∈ stmts₁ a✝¹ ∨ q₁ ∈ stmts₁ a✝ ⊢ SupportsStmt S q₁ case goto Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝ : Γ → σ → Λ hs : ∀ (a : Γ) (v : σ), a✝ a v ∈ S h : q₁ = goto a✝ ⊢ SupportsStmt S q₁ case halt Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ hs : True h : q₁ = halt ⊢ SupportsStmt S q₁ iterate 3 rcases h with (rfl \| h) <;> [exact hs; exact IH h hs] case branch Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝² : Γ → σ → Bool a✝¹ a✝ : Stmt₁ a_ih✝¹ : q₁ ∈ stmts₁ a✝¹ → SupportsStmt S a✝¹ → SupportsStmt S q₁ a_ih✝ : q₁ ∈ stmts₁ a✝ → SupportsStmt S a✝ → SupportsStmt S q₁ hs : SupportsStmt S a✝¹ ∧ SupportsStmt S a✝ h : q₁ = branch a✝² a✝¹ a✝ ∨ q₁ ∈ stmts₁ a✝¹ ∨ q₁ ∈ stmts₁ a✝ ⊢ SupportsStmt S q₁ case goto Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝ : Γ → σ → Λ hs : ∀ (a : Γ) (v : σ), a✝ a v ∈ S h : q₁ = goto a✝ ⊢ SupportsStmt S q₁ case halt Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ hs : True h : q₁ = halt ⊢ SupportsStmt S q₁ case branch p q₁ q₂ IH₁ IH₂ => rcases h with (rfl \| h \| h); exacts [hs, IH₁ h hs.1, IH₂ h hs.2] case goto Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝ : Γ → σ → Λ hs : ∀ (a : Γ) (v : σ), a✝ a v ∈ S h : q₁ = goto a✝ ⊢ SupportsStmt S q₁ case halt Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ hs : True h : q₁ = halt ⊢ SupportsStmt S q₁ case goto l => subst h; exact hs case halt Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ hs : True h : q₁ = halt ⊢ SupportsStmt S q₁ case halt => subst h; trivial case load Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝ : Γ → σ → σ q : Stmt₁ IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁ hs : SupportsStmt S q h : q₁ = load a✝ q ∨ q₁ ∈ stmts₁ q ⊢ SupportsStmt S q₁ case branch Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝² : Γ → σ → Bool a✝¹ a✝ : Stmt₁ a_ih✝¹ : q₁ ∈ stmts₁ a✝¹ → SupportsStmt S a✝¹ → SupportsStmt S q₁ a_ih✝ : q₁ ∈ stmts₁ a✝ → SupportsStmt S a✝ → SupportsStmt S q₁ hs : SupportsStmt S a✝¹ ∧ SupportsStmt S a✝ h : q₁ = branch a✝² a✝¹ a✝ ∨ q₁ ∈ stmts₁ a✝¹ ∨ q₁ ∈ stmts₁ a✝ ⊢ SupportsStmt S q₁ case goto Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝ : Γ → σ → Λ hs : ∀ (a : Γ) (v : σ), a✝ a v ∈ S h : q₁ = goto a✝ ⊢ SupportsStmt S q₁ case halt Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ hs : True h : q₁ = halt ⊢ SupportsStmt S q₁ rcases h with (rfl \| h) <;> [exact hs; exact IH h hs] Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁✝ q₂✝ : Stmt₁ h✝ : q₁✝ ∈ stmts₁ q₂✝ hs✝ : SupportsStmt S q₂✝ p : Γ → σ → Bool q₁ q₂ : Stmt₁ IH₁ : q₁✝ ∈ stmts₁ q₁ → SupportsStmt S q₁ → SupportsStmt S q₁✝ IH₂ : q₁✝ ∈ stmts₁ q₂ → SupportsStmt S q₂ → SupportsStmt S q₁✝ hs : SupportsStmt S q₁ ∧ SupportsStmt S q₂ h : q₁✝ = branch p q₁ q₂ ∨ q₁✝ ∈ stmts₁ q₁ ∨ q₁✝ ∈ stmts₁ q₂ ⊢ SupportsStmt S q₁✝ rcases h with (rfl \| h \| h) case inl Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₂✝ : Stmt₁ hs✝ : SupportsStmt S q₂✝ p : Γ → σ → Bool q₁ q₂ : Stmt₁ hs : SupportsStmt S q₁ ∧ SupportsStmt S q₂ h : branch p q₁ q₂ ∈ stmts₁ q₂✝ IH₁ : branch p q₁ q₂ ∈ stmts₁ q₁ → SupportsStmt S q₁ → SupportsStmt S (branch p q₁ q₂) IH₂ : branch p q₁ q₂ ∈ stmts₁ q₂ → SupportsStmt S q₂ → SupportsStmt S (branch p q₁ q₂) ⊢ SupportsStmt S (branch p q₁ q₂) case inr.inl Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁✝ q₂✝ : Stmt₁ h✝ : q₁✝ ∈ stmts₁ q₂✝ hs✝ : SupportsStmt S q₂✝ p : Γ → σ → Bool q₁ q₂ : Stmt₁ IH₁ : q₁✝ ∈ stmts₁ q₁ → SupportsStmt S q₁ → SupportsStmt S q₁✝ IH₂ : q₁✝ ∈ stmts₁ q₂ → SupportsStmt S q₂ → SupportsStmt S q₁✝ hs : SupportsStmt S q₁ ∧ SupportsStmt S q₂ h : q₁✝ ∈ stmts₁ q₁ ⊢ SupportsStmt S q₁✝ case inr.inr Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁✝ q₂✝ : Stmt₁ h✝ : q₁✝ ∈ stmts₁ q₂✝ hs✝ : SupportsStmt S q₂✝ p : Γ → σ → Bool q₁ q₂ : Stmt₁ IH₁ : q₁✝ ∈ stmts₁ q₁ → SupportsStmt S q₁ → SupportsStmt S q₁✝ IH₂ : q₁✝ ∈ stmts₁ q₂ → SupportsStmt S q₂ → SupportsStmt S q₁✝ hs : SupportsStmt S q₁ ∧ SupportsStmt S q₂ h : q₁✝ ∈ stmts₁ q₂ ⊢ SupportsStmt S q₁✝ exacts [hs, IH₁ h hs.1, IH₂ h hs.2] Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ l : Γ → σ → Λ hs : ∀ (a : Γ) (v : σ), l a v ∈ S h : q₁ = goto l ⊢ SupportsStmt S q₁ subst h Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₂ : Stmt₁ hs✝ : SupportsStmt S q₂ l : Γ → σ → Λ hs : ∀ (a : Γ) (v : σ), l a v ∈ S h : goto l ∈ stmts₁ q₂ ⊢ SupportsStmt S (goto l) exact hs Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₁ q₂ : Stmt₁ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ hs : True h : q₁ = halt ⊢ SupportsStmt S q₁ subst h Γ : Type u_1 inst✝ : Inhabited Γ Λ : Type u_2 σ : Type u_3 S : Finset Λ q₂ : Stmt₁ hs✝ : SupportsStmt S q₂ hs : True h : halt ∈ stmts₁ q₂ ⊢ SupportsStmt S halt trivial ** Qed
Turing.TM1.step_supports Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v : σ T : Tape Γ c' : Cfg₁ h₁ : c' ∈ step M { l := some l₁, var := v, Tape := T } h₂ : { l := some l₁, var := v, Tape := T }.l ∈ ↑Finset.insertNone S ⊢ c'.l ∈ ↑Finset.insertNone S replace h₂ := ss.2 _ (Finset.some_mem_insertNone.1 h₂) Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v : σ T : Tape Γ c' : Cfg₁ h₁ : c' ∈ step M { l := some l₁, var := v, Tape := T } h₂ : SupportsStmt S (M l₁) ⊢ c'.l ∈ ↑Finset.insertNone S simp only [step, Option.mem_def, Option.some.injEq] at h₁ Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v : σ T : Tape Γ c' : Cfg₁ h₂ : SupportsStmt S (M l₁) h₁ : stepAux (M l₁) v T = c' ⊢ c'.l ∈ ↑Finset.insertNone S subst c' Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v : σ T : Tape Γ h₂ : SupportsStmt S (M l₁) ⊢ (stepAux (M l₁) v T).l ∈ ↑Finset.insertNone S revert h₂ Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v : σ T : Tape Γ ⊢ SupportsStmt S (M l₁) → (stepAux (M l₁) v T).l ∈ ↑Finset.insertNone S induction' M l₁ with _ q IH _ q IH _ q IH generalizing v T <;> intro hs case move Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ a✝ : Dir q : Stmt₁ IH : ∀ (v : σ) (T : Tape Γ), SupportsStmt S q → (stepAux q v T).l ∈ ↑Finset.insertNone S v : σ T : Tape Γ hs : SupportsStmt S (move a✝ q) ⊢ (stepAux (move a✝ q) v T).l ∈ ↑Finset.insertNone S case write Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ a✝ : Γ → σ → Γ q : Stmt₁ IH : ∀ (v : σ) (T : Tape Γ), SupportsStmt S q → (stepAux q v T).l ∈ ↑Finset.insertNone S v : σ T : Tape Γ hs : SupportsStmt S (write a✝ q) ⊢ (stepAux (write a✝ q) v T).l ∈ ↑Finset.insertNone S case load Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ a✝ : Γ → σ → σ q : Stmt₁ IH : ∀ (v : σ) (T : Tape Γ), SupportsStmt S q → (stepAux q v T).l ∈ ↑Finset.insertNone S v : σ T : Tape Γ hs : SupportsStmt S (load a✝ q) ⊢ (stepAux (load a✝ q) v T).l ∈ ↑Finset.insertNone S case branch Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ a✝² : Γ → σ → Bool a✝¹ a✝ : Stmt₁ a_ih✝¹ : ∀ (v : σ) (T : Tape Γ), SupportsStmt S a✝¹ → (stepAux a✝¹ v T).l ∈ ↑Finset.insertNone S a_ih✝ : ∀ (v : σ) (T : Tape Γ), SupportsStmt S a✝ → (stepAux a✝ v T).l ∈ ↑Finset.insertNone S v : σ T : Tape Γ hs : SupportsStmt S (branch a✝² a✝¹ a✝) ⊢ (stepAux (branch a✝² a✝¹ a✝) v T).l ∈ ↑Finset.insertNone S case goto Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ a✝ : Γ → σ → Λ v : σ T : Tape Γ hs : SupportsStmt S (goto a✝) ⊢ (stepAux (goto a✝) v T).l ∈ ↑Finset.insertNone S case halt Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ v : σ T : Tape Γ hs : SupportsStmt S halt ⊢ (stepAux halt v T).l ∈ ↑Finset.insertNone S iterate 3 exact IH _ _ hs case goto Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ a✝ : Γ → σ → Λ v : σ T : Tape Γ hs : SupportsStmt S (goto a✝) ⊢ (stepAux (goto a✝) v T).l ∈ ↑Finset.insertNone S case halt Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ v : σ T : Tape Γ hs : SupportsStmt S halt ⊢ (stepAux halt v T).l ∈ ↑Finset.insertNone S case goto => exact Finset.some_mem_insertNone.2 (hs _ _) case halt Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ v : σ T : Tape Γ hs : SupportsStmt S halt ⊢ (stepAux halt v T).l ∈ ↑Finset.insertNone S case halt => apply Multiset.mem_cons_self case load Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ a✝ : Γ → σ → σ q : Stmt₁ IH : ∀ (v : σ) (T : Tape Γ), SupportsStmt S q → (stepAux q v T).l ∈ ↑Finset.insertNone S v : σ T : Tape Γ hs : SupportsStmt S (load a✝ q) ⊢ (stepAux (load a✝ q) v T).l ∈ ↑Finset.insertNone S case branch Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ a✝² : Γ → σ → Bool a✝¹ a✝ : Stmt₁ a_ih✝¹ : ∀ (v : σ) (T : Tape Γ), SupportsStmt S a✝¹ → (stepAux a✝¹ v T).l ∈ ↑Finset.insertNone S a_ih✝ : ∀ (v : σ) (T : Tape Γ), SupportsStmt S a✝ → (stepAux a✝ v T).l ∈ ↑Finset.insertNone S v : σ T : Tape Γ hs : SupportsStmt S (branch a✝² a✝¹ a✝) ⊢ (stepAux (branch a✝² a✝¹ a✝) v T).l ∈ ↑Finset.insertNone S case goto Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ a✝ : Γ → σ → Λ v : σ T : Tape Γ hs : SupportsStmt S (goto a✝) ⊢ (stepAux (goto a✝) v T).l ∈ ↑Finset.insertNone S case halt Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ v : σ T : Tape Γ hs : SupportsStmt S halt ⊢ (stepAux halt v T).l ∈ ↑Finset.insertNone S exact IH _ _ hs Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ p : Γ → σ → Bool q₁' q₂' : Stmt₁ IH₁ : ∀ (v : σ) (T : Tape Γ), SupportsStmt S q₁' → (stepAux q₁' v T).l ∈ ↑Finset.insertNone S IH₂ : ∀ (v : σ) (T : Tape Γ), SupportsStmt S q₂' → (stepAux q₂' v T).l ∈ ↑Finset.insertNone S v : σ T : Tape Γ hs : SupportsStmt S (branch p q₁' q₂') ⊢ (stepAux (branch p q₁' q₂') v T).l ∈ ↑Finset.insertNone S unfold stepAux Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ p : Γ → σ → Bool q₁' q₂' : Stmt₁ IH₁ : ∀ (v : σ) (T : Tape Γ), SupportsStmt S q₁' → (stepAux q₁' v T).l ∈ ↑Finset.insertNone S IH₂ : ∀ (v : σ) (T : Tape Γ), SupportsStmt S q₂' → (stepAux q₂' v T).l ∈ ↑Finset.insertNone S v : σ T : Tape Γ hs : SupportsStmt S (branch p q₁' q₂') ⊢ (bif p T.head v then stepAux q₁' v T else stepAux q₂' v T).l ∈ ↑Finset.insertNone S cases p T.1 v case false Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ p : Γ → σ → Bool q₁' q₂' : Stmt₁ IH₁ : ∀ (v : σ) (T : Tape Γ), SupportsStmt S q₁' → (stepAux q₁' v T).l ∈ ↑Finset.insertNone S IH₂ : ∀ (v : σ) (T : Tape Γ), SupportsStmt S q₂' → (stepAux q₂' v T).l ∈ ↑Finset.insertNone S v : σ T : Tape Γ hs : SupportsStmt S (branch p q₁' q₂') ⊢ (bif false then stepAux q₁' v T else stepAux q₂' v T).l ∈ ↑Finset.insertNone S exact IH₂ _ _ hs.2 case true Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ p : Γ → σ → Bool q₁' q₂' : Stmt₁ IH₁ : ∀ (v : σ) (T : Tape Γ), SupportsStmt S q₁' → (stepAux q₁' v T).l ∈ ↑Finset.insertNone S IH₂ : ∀ (v : σ) (T : Tape Γ), SupportsStmt S q₂' → (stepAux q₂' v T).l ∈ ↑Finset.insertNone S v : σ T : Tape Γ hs : SupportsStmt S (branch p q₁' q₂') ⊢ (bif true then stepAux q₁' v T else stepAux q₂' v T).l ∈ ↑Finset.insertNone S exact IH₁ _ _ hs.1 Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ a✝ : Γ → σ → Λ v : σ T : Tape Γ hs : SupportsStmt S (goto a✝) ⊢ (stepAux (goto a✝) v T).l ∈ ↑Finset.insertNone S exact Finset.some_mem_insertNone.2 (hs _ _) Γ : Type u_1 inst✝¹ : Inhabited Γ Λ : Type u_2 σ : Type u_3 inst✝ : Inhabited Λ M : Λ → Stmt₁ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : Tape Γ v : σ T : Tape Γ hs : SupportsStmt S halt ⊢ (stepAux halt v T).l ∈ ↑Finset.insertNone S apply Multiset.mem_cons_self ** Qed
Turing.TM1to1.stepAux_move Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ d : Dir q : Stmt Bool Λ' σ v : σ T : Tape Bool ⊢ stepAux (move d q) v T = stepAux q v ((Tape.move d)^[n] T) suffices ∀ i, stepAux ((Stmt.move d)^[i] q) v T = stepAux q v ((Tape.move d)^[i] T) from this n Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ d : Dir q : Stmt Bool Λ' σ v : σ T : Tape Bool ⊢ ∀ (i : ℕ), stepAux ((Stmt.move d)^[i] q) v T = stepAux q v ((Tape.move d)^[i] T) intro i Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ d : Dir q : Stmt Bool Λ' σ v : σ T : Tape Bool i : ℕ ⊢ stepAux ((Stmt.move d)^[i] q) v T = stepAux q v ((Tape.move d)^[i] T) induction' i with i IH generalizing T case succ Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ d : Dir q : Stmt Bool Λ' σ v : σ T✝ : Tape Bool i : ℕ IH : ∀ (T : Tape Bool), stepAux ((Stmt.move d)^[i] q) v T = stepAux q v ((Tape.move d)^[i] T) T : Tape Bool ⊢ stepAux ((Stmt.move d)^[Nat.succ i] q) v T = stepAux q v ((Tape.move d)^[Nat.succ i] T) rw [iterate_succ', iterate_succ] case succ Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ d : Dir q : Stmt Bool Λ' σ v : σ T✝ : Tape Bool i : ℕ IH : ∀ (T : Tape Bool), stepAux ((Stmt.move d)^[i] q) v T = stepAux q v ((Tape.move d)^[i] T) T : Tape Bool ⊢ stepAux ((Stmt.move d ∘ (Stmt.move d)^[i]) q) v T = stepAux q v (((Tape.move d)^[i] ∘ Tape.move d) T) simp only [stepAux, Function.comp_apply] case succ Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ d : Dir q : Stmt Bool Λ' σ v : σ T✝ : Tape Bool i : ℕ IH : ∀ (T : Tape Bool), stepAux ((Stmt.move d)^[i] q) v T = stepAux q v ((Tape.move d)^[i] T) T : Tape Bool ⊢ stepAux ((Stmt.move d)^[i] q) v (Tape.move d T) = stepAux q v ((Tape.move d)^[i] (Tape.move d T)) rw [IH] case zero Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ d : Dir q : Stmt Bool Λ' σ v : σ T✝ T : Tape Bool ⊢ stepAux ((Stmt.move d)^[Nat.zero] q) v T = stepAux q v ((Tape.move d)^[Nat.zero] T) rfl ** Qed
Turing.TM1to1.supportsStmt_move Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ S : Finset Λ' d : Dir q : Stmt Bool Λ' σ ⊢ SupportsStmt S (move d q) = SupportsStmt S q suffices ∀ {i}, SupportsStmt S ((Stmt.move d)^[i] q) = _ from this Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ S : Finset Λ' d : Dir q : Stmt Bool Λ' σ ⊢ ∀ {i : ℕ}, SupportsStmt S ((Stmt.move d)^[i] q) = SupportsStmt S q intro i Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ S : Finset Λ' d : Dir q : Stmt Bool Λ' σ i : ℕ ⊢ SupportsStmt S ((Stmt.move d)^[i] q) = SupportsStmt S q induction i generalizing q <;> simp only [, iterate] * case succ Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ S : Finset Λ' d : Dir n✝ : ℕ n_ih✝ : ∀ {q : Stmt Bool Λ' σ}, SupportsStmt S ((Stmt.move d)^[n✝] q) = SupportsStmt S q q : Stmt Bool Λ' σ ⊢ SupportsStmt S (Stmt.move d q) = SupportsStmt S q rfl Qed
Turing.TM1to1.supportsStmt_write Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ S : Finset Λ' l : List Bool q : Stmt Bool Λ' σ ⊢ SupportsStmt S (write l q) = SupportsStmt S q induction' l with _ l IH <;> simp only [write, SupportsStmt, ] * Qed
Turing.TM1to1.supportsStmt_read Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ S : Finset Λ' f✝ : Γ → Stmt Bool Λ' σ this : ∀ (i : ℕ) (f : Vector Bool i → Stmt Bool Λ' σ), (∀ (v : Vector Bool i), SupportsStmt S (f v)) → SupportsStmt S (readAux i f) hf : ∀ (a : Γ), SupportsStmt S (f✝ a) ⊢ ∀ (v : Vector Bool n), SupportsStmt S (move Dir.left (f✝ (dec v))) intro Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ S : Finset Λ' f✝ : Γ → Stmt Bool Λ' σ this : ∀ (i : ℕ) (f : Vector Bool i → Stmt Bool Λ' σ), (∀ (v : Vector Bool i), SupportsStmt S (f v)) → SupportsStmt S (readAux i f) hf : ∀ (a : Γ), SupportsStmt S (f✝ a) v✝ : Vector Bool n ⊢ SupportsStmt S (move Dir.left (f✝ (dec v✝))) simp only [supportsStmt_move, hf] Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ S : Finset Λ' f✝ : Γ → Stmt Bool Λ' σ i : ℕ f : Vector Bool i → Stmt Bool Λ' σ hf : ∀ (v : Vector Bool i), SupportsStmt S (f v) ⊢ SupportsStmt S (readAux i f) induction' i with i IH case succ Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ S : Finset Λ' f✝¹ : Γ → Stmt Bool Λ' σ i✝ : ℕ f✝ : Vector Bool i✝ → Stmt Bool Λ' σ hf✝ : ∀ (v : Vector Bool i✝), SupportsStmt S (f✝ v) i : ℕ IH : ∀ (f : Vector Bool i → Stmt Bool Λ' σ), (∀ (v : Vector Bool i), SupportsStmt S (f v)) → SupportsStmt S (readAux i f) f : Vector Bool (Nat.succ i) → Stmt Bool Λ' σ hf : ∀ (v : Vector Bool (Nat.succ i)), SupportsStmt S (f v) ⊢ SupportsStmt S (readAux (Nat.succ i) f) constructor <;> apply IH <;> intro <;> apply hf case zero Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ S : Finset Λ' f✝¹ : Γ → Stmt Bool Λ' σ i : ℕ f✝ : Vector Bool i → Stmt Bool Λ' σ hf✝ : ∀ (v : Vector Bool i), SupportsStmt S (f✝ v) f : Vector Bool Nat.zero → Stmt Bool Λ' σ hf : ∀ (v : Vector Bool Nat.zero), SupportsStmt S (f v) ⊢ SupportsStmt S (readAux Nat.zero f) exact hf _ ** Qed
Turing.TM1to1.trTape'_move_left Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ L R : ListBlank Γ ⊢ (Tape.move Dir.left)^[n] (trTape' enc0 L R) = trTape' enc0 (ListBlank.tail L) (ListBlank.cons (ListBlank.head L) R) obtain ⟨a, L, rfl⟩ := L.exists_cons case intro.intro Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ R : ListBlank Γ a : Γ L : ListBlank Γ ⊢ (Tape.move Dir.left)^[n] (trTape' enc0 (ListBlank.cons a L) R) = trTape' enc0 (ListBlank.tail (ListBlank.cons a L)) (ListBlank.cons (ListBlank.head (ListBlank.cons a L)) R) simp only [trTape', ListBlank.cons_bind, ListBlank.head_cons, ListBlank.tail_cons] case intro.intro Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ R : ListBlank Γ a : Γ L : ListBlank Γ ⊢ (Tape.move Dir.left)^[n] (Tape.mk' (ListBlank.append (List.reverse (Vector.toList (enc a))) (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x))) (_ : ∃ n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default))) (ListBlank.bind R (fun x => Vector.toList (enc x)) (_ : ∃ n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default))) = Tape.mk' (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x))) (_ : ∃ n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default)) (ListBlank.append (Vector.toList (enc a)) (ListBlank.bind R (fun x => Vector.toList (enc x)) (_ : ∃ n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default))) suffices ∀ {L' R' l₁ l₂} (_ : Vector.toList (enc a) = List.reverseAux l₁ l₂), (Tape.move Dir.left)^[l₁.length] (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) = Tape.mk' L' (ListBlank.append (Vector.toList (enc a)) R') by simpa only [List.length_reverse, Vector.toList_length] using this (List.reverse_reverse _).symm case intro.intro Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ R : ListBlank Γ a : Γ L : ListBlank Γ ⊢ ∀ {L' R' : ListBlank Bool} {l₁ l₂ : List Bool}, Vector.toList (enc a) = List.reverseAux l₁ l₂ → (Tape.move Dir.left)^[List.length l₁] (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) = Tape.mk' L' (ListBlank.append (Vector.toList (enc a)) R') intro _ _ l₁ l₂ e case intro.intro Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ R : ListBlank Γ a : Γ L : ListBlank Γ L'✝ R'✝ : ListBlank Bool l₁ l₂ : List Bool e : Vector.toList (enc a) = List.reverseAux l₁ l₂ ⊢ (Tape.move Dir.left)^[List.length l₁] (Tape.mk' (ListBlank.append l₁ L'✝) (ListBlank.append l₂ R'✝)) = Tape.mk' L'✝ (ListBlank.append (Vector.toList (enc a)) R'✝) induction' l₁ with b l₁ IH generalizing l₂ case intro.intro.cons Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ R : ListBlank Γ a : Γ L : ListBlank Γ L'✝ R'✝ : ListBlank Bool l₁✝ l₂✝ : List Bool e✝ : Vector.toList (enc a) = List.reverseAux l₁✝ l₂✝ b : Bool l₁ : List Bool IH : ∀ {l₂ : List Bool}, Vector.toList (enc a) = List.reverseAux l₁ l₂ → (Tape.move Dir.left)^[List.length l₁] (Tape.mk' (ListBlank.append l₁ L'✝) (ListBlank.append l₂ R'✝)) = Tape.mk' L'✝ (ListBlank.append (Vector.toList (enc a)) R'✝) l₂ : List Bool e : Vector.toList (enc a) = List.reverseAux (b :: l₁) l₂ ⊢ (Tape.move Dir.left)^[List.length (b :: l₁)] (Tape.mk' (ListBlank.append (b :: l₁) L'✝) (ListBlank.append l₂ R'✝)) = Tape.mk' L'✝ (ListBlank.append (Vector.toList (enc a)) R'✝) simp only [List.length, List.cons_append, iterate_succ_apply] case intro.intro.cons Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ R : ListBlank Γ a : Γ L : ListBlank Γ L'✝ R'✝ : ListBlank Bool l₁✝ l₂✝ : List Bool e✝ : Vector.toList (enc a) = List.reverseAux l₁✝ l₂✝ b : Bool l₁ : List Bool IH : ∀ {l₂ : List Bool}, Vector.toList (enc a) = List.reverseAux l₁ l₂ → (Tape.move Dir.left)^[List.length l₁] (Tape.mk' (ListBlank.append l₁ L'✝) (ListBlank.append l₂ R'✝)) = Tape.mk' L'✝ (ListBlank.append (Vector.toList (enc a)) R'✝) l₂ : List Bool e : Vector.toList (enc a) = List.reverseAux (b :: l₁) l₂ ⊢ (Tape.move Dir.left)^[List.length l₁] (Tape.move Dir.left (Tape.mk' (ListBlank.append (b :: l₁) L'✝) (ListBlank.append l₂ R'✝))) = Tape.mk' L'✝ (ListBlank.append (Vector.toList (enc a)) R'✝) convert IH e case h.e'_2.h.e'_4 Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ R : ListBlank Γ a : Γ L : ListBlank Γ L'✝ R'✝ : ListBlank Bool l₁✝ l₂✝ : List Bool e✝ : Vector.toList (enc a) = List.reverseAux l₁✝ l₂✝ b : Bool l₁ : List Bool IH : ∀ {l₂ : List Bool}, Vector.toList (enc a) = List.reverseAux l₁ l₂ → (Tape.move Dir.left)^[List.length l₁] (Tape.mk' (ListBlank.append l₁ L'✝) (ListBlank.append l₂ R'✝)) = Tape.mk' L'✝ (ListBlank.append (Vector.toList (enc a)) R'✝) l₂ : List Bool e : Vector.toList (enc a) = List.reverseAux (b :: l₁) l₂ ⊢ Tape.move Dir.left (Tape.mk' (ListBlank.append (b :: l₁) L'✝) (ListBlank.append l₂ R'✝)) = Tape.mk' (ListBlank.append l₁ L'✝) (ListBlank.append (b :: l₂) R'✝) simp only [ListBlank.tail_cons, ListBlank.append, Tape.move_left_mk', ListBlank.head_cons] Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ R : ListBlank Γ a : Γ L : ListBlank Γ this : ∀ {L' R' : ListBlank Bool} {l₁ l₂ : List Bool}, Vector.toList (enc a) = List.reverseAux l₁ l₂ → (Tape.move Dir.left)^[List.length l₁] (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) = Tape.mk' L' (ListBlank.append (Vector.toList (enc a)) R') ⊢ (Tape.move Dir.left)^[n] (Tape.mk' (ListBlank.append (List.reverse (Vector.toList (enc a))) (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x))) (_ : ∃ n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default))) (ListBlank.bind R (fun x => Vector.toList (enc x)) (_ : ∃ n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default))) = Tape.mk' (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x))) (_ : ∃ n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default)) (ListBlank.append (Vector.toList (enc a)) (ListBlank.bind R (fun x => Vector.toList (enc x)) (_ : ∃ n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default))) simpa only [List.length_reverse, Vector.toList_length] using this (List.reverse_reverse _).symm case intro.intro.nil Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ R : ListBlank Γ a : Γ L : ListBlank Γ L'✝ R'✝ : ListBlank Bool l₁ l₂✝ : List Bool e✝ : Vector.toList (enc a) = List.reverseAux l₁ l₂✝ l₂ : List Bool e : Vector.toList (enc a) = List.reverseAux [] l₂ ⊢ (Tape.move Dir.left)^[List.length []] (Tape.mk' (ListBlank.append [] L'✝) (ListBlank.append l₂ R'✝)) = Tape.mk' L'✝ (ListBlank.append (Vector.toList (enc a)) R'✝) cases e case intro.intro.nil.refl Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ R : ListBlank Γ a : Γ L : ListBlank Γ L'✝ R'✝ : ListBlank Bool l₁ l₂ : List Bool e : Vector.toList (enc a) = List.reverseAux l₁ l₂ ⊢ (Tape.move Dir.left)^[List.length []] (Tape.mk' (ListBlank.append [] L'✝) (ListBlank.append (enc a).1 R'✝)) = Tape.mk' L'✝ (ListBlank.append (Vector.toList (enc a)) R'✝) rfl ** Qed
Turing.TM1to1.trTape'_move_right Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ L R : ListBlank Γ ⊢ (Tape.move Dir.right)^[n] (trTape' enc0 L R) = trTape' enc0 (ListBlank.cons (ListBlank.head R) L) (ListBlank.tail R) suffices ∀ i L, (Tape.move Dir.right)^[i] ((Tape.move Dir.left)^[i] L) = L by refine' (Eq.symm _).trans (this n _) simp only [trTape'_move_left, ListBlank.cons_head_tail, ListBlank.head_cons, ListBlank.tail_cons] Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ L R : ListBlank Γ ⊢ ∀ (i : ℕ) (L : Tape Bool), (Tape.move Dir.right)^[i] ((Tape.move Dir.left)^[i] L) = L intro i _ Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ L R : ListBlank Γ i : ℕ L✝ : Tape Bool ⊢ (Tape.move Dir.right)^[i] ((Tape.move Dir.left)^[i] L✝) = L✝ induction' i with i IH case succ Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ L R : ListBlank Γ L✝ : Tape Bool i : ℕ IH : (Tape.move Dir.right)^[i] ((Tape.move Dir.left)^[i] L✝) = L✝ ⊢ (Tape.move Dir.right)^[Nat.succ i] ((Tape.move Dir.left)^[Nat.succ i] L✝) = L✝ rw [iterate_succ_apply, iterate_succ_apply', Tape.move_left_right, IH] Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ L R : ListBlank Γ this : ∀ (i : ℕ) (L : Tape (?m.376782 i)), (Tape.move Dir.right)^[i] ((Tape.move Dir.left)^[i] L) = L ⊢ (Tape.move Dir.right)^[n] (trTape' enc0 L R) = trTape' enc0 (ListBlank.cons (ListBlank.head R) L) (ListBlank.tail R) refine' (Eq.symm _).trans (this n _) Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ L R : ListBlank Γ this : ∀ (i : ℕ) (L : Tape Bool), (Tape.move Dir.right)^[i] ((Tape.move Dir.left)^[i] L) = L ⊢ (Tape.move Dir.right)^[n] ((Tape.move Dir.left)^[n] (trTape' enc0 (ListBlank.cons (ListBlank.head R) L) (ListBlank.tail R))) = (Tape.move Dir.right)^[n] (trTape' enc0 L R) simp only [trTape'_move_left, ListBlank.cons_head_tail, ListBlank.head_cons, ListBlank.tail_cons] case zero Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ L R : ListBlank Γ L✝ : Tape Bool ⊢ (Tape.move Dir.right)^[Nat.zero] ((Tape.move Dir.left)^[Nat.zero] L✝) = L✝ rfl ** Qed
Turing.TM1to1.stepAux_write Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ q : Stmt Bool Λ' σ v : σ a b : Γ L R : ListBlank Γ ⊢ stepAux (write (Vector.toList (enc a)) q) v (trTape' enc0 L (ListBlank.cons b R)) = stepAux q v (trTape' enc0 (ListBlank.cons a L) R) simp only [trTape', ListBlank.cons_bind] Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ q : Stmt Bool Λ' σ v : σ a b : Γ L R : ListBlank Γ ⊢ stepAux (write (Vector.toList (enc a)) q) v (Tape.mk' (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x))) (_ : ∃ n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default)) (ListBlank.append (Vector.toList (enc b)) (ListBlank.bind R (fun x => Vector.toList (enc x)) (_ : ∃ n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))) = stepAux q v (Tape.mk' (ListBlank.append (List.reverse (Vector.toList (enc a))) (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x))) (_ : ∃ n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default))) (ListBlank.bind R (fun x => Vector.toList (enc x)) (_ : ∃ n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default))) suffices ∀ {L' R'} (l₁ l₂ l₂' : List Bool) (_ : l₂'.length = l₂.length), stepAux (write l₂ q) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂' R')) = stepAux q v (Tape.mk' (L'.append (List.reverseAux l₂ l₁)) R') by refine' this [] _ _ ((enc b).2.trans (enc a).2.symm) Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ q : Stmt Bool Λ' σ v : σ a b : Γ L R : ListBlank Γ ⊢ ∀ {L' R' : ListBlank Bool} (l₁ l₂ l₂' : List Bool), List.length l₂' = List.length l₂ → stepAux (write l₂ q) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂' R')) = stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R') clear a b L R Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ q : Stmt Bool Λ' σ v : σ ⊢ ∀ {L' R' : ListBlank Bool} (l₁ l₂ l₂' : List Bool), List.length l₂' = List.length l₂ → stepAux (write l₂ q) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂' R')) = stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R') intro L' R' l₁ l₂ l₂' e Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ q : Stmt Bool Λ' σ v : σ L' R' : ListBlank Bool l₁ l₂ l₂' : List Bool e : List.length l₂' = List.length l₂ ⊢ stepAux (write l₂ q) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂' R')) = stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R') induction' l₂ with a l₂ IH generalizing l₁ l₂' case cons Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ q : Stmt Bool Λ' σ v : σ L' R' : ListBlank Bool l₁✝ l₂✝ l₂'✝ : List Bool e✝ : List.length l₂'✝ = List.length l₂✝ a : Bool l₂ : List Bool IH : ∀ (l₁ l₂' : List Bool), List.length l₂' = List.length l₂ → stepAux (write l₂ q) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂' R')) = stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R') l₁ l₂' : List Bool e : List.length l₂' = List.length (a :: l₂) ⊢ stepAux (write (a :: l₂) q) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂' R')) = stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux (a :: l₂) l₁) L') R') cases' l₂' with b l₂' <;> simp only [List.length_nil, List.length_cons, Nat.succ_inj'] at e case cons.cons Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ q : Stmt Bool Λ' σ v : σ L' R' : ListBlank Bool l₁✝ l₂✝ l₂'✝ : List Bool e✝ : List.length l₂'✝ = List.length l₂✝ a : Bool l₂ : List Bool IH : ∀ (l₁ l₂' : List Bool), List.length l₂' = List.length l₂ → stepAux (write l₂ q) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂' R')) = stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R') l₁ : List Bool b : Bool l₂' : List Bool e : List.length l₂' = List.length l₂ ⊢ stepAux (write (a :: l₂) q) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append (b :: l₂') R')) = stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux (a :: l₂) l₁) L') R') rw [List.reverseAux, ← IH (a :: l₁) l₂' e] case cons.cons Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ q : Stmt Bool Λ' σ v : σ L' R' : ListBlank Bool l₁✝ l₂✝ l₂'✝ : List Bool e✝ : List.length l₂'✝ = List.length l₂✝ a : Bool l₂ : List Bool IH : ∀ (l₁ l₂' : List Bool), List.length l₂' = List.length l₂ → stepAux (write l₂ q) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂' R')) = stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R') l₁ : List Bool b : Bool l₂' : List Bool e : List.length l₂' = List.length l₂ ⊢ stepAux (write (a :: l₂) q) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append (b :: l₂') R')) = stepAux (write l₂ q) v (Tape.mk' (ListBlank.append (a :: l₁) L') (ListBlank.append l₂' R')) simp only [stepAux, ListBlank.append, Tape.write_mk', Tape.move_right_mk', ListBlank.head_cons, ListBlank.tail_cons] Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ q : Stmt Bool Λ' σ v : σ a b : Γ L R : ListBlank Γ this : ∀ {L' R' : ListBlank Bool} (l₁ l₂ l₂' : List Bool), List.length l₂' = List.length l₂ → stepAux (write l₂ q) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂' R')) = stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux l₂ l₁) L') R') ⊢ stepAux (write (Vector.toList (enc a)) q) v (Tape.mk' (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x))) (_ : ∃ n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default)) (ListBlank.append (Vector.toList (enc b)) (ListBlank.bind R (fun x => Vector.toList (enc x)) (_ : ∃ n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default)))) = stepAux q v (Tape.mk' (ListBlank.append (List.reverse (Vector.toList (enc a))) (ListBlank.bind L (fun x => List.reverse (Vector.toList (enc x))) (_ : ∃ n_1, (fun x => List.reverse (Vector.toList (enc x))) default = List.replicate n_1 default))) (ListBlank.bind R (fun x => Vector.toList (enc x)) (_ : ∃ n_1, (fun x => Vector.toList (enc x)) default = List.replicate n_1 default))) refine' this [] _ _ ((enc b).2.trans (enc a).2.symm) case nil Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ q : Stmt Bool Λ' σ v : σ L' R' : ListBlank Bool l₁✝ l₂ l₂'✝ : List Bool e✝ : List.length l₂'✝ = List.length l₂ l₁ l₂' : List Bool e : List.length l₂' = List.length [] ⊢ stepAux (write [] q) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂' R')) = stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux [] l₁) L') R') cases List.length_eq_zero.1 e case nil.refl Γ : Type u_1 inst✝² : Inhabited Γ Λ : Type u_2 inst✝¹ : Inhabited Λ σ : Type u_3 inst✝ : Inhabited σ n : ℕ enc : Γ → Vector Bool n dec : Vector Bool n → Γ enc0 : enc default = Vector.replicate n false M : Λ → Stmt₁ q : Stmt Bool Λ' σ v : σ L' R' : ListBlank Bool l₁✝ l₂ l₂' : List Bool e✝ : List.length l₂' = List.length l₂ l₁ : List Bool e : List.length [] = List.length [] ⊢ stepAux (write [] q) v (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append [] R')) = stepAux q v (Tape.mk' (ListBlank.append (List.reverseAux [] l₁) L') R') rfl ** Qed
Turing.TM2.stmts₁_self K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q : Stmt₂ ⊢ q ∈ stmts₁ q cases q <;> simp only [Finset.mem_insert_self, Finset.mem_singleton_self, stmts₁] ** Qed
Turing.TM2.stmts₁_trans K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁ q₂ : Stmt₂ ⊢ q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ intro h₁₂ q₀ h₀₁ K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁ q₂ : Stmt₂ h₁₂ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₂ h₀₁ : q₀ ∈ stmts₁ q₁ ⊢ q₀ ∈ stmts₁ q₂ induction' q₂ with _ _ q IH _ _ q IH _ _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;> simp only [Finset.mem_insert, Finset.mem_singleton, Finset.mem_union] at h₁₂ case goto K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁ q₂ : Stmt₂ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₂ h₀₁ : q₀ ∈ stmts₁ q₁ a✝ : σ → Λ h₁₂ : q₁ = goto a✝ ⊢ q₀ ∈ {goto a✝} case halt K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁ q₂ : Stmt₂ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₂ h₀₁ : q₀ ∈ stmts₁ q₁ h₁₂ : q₁ = halt ⊢ q₀ ∈ {halt} case goto l => subst h₁₂; exact h₀₁ case halt K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁ q₂ : Stmt₂ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₂ h₀₁ : q₀ ∈ stmts₁ q₁ h₁₂ : q₁ = halt ⊢ q₀ ∈ {halt} case halt => subst h₁₂; exact h₀₁ case load K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁ q₂ : Stmt₂ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₂ h₀₁ : q₀ ∈ stmts₁ q₁ a✝ : σ → σ q : Stmt₂ IH : q₁ ∈ stmts₁ q → q₀ ∈ stmts₁ q h₁₂ : q₁ = load a✝ q ∨ q₁ ∈ stmts₁ q ⊢ q₀ ∈ insert (load a✝ q) (stmts₁ q) case branch K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁ q₂ : Stmt₂ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₂ h₀₁ : q₀ ∈ stmts₁ q₁ a✝² : σ → Bool a✝¹ a✝ : Stmt₂ a_ih✝¹ : q₁ ∈ stmts₁ a✝¹ → q₀ ∈ stmts₁ a✝¹ a_ih✝ : q₁ ∈ stmts₁ a✝ → q₀ ∈ stmts₁ a✝ h₁₂ : q₁ = branch a✝² a✝¹ a✝ ∨ q₁ ∈ stmts₁ a✝¹ ∨ q₁ ∈ stmts₁ a✝ ⊢ q₀ ∈ insert (branch a✝² a✝¹ a✝) (stmts₁ a✝¹ ∪ stmts₁ a✝) case goto K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁ q₂ : Stmt₂ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₂ h₀₁ : q₀ ∈ stmts₁ q₁ a✝ : σ → Λ h₁₂ : q₁ = goto a✝ ⊢ q₀ ∈ {goto a✝} case halt K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁ q₂ : Stmt₂ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₂ h₀₁ : q₀ ∈ stmts₁ q₁ h₁₂ : q₁ = halt ⊢ q₀ ∈ {halt} rcases h₁₂ with (rfl \| h₁₂) case load.inl K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₂ q₀ : Stmt₂ a✝ : σ → σ q : Stmt₂ h₁₂ : load a✝ q ∈ stmts₁ q₂ h₀₁ : q₀ ∈ stmts₁ (load a✝ q) IH : load a✝ q ∈ stmts₁ q → q₀ ∈ stmts₁ q ⊢ q₀ ∈ insert (load a✝ q) (stmts₁ q) unfold stmts₁ at h₀₁ case load.inl K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₂ q₀ : Stmt₂ a✝ : σ → σ q : Stmt₂ h₁₂ : load a✝ q ∈ stmts₁ q₂ IH : load a✝ q ∈ stmts₁ q → q₀ ∈ stmts₁ q h₀₁ : q₀ ∈ insert (load a✝ q) (stmts₁ q) ⊢ q₀ ∈ insert (load a✝ q) (stmts₁ q) exact h₀₁ case load.inr K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁ q₂ : Stmt₂ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₂ h₀₁ : q₀ ∈ stmts₁ q₁ a✝ : σ → σ q : Stmt₂ IH : q₁ ∈ stmts₁ q → q₀ ∈ stmts₁ q h₁₂ : q₁ ∈ stmts₁ q ⊢ q₀ ∈ insert (load a✝ q) (stmts₁ q) exact Finset.mem_insert_of_mem (IH h₁₂) K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁✝ q₂✝ : Stmt₂ h₁₂✝ : q₁✝ ∈ stmts₁ q₂✝ q₀ : Stmt₂ h₀₁ : q₀ ∈ stmts₁ q₁✝ f : σ → Bool q₁ q₂ : Stmt₂ IH₁ : q₁✝ ∈ stmts₁ q₁ → q₀ ∈ stmts₁ q₁ IH₂ : q₁✝ ∈ stmts₁ q₂ → q₀ ∈ stmts₁ q₂ h₁₂ : q₁✝ = branch f q₁ q₂ ∨ q₁✝ ∈ stmts₁ q₁ ∨ q₁✝ ∈ stmts₁ q₂ ⊢ q₀ ∈ insert (branch f q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂) rcases h₁₂ with (rfl \| h₁₂ \| h₁₂) case inl K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₂✝ q₀ : Stmt₂ f : σ → Bool q₁ q₂ : Stmt₂ h₁₂ : branch f q₁ q₂ ∈ stmts₁ q₂✝ h₀₁ : q₀ ∈ stmts₁ (branch f q₁ q₂) IH₁ : branch f q₁ q₂ ∈ stmts₁ q₁ → q₀ ∈ stmts₁ q₁ IH₂ : branch f q₁ q₂ ∈ stmts₁ q₂ → q₀ ∈ stmts₁ q₂ ⊢ q₀ ∈ insert (branch f q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂) unfold stmts₁ at h₀₁ case inl K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₂✝ q₀ : Stmt₂ f : σ → Bool q₁ q₂ : Stmt₂ h₁₂ : branch f q₁ q₂ ∈ stmts₁ q₂✝ IH₁ : branch f q₁ q₂ ∈ stmts₁ q₁ → q₀ ∈ stmts₁ q₁ IH₂ : branch f q₁ q₂ ∈ stmts₁ q₂ → q₀ ∈ stmts₁ q₂ h₀₁ : q₀ ∈ insert (branch f q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂) ⊢ q₀ ∈ insert (branch f q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂) exact h₀₁ case inr.inl K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁✝ q₂✝ : Stmt₂ h₁₂✝ : q₁✝ ∈ stmts₁ q₂✝ q₀ : Stmt₂ h₀₁ : q₀ ∈ stmts₁ q₁✝ f : σ → Bool q₁ q₂ : Stmt₂ IH₁ : q₁✝ ∈ stmts₁ q₁ → q₀ ∈ stmts₁ q₁ IH₂ : q₁✝ ∈ stmts₁ q₂ → q₀ ∈ stmts₁ q₂ h₁₂ : q₁✝ ∈ stmts₁ q₁ ⊢ q₀ ∈ insert (branch f q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂) exact Finset.mem_insert_of_mem (Finset.mem_union_left _ (IH₁ h₁₂)) case inr.inr K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁✝ q₂✝ : Stmt₂ h₁₂✝ : q₁✝ ∈ stmts₁ q₂✝ q₀ : Stmt₂ h₀₁ : q₀ ∈ stmts₁ q₁✝ f : σ → Bool q₁ q₂ : Stmt₂ IH₁ : q₁✝ ∈ stmts₁ q₁ → q₀ ∈ stmts₁ q₁ IH₂ : q₁✝ ∈ stmts₁ q₂ → q₀ ∈ stmts₁ q₂ h₁₂ : q₁✝ ∈ stmts₁ q₂ ⊢ q₀ ∈ insert (branch f q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂) exact Finset.mem_insert_of_mem (Finset.mem_union_right _ (IH₂ h₁₂)) K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁ q₂ : Stmt₂ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₂ h₀₁ : q₀ ∈ stmts₁ q₁ l : σ → Λ h₁₂ : q₁ = goto l ⊢ q₀ ∈ {goto l} subst h₁₂ K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₂ q₀ : Stmt₂ l : σ → Λ h₁₂ : goto l ∈ stmts₁ q₂ h₀₁ : q₀ ∈ stmts₁ (goto l) ⊢ q₀ ∈ {goto l} exact h₀₁ K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₁ q₂ : Stmt₂ h₁₂✝ : q₁ ∈ stmts₁ q₂ q₀ : Stmt₂ h₀₁ : q₀ ∈ stmts₁ q₁ h₁₂ : q₁ = halt ⊢ q₀ ∈ {halt} subst h₁₂ K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 q₂ q₀ : Stmt₂ h₁₂ : halt ∈ stmts₁ q₂ h₀₁ : q₀ ∈ stmts₁ halt ⊢ q₀ ∈ {halt} exact h₀₁ ** Qed
Turing.TM2.stmts₁_supportsStmt_mono K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h : q₁ ∈ stmts₁ q₂ hs : SupportsStmt S q₂ ⊢ SupportsStmt S q₁ induction' q₂ with _ _ q IH _ _ q IH _ _ q IH _ q IH <;> simp only [stmts₁, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h hs case push K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ k✝ : K a✝ : σ → Γ k✝ q : Stmt₂ IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁ hs : SupportsStmt S q h : q₁ = push k✝ a✝ q ∨ q₁ ∈ stmts₁ q ⊢ SupportsStmt S q₁ case peek K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ k✝ : K a✝ : σ → Option (Γ k✝) → σ q : Stmt₂ IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁ hs : SupportsStmt S q h : q₁ = peek k✝ a✝ q ∨ q₁ ∈ stmts₁ q ⊢ SupportsStmt S q₁ case pop K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ k✝ : K a✝ : σ → Option (Γ k✝) → σ q : Stmt₂ IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁ hs : SupportsStmt S q h : q₁ = pop k✝ a✝ q ∨ q₁ ∈ stmts₁ q ⊢ SupportsStmt S q₁ case load K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝ : σ → σ q : Stmt₂ IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁ hs : SupportsStmt S q h : q₁ = load a✝ q ∨ q₁ ∈ stmts₁ q ⊢ SupportsStmt S q₁ case branch K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝² : σ → Bool a✝¹ a✝ : Stmt₂ a_ih✝¹ : q₁ ∈ stmts₁ a✝¹ → SupportsStmt S a✝¹ → SupportsStmt S q₁ a_ih✝ : q₁ ∈ stmts₁ a✝ → SupportsStmt S a✝ → SupportsStmt S q₁ hs : SupportsStmt S a✝¹ ∧ SupportsStmt S a✝ h : q₁ = branch a✝² a✝¹ a✝ ∨ q₁ ∈ stmts₁ a✝¹ ∨ q₁ ∈ stmts₁ a✝ ⊢ SupportsStmt S q₁ case goto K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝ : σ → Λ hs : ∀ (v : σ), a✝ v ∈ S h : q₁ = goto a✝ ⊢ SupportsStmt S q₁ case halt K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ hs : True h : q₁ = halt ⊢ SupportsStmt S q₁ iterate 4 rcases h with (rfl \| h) <;> [exact hs; exact IH h hs] case branch K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝² : σ → Bool a✝¹ a✝ : Stmt₂ a_ih✝¹ : q₁ ∈ stmts₁ a✝¹ → SupportsStmt S a✝¹ → SupportsStmt S q₁ a_ih✝ : q₁ ∈ stmts₁ a✝ → SupportsStmt S a✝ → SupportsStmt S q₁ hs : SupportsStmt S a✝¹ ∧ SupportsStmt S a✝ h : q₁ = branch a✝² a✝¹ a✝ ∨ q₁ ∈ stmts₁ a✝¹ ∨ q₁ ∈ stmts₁ a✝ ⊢ SupportsStmt S q₁ case goto K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝ : σ → Λ hs : ∀ (v : σ), a✝ v ∈ S h : q₁ = goto a✝ ⊢ SupportsStmt S q₁ case halt K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ hs : True h : q₁ = halt ⊢ SupportsStmt S q₁ case branch f q₁ q₂ IH₁ IH₂ => rcases h with (rfl \| h \| h); exacts [hs, IH₁ h hs.1, IH₂ h hs.2] case goto K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝ : σ → Λ hs : ∀ (v : σ), a✝ v ∈ S h : q₁ = goto a✝ ⊢ SupportsStmt S q₁ case halt K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ hs : True h : q₁ = halt ⊢ SupportsStmt S q₁ case goto l => subst h; exact hs case halt K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ hs : True h : q₁ = halt ⊢ SupportsStmt S q₁ case halt => subst h; trivial case load K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝ : σ → σ q : Stmt₂ IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁ hs : SupportsStmt S q h : q₁ = load a✝ q ∨ q₁ ∈ stmts₁ q ⊢ SupportsStmt S q₁ case branch K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝² : σ → Bool a✝¹ a✝ : Stmt₂ a_ih✝¹ : q₁ ∈ stmts₁ a✝¹ → SupportsStmt S a✝¹ → SupportsStmt S q₁ a_ih✝ : q₁ ∈ stmts₁ a✝ → SupportsStmt S a✝ → SupportsStmt S q₁ hs : SupportsStmt S a✝¹ ∧ SupportsStmt S a✝ h : q₁ = branch a✝² a✝¹ a✝ ∨ q₁ ∈ stmts₁ a✝¹ ∨ q₁ ∈ stmts₁ a✝ ⊢ SupportsStmt S q₁ case goto K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ a✝ : σ → Λ hs : ∀ (v : σ), a✝ v ∈ S h : q₁ = goto a✝ ⊢ SupportsStmt S q₁ case halt K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ hs : True h : q₁ = halt ⊢ SupportsStmt S q₁ rcases h with (rfl \| h) <;> [exact hs; exact IH h hs] K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁✝ q₂✝ : Stmt₂ h✝ : q₁✝ ∈ stmts₁ q₂✝ hs✝ : SupportsStmt S q₂✝ f : σ → Bool q₁ q₂ : Stmt₂ IH₁ : q₁✝ ∈ stmts₁ q₁ → SupportsStmt S q₁ → SupportsStmt S q₁✝ IH₂ : q₁✝ ∈ stmts₁ q₂ → SupportsStmt S q₂ → SupportsStmt S q₁✝ hs : SupportsStmt S q₁ ∧ SupportsStmt S q₂ h : q₁✝ = branch f q₁ q₂ ∨ q₁✝ ∈ stmts₁ q₁ ∨ q₁✝ ∈ stmts₁ q₂ ⊢ SupportsStmt S q₁✝ rcases h with (rfl \| h \| h) case inl K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₂✝ : Stmt₂ hs✝ : SupportsStmt S q₂✝ f : σ → Bool q₁ q₂ : Stmt₂ hs : SupportsStmt S q₁ ∧ SupportsStmt S q₂ h : branch f q₁ q₂ ∈ stmts₁ q₂✝ IH₁ : branch f q₁ q₂ ∈ stmts₁ q₁ → SupportsStmt S q₁ → SupportsStmt S (branch f q₁ q₂) IH₂ : branch f q₁ q₂ ∈ stmts₁ q₂ → SupportsStmt S q₂ → SupportsStmt S (branch f q₁ q₂) ⊢ SupportsStmt S (branch f q₁ q₂) case inr.inl K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁✝ q₂✝ : Stmt₂ h✝ : q₁✝ ∈ stmts₁ q₂✝ hs✝ : SupportsStmt S q₂✝ f : σ → Bool q₁ q₂ : Stmt₂ IH₁ : q₁✝ ∈ stmts₁ q₁ → SupportsStmt S q₁ → SupportsStmt S q₁✝ IH₂ : q₁✝ ∈ stmts₁ q₂ → SupportsStmt S q₂ → SupportsStmt S q₁✝ hs : SupportsStmt S q₁ ∧ SupportsStmt S q₂ h : q₁✝ ∈ stmts₁ q₁ ⊢ SupportsStmt S q₁✝ case inr.inr K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁✝ q₂✝ : Stmt₂ h✝ : q₁✝ ∈ stmts₁ q₂✝ hs✝ : SupportsStmt S q₂✝ f : σ → Bool q₁ q₂ : Stmt₂ IH₁ : q₁✝ ∈ stmts₁ q₁ → SupportsStmt S q₁ → SupportsStmt S q₁✝ IH₂ : q₁✝ ∈ stmts₁ q₂ → SupportsStmt S q₂ → SupportsStmt S q₁✝ hs : SupportsStmt S q₁ ∧ SupportsStmt S q₂ h : q₁✝ ∈ stmts₁ q₂ ⊢ SupportsStmt S q₁✝ exacts [hs, IH₁ h hs.1, IH₂ h hs.2] K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ l : σ → Λ hs : ∀ (v : σ), l v ∈ S h : q₁ = goto l ⊢ SupportsStmt S q₁ subst h K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₂ : Stmt₂ hs✝ : SupportsStmt S q₂ l : σ → Λ hs : ∀ (v : σ), l v ∈ S h : goto l ∈ stmts₁ q₂ ⊢ SupportsStmt S (goto l) exact hs K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₁ q₂ : Stmt₂ h✝ : q₁ ∈ stmts₁ q₂ hs✝ : SupportsStmt S q₂ hs : True h : q₁ = halt ⊢ SupportsStmt S q₁ subst h K : Type u_1 inst✝ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 S : Finset Λ q₂ : Stmt₂ hs✝ : SupportsStmt S q₂ hs : True h : halt ∈ stmts₁ q₂ ⊢ SupportsStmt S halt trivial ** Qed
Turing.TM2.step_supports K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v : σ T : (k : K) → List (Γ k) c' : Cfg₂ h₁ : c' ∈ step M { l := some l₁, var := v, stk := T } h₂ : { l := some l₁, var := v, stk := T }.l ∈ ↑Finset.insertNone S ⊢ c'.l ∈ ↑Finset.insertNone S replace h₂ := ss.2 _ (Finset.some_mem_insertNone.1 h₂) K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v : σ T : (k : K) → List (Γ k) c' : Cfg₂ h₁ : c' ∈ step M { l := some l₁, var := v, stk := T } h₂ : SupportsStmt S (M l₁) ⊢ c'.l ∈ ↑Finset.insertNone S simp only [step, Option.mem_def, Option.some.injEq] at h₁ K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v : σ T : (k : K) → List (Γ k) c' : Cfg₂ h₂ : SupportsStmt S (M l₁) h₁ : stepAux (M l₁) v T = c' ⊢ c'.l ∈ ↑Finset.insertNone S subst c' K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v : σ T : (k : K) → List (Γ k) h₂ : SupportsStmt S (M l₁) ⊢ (stepAux (M l₁) v T).l ∈ ↑Finset.insertNone S revert h₂ K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v : σ T : (k : K) → List (Γ k) ⊢ SupportsStmt S (M l₁) → (stepAux (M l₁) v T).l ∈ ↑Finset.insertNone S induction' M l₁ with _ _ q IH _ _ q IH _ _ q IH _ q IH generalizing v T <;> intro hs case push K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) k✝ : K a✝ : σ → Γ k✝ q : Stmt₂ IH : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S q → (stepAux q v T).l ∈ ↑Finset.insertNone S v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (push k✝ a✝ q) ⊢ (stepAux (push k✝ a✝ q) v T).l ∈ ↑Finset.insertNone S case peek K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) k✝ : K a✝ : σ → Option (Γ k✝) → σ q : Stmt₂ IH : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S q → (stepAux q v T).l ∈ ↑Finset.insertNone S v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (peek k✝ a✝ q) ⊢ (stepAux (peek k✝ a✝ q) v T).l ∈ ↑Finset.insertNone S case pop K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) k✝ : K a✝ : σ → Option (Γ k✝) → σ q : Stmt₂ IH : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S q → (stepAux q v T).l ∈ ↑Finset.insertNone S v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (pop k✝ a✝ q) ⊢ (stepAux (pop k✝ a✝ q) v T).l ∈ ↑Finset.insertNone S case load K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) a✝ : σ → σ q : Stmt₂ IH : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S q → (stepAux q v T).l ∈ ↑Finset.insertNone S v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (load a✝ q) ⊢ (stepAux (load a✝ q) v T).l ∈ ↑Finset.insertNone S case branch K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) a✝² : σ → Bool a✝¹ a✝ : Stmt₂ a_ih✝¹ : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S a✝¹ → (stepAux a✝¹ v T).l ∈ ↑Finset.insertNone S a_ih✝ : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S a✝ → (stepAux a✝ v T).l ∈ ↑Finset.insertNone S v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (branch a✝² a✝¹ a✝) ⊢ (stepAux (branch a✝² a✝¹ a✝) v T).l ∈ ↑Finset.insertNone S case goto K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) a✝ : σ → Λ v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (goto a✝) ⊢ (stepAux (goto a✝) v T).l ∈ ↑Finset.insertNone S case halt K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S halt ⊢ (stepAux halt v T).l ∈ ↑Finset.insertNone S iterate 4 exact IH _ _ hs case goto K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) a✝ : σ → Λ v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (goto a✝) ⊢ (stepAux (goto a✝) v T).l ∈ ↑Finset.insertNone S case halt K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S halt ⊢ (stepAux halt v T).l ∈ ↑Finset.insertNone S case goto => exact Finset.some_mem_insertNone.2 (hs _) case halt K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S halt ⊢ (stepAux halt v T).l ∈ ↑Finset.insertNone S case halt => apply Multiset.mem_cons_self case load K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) a✝ : σ → σ q : Stmt₂ IH : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S q → (stepAux q v T).l ∈ ↑Finset.insertNone S v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (load a✝ q) ⊢ (stepAux (load a✝ q) v T).l ∈ ↑Finset.insertNone S case branch K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) a✝² : σ → Bool a✝¹ a✝ : Stmt₂ a_ih✝¹ : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S a✝¹ → (stepAux a✝¹ v T).l ∈ ↑Finset.insertNone S a_ih✝ : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S a✝ → (stepAux a✝ v T).l ∈ ↑Finset.insertNone S v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (branch a✝² a✝¹ a✝) ⊢ (stepAux (branch a✝² a✝¹ a✝) v T).l ∈ ↑Finset.insertNone S case goto K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) a✝ : σ → Λ v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (goto a✝) ⊢ (stepAux (goto a✝) v T).l ∈ ↑Finset.insertNone S case halt K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S halt ⊢ (stepAux halt v T).l ∈ ↑Finset.insertNone S exact IH _ _ hs K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) p : σ → Bool q₁' q₂' : Stmt₂ IH₁ : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S q₁' → (stepAux q₁' v T).l ∈ ↑Finset.insertNone S IH₂ : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S q₂' → (stepAux q₂' v T).l ∈ ↑Finset.insertNone S v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (branch p q₁' q₂') ⊢ (stepAux (branch p q₁' q₂') v T).l ∈ ↑Finset.insertNone S unfold stepAux K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) p : σ → Bool q₁' q₂' : Stmt₂ IH₁ : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S q₁' → (stepAux q₁' v T).l ∈ ↑Finset.insertNone S IH₂ : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S q₂' → (stepAux q₂' v T).l ∈ ↑Finset.insertNone S v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (branch p q₁' q₂') ⊢ (bif p v then stepAux q₁' v T else stepAux q₂' v T).l ∈ ↑Finset.insertNone S cases p v case false K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) p : σ → Bool q₁' q₂' : Stmt₂ IH₁ : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S q₁' → (stepAux q₁' v T).l ∈ ↑Finset.insertNone S IH₂ : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S q₂' → (stepAux q₂' v T).l ∈ ↑Finset.insertNone S v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (branch p q₁' q₂') ⊢ (bif false then stepAux q₁' v T else stepAux q₂' v T).l ∈ ↑Finset.insertNone S exact IH₂ _ _ hs.2 case true K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) p : σ → Bool q₁' q₂' : Stmt₂ IH₁ : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S q₁' → (stepAux q₁' v T).l ∈ ↑Finset.insertNone S IH₂ : ∀ (v : σ) (T : (k : K) → List (Γ k)), SupportsStmt S q₂' → (stepAux q₂' v T).l ∈ ↑Finset.insertNone S v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (branch p q₁' q₂') ⊢ (bif true then stepAux q₁' v T else stepAux q₂' v T).l ∈ ↑Finset.insertNone S exact IH₁ _ _ hs.1 K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) a✝ : σ → Λ v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S (goto a✝) ⊢ (stepAux (goto a✝) v T).l ∈ ↑Finset.insertNone S exact Finset.some_mem_insertNone.2 (hs _) K : Type u_1 inst✝¹ : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 σ : Type u_4 inst✝ : Inhabited Λ M : Λ → Stmt₂ S : Finset Λ ss : Supports M S l₁ : Λ v✝ : σ T✝ : (k : K) → List (Γ k) v : σ T : (k : K) → List (Γ k) hs : SupportsStmt S halt ⊢ (stepAux halt v T).l ∈ ↑Finset.insertNone S apply Multiset.mem_cons_self ** Qed
Turing.TM2to1.stk_nth_val K : Type u_1 Γ : K → Type u_2 L : ListBlank ((k : K) → Option (Γ k)) k : K S : List (Γ k) n : ℕ hL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S)) ⊢ ListBlank.nth L n k = List.get? (List.reverse S) n rw [← proj_map_nth, hL, ← List.map_reverse, ListBlank.nth_mk, List.getI_eq_iget_get?, List.get?_map] K : Type u_1 Γ : K → Type u_2 L : ListBlank ((k : K) → Option (Γ k)) k : K S : List (Γ k) n : ℕ hL : ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some S)) ⊢ Option.iget (Option.map some (List.get? (List.reverse S) n)) = List.get? (List.reverse S) n cases S.reverse.get? n <;> rfl ** Qed
Turing.TM2to1.addBottom_map K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ L : ListBlank ((k : K) → Option (Γ k)) ⊢ default.2 = default rfl K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ L : ListBlank ((k : K) → Option (Γ k)) ⊢ ListBlank.map { f := Prod.snd, map_pt' := (_ : default.2 = default.2) } (addBottom L) = L simp only [addBottom, ListBlank.map_cons] K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ L : ListBlank ((k : K) → Option (Γ k)) ⊢ ListBlank.cons (ListBlank.head L) (ListBlank.map { f := Prod.snd, map_pt' := (_ : default.2 = default.2) } (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } (ListBlank.tail L))) = L convert ListBlank.cons_head_tail L case h.e'_2.h.e'_4 K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ L : ListBlank ((k : K) → Option (Γ k)) ⊢ ListBlank.map { f := Prod.snd, map_pt' := (_ : default.2 = default.2) } (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } (ListBlank.tail L)) = ListBlank.tail L generalize ListBlank.tail L = L' case h.e'_2.h.e'_4 K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ L L' : ListBlank ((k : K) → Option (Γ k)) ⊢ ListBlank.map { f := Prod.snd, map_pt' := (_ : default.2 = default.2) } (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } L') = L' refine' L'.induction_on fun l ↦ _ case h.e'_2.h.e'_4 K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ L L' : ListBlank ((k : K) → Option (Γ k)) l : List ((k : K) → Option (Γ k)) ⊢ ListBlank.map { f := Prod.snd, map_pt' := (_ : default.2 = default.2) } (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } (ListBlank.mk l)) = ListBlank.mk l simp ** Qed
Turing.TM2to1.addBottom_nth_succ_fst K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ L : ListBlank ((k : K) → Option (Γ k)) n : ℕ ⊢ (ListBlank.nth (addBottom L) (n + 1)).1 = false rw [ListBlank.nth_succ, addBottom, ListBlank.tail_cons, ListBlank.nth_map] ** Qed
Turing.TM2to1.addBottom_head_fst K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ L : ListBlank ((k : K) → Option (Γ k)) ⊢ (ListBlank.head (addBottom L)).1 = true rw [addBottom, ListBlank.head_cons] ** Qed
Turing.TM2to1.supports_run K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ S : Finset Λ k : K s : StAct k q : Stmt₂ ⊢ TM2.SupportsStmt S (stRun s q) ↔ TM2.SupportsStmt S q cases s <;> rfl ** Qed
Turing.TM2to1.trStmts₁_run K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K s : StAct k q : Stmt₂ ⊢ trStmts₁ (stRun s q) = {go k s q, ret q} ∪ trStmts₁ q cases s <;> simp only [trStmts₁] ** Qed
Turing.TM2to1.tr_respects_aux₂ K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) o : StAct k ⊢ let v' := stVar v (S k) o; let Sk' := stWrite v (S k) o; let S' := update S k Sk'; ∃ L', (∀ (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S' k)))) ∧ TM1.stepAux (trStAct q o) v ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))) = TM1.stepAux q v' ((Tape.move Dir.right)^[List.length (S' k)] (Tape.mk' ∅ (addBottom L'))) dsimp only K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) o : StAct k ⊢ ∃ L', (∀ (k_1 : K), ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (stWrite v (S k) o) k_1)))) ∧ TM1.stepAux (trStAct q o) v ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))) = TM1.stepAux q (stVar v (S k) o) ((Tape.move Dir.right)^[List.length (update S k (stWrite v (S k) o) k)] (Tape.mk' ∅ (addBottom L'))) simp K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) o : StAct k ⊢ ∃ L', (∀ (k_1 : K), ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (stWrite v (S k) o) k_1)))) ∧ TM1.stepAux (trStAct q o) v ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))) = TM1.stepAux q (stVar v (S k) o) ((Tape.move Dir.right)^[List.length (stWrite v (S k) o)] (Tape.mk' ∅ (addBottom L'))) cases o <;> simp only [stWrite, stVar, trStAct, TM1.stepAux] K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k ⊢ ∃ L', (∀ (k_1 : K), ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k_1)))) ∧ TM1.stepAux q v (Tape.move Dir.right (Tape.write (((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))).head.1, update ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))).head.2 k (some (f v))) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))))) = TM1.stepAux q v ((Tape.move Dir.right)^[List.length (f v :: S k)] (Tape.mk' ∅ (addBottom L'))) have := Tape.write_move_right_n fun a : Γ' ↦ (a.1, update a.2 k (some (f v))) K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) ⊢ ∃ L', (∀ (k_1 : K), ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k_1)))) ∧ TM1.stepAux q v (Tape.move Dir.right (Tape.write (((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))).head.1, update ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))).head.2 k (some (f v))) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))))) = TM1.stepAux q v ((Tape.move Dir.right)^[List.length (f v :: S k)] (Tape.mk' ∅ (addBottom L'))) refine' ⟨_, fun k' ↦ _, by erw [Tape.move_right_n_head, List.length, Tape.mk'_nth_nat, this, addBottom_modifyNth fun a ↦ update a k (some (f v)), Nat.add_one, iterate_succ'] rfl⟩ K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) k' : K ⊢ ListBlank.map (proj k') (ListBlank.modifyNth (fun a => update a k (some (f v))) (List.length (S k)) L) = ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k'))) refine' ListBlank.ext fun i ↦ _ K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) k' : K i : ℕ ⊢ ListBlank.nth (ListBlank.map (proj k') (ListBlank.modifyNth (fun a => update a k (some (f v))) (List.length (S k)) L)) i = ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k')))) i rw [ListBlank.nth_map, ListBlank.nth_modifyNth, proj, PointedMap.mk_val] K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) k' : K i : ℕ ⊢ ite (i = List.length (S k)) (update (ListBlank.nth L i) k (some (f v))) (ListBlank.nth L i) k' = ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k')))) i by_cases h' : k' = k K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) ⊢ TM1.stepAux q v (Tape.move Dir.right (Tape.write (((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))).head.1, update ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))).head.2 k (some (f v))) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))))) = TM1.stepAux q v ((Tape.move Dir.right)^[List.length (f v :: S k)] (Tape.mk' ∅ (addBottom ?m.673260))) erw [Tape.move_right_n_head, List.length, Tape.mk'_nth_nat, this, addBottom_modifyNth fun a ↦ update a k (some (f v)), Nat.add_one, iterate_succ'] K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) ⊢ TM1.stepAux q v (Tape.move Dir.right ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom (ListBlank.modifyNth (fun a => update a k (some (f v))) (List.length (S k)) L))))) = TM1.stepAux q v ((Tape.move Dir.right ∘ (Tape.move Dir.right)^[List.length (S k)]) (Tape.mk' ∅ (addBottom ?m.673260))) K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) ⊢ ListBlank ((k : K) → Option (Γ k)) K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) ⊢ ListBlank ((k : K) → Option (Γ k)) K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) ⊢ ListBlank ((k : K) → Option (Γ k)) K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) ⊢ ListBlank ((k : K) → Option (Γ k)) K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) ⊢ ListBlank ((k : K) → Option (Γ k)) K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) ⊢ ListBlank ((k : K) → Option (Γ k)) K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) ⊢ ListBlank ((k : K) → Option (Γ k)) rfl case pos K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) k' : K i : ℕ h' : k' = k ⊢ ite (i = List.length (S k)) (update (ListBlank.nth L i) k (some (f v))) (ListBlank.nth L i) k' = ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k')))) i subst k' case pos K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) i : ℕ ⊢ ite (i = List.length (S k)) (update (ListBlank.nth L i) k (some (f v))) (ListBlank.nth L i) k = ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k)))) i split_ifs with h <;> simp only [List.reverse_cons, Function.update_same, ListBlank.nth_mk, List.map] case neg K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) i : ℕ h : ¬i = List.length (S k) ⊢ ListBlank.nth L i k = List.getI (List.reverse (List.map some (S k)) ++ [some (f v)]) i rw [← proj_map_nth, hL, ListBlank.nth_mk] case neg K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) i : ℕ h : ¬i = List.length (S k) ⊢ List.getI (List.reverse (List.map some (S k))) i = List.getI (List.reverse (List.map some (S k)) ++ [some (f v)]) i cases' lt_or_gt_of_ne h with h h case pos K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) i : ℕ h : i = List.length (S k) ⊢ some (f v) = List.getI (List.reverse (List.map some (S k)) ++ [some (f v)]) i rw [List.getI_eq_get, List.get_append_right'] <;> simp only [h, List.get_singleton, List.length_map, List.length_reverse, Nat.succ_pos', List.length_append, lt_add_iff_pos_right, List.length, le_refl] case neg.inl K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) i : ℕ h✝ : ¬i = List.length (S k) h : i < List.length (S k) ⊢ List.getI (List.reverse (List.map some (S k))) i = List.getI (List.reverse (List.map some (S k)) ++ [some (f v)]) i rw [List.getI_append] case neg.inl.h K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) i : ℕ h✝ : ¬i = List.length (S k) h : i < List.length (S k) ⊢ i < List.length (List.reverse (List.map some (S k))) simpa only [List.length_map, List.length_reverse] using h case neg.inr K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) i : ℕ h✝ : ¬i = List.length (S k) h : i > List.length (S k) ⊢ List.getI (List.reverse (List.map some (S k))) i = List.getI (List.reverse (List.map some (S k)) ++ [some (f v)]) i rw [gt_iff_lt] at h case neg.inr K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) i : ℕ h✝ : ¬i = List.length (S k) h : List.length (S k) < i ⊢ List.getI (List.reverse (List.map some (S k))) i = List.getI (List.reverse (List.map some (S k)) ++ [some (f v)]) i rw [List.getI_eq_default, List.getI_eq_default] <;> simp only [Nat.add_one_le_iff, h, List.length, le_of_lt, List.length_reverse, List.length_append, List.length_map] case neg K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) k' : K i : ℕ h' : ¬k' = k ⊢ ite (i = List.length (S k)) (update (ListBlank.nth L i) k (some (f v))) (ListBlank.nth L i) k' = ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k')))) i split_ifs <;> rw [Function.update_noteq h', ← proj_map_nth, hL] case pos K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Γ k this : ∀ (L R : ListBlank Γ') (n : ℕ), Tape.write ((ListBlank.nth R n).1, update (ListBlank.nth R n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) = (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R)) k' : K i : ℕ h' : ¬k' = k h✝ : i = List.length (S k) ⊢ ListBlank.nth (ListBlank.mk (List.reverse (List.map some (S k')))) i = ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (f v :: S k) k')))) i rw [Function.update_noteq h'] K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ ⊢ ∃ L', (∀ (k_1 : K), ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (S k) k_1)))) ∧ TM1.stepAux q (f v ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L)))).head.2 k)) (Tape.move Dir.right (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))))) = TM1.stepAux q (f v (List.head? (S k))) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L'))) rw [Function.update_eq_self] K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ ⊢ ∃ L', (∀ (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S k)))) ∧ TM1.stepAux q (f v ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L)))).head.2 k)) (Tape.move Dir.right (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))))) = TM1.stepAux q (f v (List.head? (S k))) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L'))) use L, hL case right K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ ⊢ TM1.stepAux q (f v ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L)))).head.2 k)) (Tape.move Dir.right (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))))) = TM1.stepAux q (f v (List.head? (S k))) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))) rw [Tape.move_left_right] case right K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ ⊢ TM1.stepAux q (f v ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L)))).head.2 k)) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))) = TM1.stepAux q (f v (List.head? (S k))) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))) congr case right.e_a.e_a K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ ⊢ (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L)))).head.2 k = List.head? (S k) cases e : S k case right.e_a.e_a.cons K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ head✝ : Γ k tail✝ : List (Γ k) e : S k = head✝ :: tail✝ ⊢ (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (head✝ :: tail✝)] (Tape.mk' ∅ (addBottom L)))).head.2 k = List.head? (head✝ :: tail✝) rw [List.length_cons, iterate_succ', Function.comp, Tape.move_right_left, Tape.move_right_n_head, Tape.mk'_nth_nat, addBottom_nth_snd, stk_nth_val _ (hL k), e, List.reverse_cons, ← List.length_reverse, List.get?_concat_length] case right.e_a.e_a.cons K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ head✝ : Γ k tail✝ : List (Γ k) e : S k = head✝ :: tail✝ ⊢ some head✝ = List.head? (head✝ :: tail✝) rfl case right.e_a.e_a.nil K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ e : S k = [] ⊢ (Tape.move Dir.left ((Tape.move Dir.right)^[List.length []] (Tape.mk' ∅ (addBottom L)))).head.2 k = List.head? [] rfl K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ ⊢ ∃ L', (∀ (k_1 : K), ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (List.tail (S k)) k_1)))) ∧ (bif ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))).head.1 then TM1.stepAux q (f v none) ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L))) else TM1.stepAux q (f v ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L)))).head.2 k)) (Tape.write ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L)))).head.1, update (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L)))).head.2 k none) (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (S k)] (Tape.mk' ∅ (addBottom L)))))) = TM1.stepAux q (f v (List.head? (S k))) ((Tape.move Dir.right)^[List.length (List.tail (S k))] (Tape.mk' ∅ (addBottom L'))) cases' e : S k with hd tl case nil K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ e : S k = [] ⊢ ∃ L', (∀ (k_1 : K), ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (List.tail []) k_1)))) ∧ (bif ((Tape.move Dir.right)^[List.length []] (Tape.mk' ∅ (addBottom L))).head.1 then TM1.stepAux q (f v none) ((Tape.move Dir.right)^[List.length []] (Tape.mk' ∅ (addBottom L))) else TM1.stepAux q (f v ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length []] (Tape.mk' ∅ (addBottom L)))).head.2 k)) (Tape.write ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length []] (Tape.mk' ∅ (addBottom L)))).head.1, update (Tape.move Dir.left ((Tape.move Dir.right)^[List.length []] (Tape.mk' ∅ (addBottom L)))).head.2 k none) (Tape.move Dir.left ((Tape.move Dir.right)^[List.length []] (Tape.mk' ∅ (addBottom L)))))) = TM1.stepAux q (f v (List.head? [])) ((Tape.move Dir.right)^[List.length (List.tail [])] (Tape.mk' ∅ (addBottom L'))) simp only [Tape.mk'_head, ListBlank.head_cons, Tape.move_left_mk', List.length, Tape.write_mk', List.head?, iterate_zero_apply, List.tail_nil] case nil K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ e : S k = [] ⊢ ∃ L', (∀ (k_1 : K), ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k [] k_1)))) ∧ (bif (ListBlank.head (addBottom L)).1 then TM1.stepAux q (f v none) (Tape.mk' ∅ (addBottom L)) else TM1.stepAux q (f v ((ListBlank.head ∅).2 k)) (Tape.mk' (ListBlank.tail ∅) (ListBlank.cons ((ListBlank.head ∅).1, update (ListBlank.head ∅).2 k none) (addBottom L)))) = TM1.stepAux q (f v none) (Tape.mk' ∅ (addBottom L')) rw [← e, Function.update_eq_self] case nil K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ e : S k = [] ⊢ ∃ L', (∀ (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S k)))) ∧ (bif (ListBlank.head (addBottom L)).1 then TM1.stepAux q (f v none) (Tape.mk' ∅ (addBottom L)) else TM1.stepAux q (f v ((ListBlank.head ∅).2 k)) (Tape.mk' (ListBlank.tail ∅) (ListBlank.cons ((ListBlank.head ∅).1, update (ListBlank.head ∅).2 k none) (addBottom L)))) = TM1.stepAux q (f v none) (Tape.mk' ∅ (addBottom L')) exact ⟨L, hL, by rw [addBottom_head_fst, cond]⟩ K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ e : S k = [] ⊢ (bif (ListBlank.head (addBottom L)).1 then TM1.stepAux q (f v none) (Tape.mk' ∅ (addBottom L)) else TM1.stepAux q (f v ((ListBlank.head ∅).2 k)) (Tape.mk' (ListBlank.tail ∅) (ListBlank.cons ((ListBlank.head ∅).1, update (ListBlank.head ∅).2 k none) (addBottom L)))) = TM1.stepAux q (f v none) (Tape.mk' ∅ (addBottom L)) rw [addBottom_head_fst, cond] case cons K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl ⊢ ∃ L', (∀ (k_1 : K), ListBlank.map (proj k_1) L' = ListBlank.mk (List.reverse (List.map some (update S k (List.tail (hd :: tl)) k_1)))) ∧ (bif ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' ∅ (addBottom L))).head.1 then TM1.stepAux q (f v none) ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' ∅ (addBottom L))) else TM1.stepAux q (f v ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' ∅ (addBottom L)))).head.2 k)) (Tape.write ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' ∅ (addBottom L)))).head.1, update (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' ∅ (addBottom L)))).head.2 k none) (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' ∅ (addBottom L)))))) = TM1.stepAux q (f v (List.head? (hd :: tl))) ((Tape.move Dir.right)^[List.length (List.tail (hd :: tl))] (Tape.mk' ∅ (addBottom L'))) refine' ⟨_, fun k' ↦ _, by erw [List.length_cons, Tape.move_right_n_head, Tape.mk'_nth_nat, addBottom_nth_succ_fst, cond, iterate_succ', Function.comp, Tape.move_right_left, Tape.move_right_n_head, Tape.mk'_nth_nat, Tape.write_move_right_n fun a : Γ' ↦ (a.1, update a.2 k none), addBottom_modifyNth fun a ↦ update a k none, addBottom_nth_snd, stk_nth_val _ (hL k), e, show (List.cons hd tl).reverse.get? tl.length = some hd by rw [List.reverse_cons, ← List.length_reverse, List.get?_concat_length], List.head?, List.tail]⟩ case cons K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl k' : K ⊢ ListBlank.map (proj k') (ListBlank.modifyNth (fun a => update a k none) (List.length tl) L) = ListBlank.mk (List.reverse (List.map some (update S k (List.tail (hd :: tl)) k'))) refine' ListBlank.ext fun i ↦ _ case cons K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl k' : K i : ℕ ⊢ ListBlank.nth (ListBlank.map (proj k') (ListBlank.modifyNth (fun a => update a k none) (List.length tl) L)) i = ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (List.tail (hd :: tl)) k')))) i rw [ListBlank.nth_map, ListBlank.nth_modifyNth, proj, PointedMap.mk_val] case cons K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl k' : K i : ℕ ⊢ ite (i = List.length tl) (update (ListBlank.nth L i) k none) (ListBlank.nth L i) k' = ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (List.tail (hd :: tl)) k')))) i by_cases h' : k' = k K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl ⊢ (bif ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' ∅ (addBottom L))).head.1 then TM1.stepAux q (f v none) ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' ∅ (addBottom L))) else TM1.stepAux q (f v ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' ∅ (addBottom L)))).head.2 k)) (Tape.write ((Tape.move Dir.left ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' ∅ (addBottom L)))).head.1, update (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' ∅ (addBottom L)))).head.2 k none) (Tape.move Dir.left ((Tape.move Dir.right)^[List.length (hd :: tl)] (Tape.mk' ∅ (addBottom L)))))) = TM1.stepAux q (f v (List.head? (hd :: tl))) ((Tape.move Dir.right)^[List.length (List.tail (hd :: tl))] (Tape.mk' ∅ (addBottom ?m.690014))) erw [List.length_cons, Tape.move_right_n_head, Tape.mk'_nth_nat, addBottom_nth_succ_fst, cond, iterate_succ', Function.comp, Tape.move_right_left, Tape.move_right_n_head, Tape.mk'_nth_nat, Tape.write_move_right_n fun a : Γ' ↦ (a.1, update a.2 k none), addBottom_modifyNth fun a ↦ update a k none, addBottom_nth_snd, stk_nth_val _ (hL k), e, show (List.cons hd tl).reverse.get? tl.length = some hd by rw [List.reverse_cons, ← List.length_reverse, List.get?_concat_length], List.head?, List.tail] K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl ⊢ List.get? (List.reverse (hd :: tl)) (List.length tl) = some hd rw [List.reverse_cons, ← List.length_reverse, List.get?_concat_length] case pos K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl k' : K i : ℕ h' : k' = k ⊢ ite (i = List.length tl) (update (ListBlank.nth L i) k none) (ListBlank.nth L i) k' = ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (List.tail (hd :: tl)) k')))) i subst k' case pos K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl i : ℕ ⊢ ite (i = List.length tl) (update (ListBlank.nth L i) k none) (ListBlank.nth L i) k = ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (List.tail (hd :: tl)) k)))) i split_ifs with h <;> simp only [Function.update_same, ListBlank.nth_mk, List.tail] case neg K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl i : ℕ h : ¬i = List.length tl ⊢ ListBlank.nth L i k = List.getI (List.reverse (List.map some tl)) i rw [← proj_map_nth, hL, ListBlank.nth_mk, e, List.map, List.reverse_cons] case neg K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl i : ℕ h : ¬i = List.length tl ⊢ List.getI (List.reverse (List.map some tl) ++ [some hd]) i = List.getI (List.reverse (List.map some tl)) i cases' lt_or_gt_of_ne h with h h case pos K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl i : ℕ h : i = List.length tl ⊢ none = List.getI (List.reverse (List.map some tl)) i rw [List.getI_eq_default] case pos.hn K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl i : ℕ h : i = List.length tl ⊢ List.length (List.reverse (List.map some tl)) ≤ i rw [h, List.length_reverse, List.length_map] case pos K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl i : ℕ h : i = List.length tl ⊢ none = default rfl case neg.inl K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl i : ℕ h✝ : ¬i = List.length tl h : i < List.length tl ⊢ List.getI (List.reverse (List.map some tl) ++ [some hd]) i = List.getI (List.reverse (List.map some tl)) i rw [List.getI_append] case neg.inl.h K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl i : ℕ h✝ : ¬i = List.length tl h : i < List.length tl ⊢ i < List.length (List.reverse (List.map some tl)) simpa only [List.length_map, List.length_reverse] using h case neg.inr K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl i : ℕ h✝ : ¬i = List.length tl h : i > List.length tl ⊢ List.getI (List.reverse (List.map some tl) ++ [some hd]) i = List.getI (List.reverse (List.map some tl)) i rw [gt_iff_lt] at h case neg.inr K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl i : ℕ h✝ : ¬i = List.length tl h : List.length tl < i ⊢ List.getI (List.reverse (List.map some tl) ++ [some hd]) i = List.getI (List.reverse (List.map some tl)) i rw [List.getI_eq_default, List.getI_eq_default] <;> simp only [Nat.add_one_le_iff, h, List.length, le_of_lt, List.length_reverse, List.length_append, List.length_map] case neg K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl k' : K i : ℕ h' : ¬k' = k ⊢ ite (i = List.length tl) (update (ListBlank.nth L i) k none) (ListBlank.nth L i) k' = ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (List.tail (hd :: tl)) k')))) i split_ifs <;> rw [Function.update_noteq h', ← proj_map_nth, hL] case pos K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ k : K q : TM1.Stmt Γ' Λ' σ v : σ S : (k : K) → List (Γ k) L : ListBlank ((k : K) → Option (Γ k)) hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.reverse (List.map some (S k))) f : σ → Option (Γ k) → σ hd : Γ k tl : List (Γ k) e : S k = hd :: tl k' : K i : ℕ h' : ¬k' = k h✝ : i = List.length tl ⊢ ListBlank.nth (ListBlank.mk (List.reverse (List.map some (S k')))) i = ListBlank.nth (ListBlank.mk (List.reverse (List.map some (update S k (List.tail (hd :: tl)) k')))) i rw [Function.update_noteq h'] ** Qed
Turing.TM2to1.tr_eval K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ M : Λ → Stmt₂ k : K L : List (Γ k) L₁ : ListBlank Γ' L₂ : List (Γ k) H₁ : L₁ ∈ TM1.eval (tr M) (trInit k L) H₂ : L₂ ∈ TM2.eval M k L ⊢ ∃ S L', addBottom L' = L₁ ∧ (∀ (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S k)))) ∧ S k = L₂ obtain ⟨c₁, h₁, rfl⟩ := (Part.mem_map_iff _).1 H₁ case intro.intro K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ M : Λ → Stmt₂ k : K L L₂ : List (Γ k) H₂ : L₂ ∈ TM2.eval M k L c₁ : TM1.Cfg Γ' Λ' σ h₁ : c₁ ∈ eval (TM1.step (tr M)) (TM1.init (trInit k L)) H₁ : Tape.right₀ c₁.Tape ∈ TM1.eval (tr M) (trInit k L) ⊢ ∃ S L', addBottom L' = Tape.right₀ c₁.Tape ∧ (∀ (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S k)))) ∧ S k = L₂ obtain ⟨c₂, h₂, rfl⟩ := (Part.mem_map_iff _).1 H₂ case intro.intro.intro.intro K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ M : Λ → Stmt₂ k : K L : List (Γ k) c₁ : TM1.Cfg Γ' Λ' σ h₁ : c₁ ∈ eval (TM1.step (tr M)) (TM1.init (trInit k L)) H₁ : Tape.right₀ c₁.Tape ∈ TM1.eval (tr M) (trInit k L) c₂ : Cfg₂ h₂ : c₂ ∈ eval (TM2.step M) (TM2.init k L) H₂ : TM2.Cfg.stk c₂ k ∈ TM2.eval M k L ⊢ ∃ S L', addBottom L' = Tape.right₀ c₁.Tape ∧ (∀ (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S k)))) ∧ S k = TM2.Cfg.stk c₂ k obtain ⟨_, ⟨L', hT⟩, h₃⟩ := Turing.tr_eval (tr_respects M) (trCfg_init k L) h₂ case intro.intro.intro.intro.intro.intro.mk K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ M : Λ → Stmt₂ k : K L : List (Γ k) c₁ : TM1.Cfg Γ' Λ' σ h₁ : c₁ ∈ eval (TM1.step (tr M)) (TM1.init (trInit k L)) H₁ : Tape.right₀ c₁.Tape ∈ TM1.eval (tr M) (trInit k L) q✝ : Option Λ v✝ : σ S✝ : (k : K) → List (Γ k) L' : ListBlank ((k : K) → Option (Γ k)) hT : ∀ (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S✝ k))) h₂ : { l := q✝, var := v✝, stk := S✝ } ∈ eval (TM2.step M) (TM2.init k L) H₂ : TM2.Cfg.stk { l := q✝, var := v✝, stk := S✝ } k ∈ TM2.eval M k L h₃ : { l := Option.map normal q✝, var := v✝, Tape := Tape.mk' ∅ (addBottom L') } ∈ eval (TM1.step (tr M)) (TM1.init (trInit k L)) ⊢ ∃ S L', addBottom L' = Tape.right₀ c₁.Tape ∧ (∀ (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S k)))) ∧ S k = TM2.Cfg.stk { l := q✝, var := v✝, stk := S✝ } k cases Part.mem_unique h₁ h₃ case intro.intro.intro.intro.intro.intro.mk.refl K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ M : Λ → Stmt₂ k : K L : List (Γ k) q✝ : Option Λ v✝ : σ S✝ : (k : K) → List (Γ k) L' : ListBlank ((k : K) → Option (Γ k)) hT : ∀ (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S✝ k))) h₂ : { l := q✝, var := v✝, stk := S✝ } ∈ eval (TM2.step M) (TM2.init k L) H₂ : TM2.Cfg.stk { l := q✝, var := v✝, stk := S✝ } k ∈ TM2.eval M k L h₃ : { l := Option.map normal q✝, var := v✝, Tape := Tape.mk' ∅ (addBottom L') } ∈ eval (TM1.step (tr M)) (TM1.init (trInit k L)) h₁ : { l := Option.map normal q✝, var := v✝, Tape := Tape.mk' ∅ (addBottom L') } ∈ eval (TM1.step (tr M)) (TM1.init (trInit k L)) H₁ : Tape.right₀ { l := Option.map normal q✝, var := v✝, Tape := Tape.mk' ∅ (addBottom L') }.Tape ∈ TM1.eval (tr M) (trInit k L) ⊢ ∃ S L'_1, addBottom L'_1 = Tape.right₀ { l := Option.map normal q✝, var := v✝, Tape := Tape.mk' ∅ (addBottom L') }.Tape ∧ (∀ (k : K), ListBlank.map (proj k) L'_1 = ListBlank.mk (List.reverse (List.map some (S k)))) ∧ S k = TM2.Cfg.stk { l := q✝, var := v✝, stk := S✝ } k exact ⟨_, L', by simp only [Tape.mk'_right₀], hT, rfl⟩ K : Type u_1 inst✝² : DecidableEq K Γ : K → Type u_2 Λ : Type u_3 inst✝¹ : Inhabited Λ σ : Type u_4 inst✝ : Inhabited σ M : Λ → Stmt₂ k : K L : List (Γ k) q✝ : Option Λ v✝ : σ S✝ : (k : K) → List (Γ k) L' : ListBlank ((k : K) → Option (Γ k)) hT : ∀ (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.reverse (List.map some (S✝ k))) h₂ : { l := q✝, var := v✝, stk := S✝ } ∈ eval (TM2.step M) (TM2.init k L) H₂ : TM2.Cfg.stk { l := q✝, var := v✝, stk := S✝ } k ∈ TM2.eval M k L h₃ : { l := Option.map normal q✝, var := v✝, Tape := Tape.mk' ∅ (addBottom L') } ∈ eval (TM1.step (tr M)) (TM1.init (trInit k L)) h₁ : { l := Option.map normal q✝, var := v✝, Tape := Tape.mk' ∅ (addBottom L') } ∈ eval (TM1.step (tr M)) (TM1.init (trInit k L)) H₁ : Tape.right₀ { l := Option.map normal q✝, var := v✝, Tape := Tape.mk' ∅ (addBottom L') }.Tape ∈ TM1.eval (tr M) (trInit k L) ⊢ addBottom L' = Tape.right₀ { l := Option.map normal q✝, var := v✝, Tape := Tape.mk' ∅ (addBottom L') }.Tape simp only [Tape.mk'_right₀] ** Qed
borel_eq_top_of_countable α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝² : TopologicalSpace α inst✝¹ : T1Space α inst✝ : Countable α ⊢ borel α = ⊤ refine' top_le_iff.1 fun s _ => biUnion_of_singleton s ▸ _ α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝² : TopologicalSpace α inst✝¹ : T1Space α inst✝ : Countable α s : Set α x✝ : MeasurableSet s ⊢ MeasurableSet (⋃ x ∈ s, {x}) apply MeasurableSet.biUnion s.to_countable α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝² : TopologicalSpace α inst✝¹ : T1Space α inst✝ : Countable α s : Set α x✝ : MeasurableSet s ⊢ ∀ (b : α), b ∈ s → MeasurableSet {b} intro x _ α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝² : TopologicalSpace α inst✝¹ : T1Space α inst✝ : Countable α s : Set α x✝ : MeasurableSet s x : α a✝ : x ∈ s ⊢ MeasurableSet {x} apply MeasurableSet.of_compl case h α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝² : TopologicalSpace α inst✝¹ : T1Space α inst✝ : Countable α s : Set α x✝ : MeasurableSet s x : α a✝ : x ∈ s ⊢ MeasurableSet {x}ᶜ apply GenerateMeasurable.basic case h.a α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝² : TopologicalSpace α inst✝¹ : T1Space α inst✝ : Countable α s : Set α x✝ : MeasurableSet s x : α a✝ : x ∈ s ⊢ {x}ᶜ ∈ {s \| IsOpen s} exact isClosed_singleton.isOpen_compl ** Qed
borel_eq_generateFrom_Iio α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α ⊢ borel α = MeasurableSpace.generateFrom (range Iio) refine' le_antisymm _ (generateFrom_le _) case refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α ⊢ borel α ≤ MeasurableSpace.generateFrom (range Iio) rw [borel_eq_generateFrom_of_subbasis (@OrderTopology.topology_eq_generate_intervals α _ _ _)] case refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α ⊢ MeasurableSpace.generateFrom {s \| ∃ a, s = Ioi a ∨ s = Iio a} ≤ MeasurableSpace.generateFrom (range Iio) letI : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) case refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) ⊢ MeasurableSpace.generateFrom {s \| ∃ a, s = Ioi a ∨ s = Iio a} ≤ MeasurableSpace.generateFrom (range Iio) have H : ∀ a : α, MeasurableSet (Iio a) := fun a => GenerateMeasurable.basic _ ⟨_, rfl⟩ case refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) ⊢ MeasurableSpace.generateFrom {s \| ∃ a, s = Ioi a ∨ s = Iio a} ≤ MeasurableSpace.generateFrom (range Iio) refine' generateFrom_le _ case refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) ⊢ ∀ (t : Set α), t ∈ {s \| ∃ a, s = Ioi a ∨ s = Iio a} → MeasurableSet t rintro _ ⟨a, rfl \| rfl⟩ <;> [skip; apply H] case refine'_1.intro.inl α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a : α ⊢ MeasurableSet (Ioi a) by_cases h : ∃ a', ∀ b, a < b ↔ a' ≤ b case pos α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a : α h : ∃ a', ∀ (b : α), a < b ↔ a' ≤ b ⊢ MeasurableSet (Ioi a) rcases h with ⟨a', ha'⟩ case pos.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a a' : α ha' : ∀ (b : α), a < b ↔ a' ≤ b ⊢ MeasurableSet (Ioi a) rw [(_ : Ioi a = (Iio a')ᶜ)] α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a a' : α ha' : ∀ (b : α), a < b ↔ a' ≤ b ⊢ Ioi a = (Iio a')ᶜ simp [Set.ext_iff, ha'] case pos.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a a' : α ha' : ∀ (b : α), a < b ↔ a' ≤ b ⊢ MeasurableSet (Iio a')ᶜ exact (H _).compl case neg α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a : α h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b ⊢ MeasurableSet (Ioi a) rcases isOpen_iUnion_countable (fun a' : { a' : α // a < a' } => { b \| a'.1 < b }) fun a' => isOpen_lt' _ with ⟨v, ⟨hv⟩, vu⟩ case neg.intro.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a : α h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b v : Set { a' // a < a' } vu : ⋃ i ∈ v, {b \| ↑i < b} = ⋃ i, {b \| ↑i < b} hv : Encodable ↑v ⊢ MeasurableSet (Ioi a) simp [Set.ext_iff] at vu case neg.intro.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this✝ : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a : α h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b v : Set { a' // a < a' } hv : Encodable ↑v vu : ∀ (x : α), (∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 < x) ↔ ∃ a_1, a < a_1 ∧ a_1 < x this : Ioi a = ⋃ x, (Iio ↑↑x)ᶜ ⊢ MeasurableSet (Ioi a) rw [this] case neg.intro.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this✝ : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a : α h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b v : Set { a' // a < a' } hv : Encodable ↑v vu : ∀ (x : α), (∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 < x) ↔ ∃ a_1, a < a_1 ∧ a_1 < x this : Ioi a = ⋃ x, (Iio ↑↑x)ᶜ ⊢ MeasurableSet (⋃ x, (Iio ↑↑x)ᶜ) apply MeasurableSet.iUnion case neg.intro.intro.intro.h α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this✝ : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a : α h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b v : Set { a' // a < a' } hv : Encodable ↑v vu : ∀ (x : α), (∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 < x) ↔ ∃ a_1, a < a_1 ∧ a_1 < x this : Ioi a = ⋃ x, (Iio ↑↑x)ᶜ ⊢ ∀ (b : ↑v), MeasurableSet (Iio ↑↑b)ᶜ exact fun _ => (H _).compl α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a : α h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b v : Set { a' // a < a' } hv : Encodable ↑v vu : ∀ (x : α), (∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 < x) ↔ ∃ a_1, a < a_1 ∧ a_1 < x ⊢ Ioi a = ⋃ x, (Iio ↑↑x)ᶜ simp only [compl_Iio, iUnion_coe_set, Set.ext_iff, mem_Ioi, mem_iUnion, mem_Ici, exists_prop, Subtype.exists, exists_and_right] α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a : α h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b v : Set { a' // a < a' } hv : Encodable ↑v vu : ∀ (x : α), (∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 < x) ↔ ∃ a_1, a < a_1 ∧ a_1 < x ⊢ ∀ (x : α), a < x ↔ ∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 ≤ x refine' fun x => ⟨fun ax => _, fun ⟨a', ⟨h, _⟩, ax⟩ => lt_of_lt_of_le h ax⟩ α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a : α h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b v : Set { a' // a < a' } hv : Encodable ↑v vu : ∀ (x : α), (∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 < x) ↔ ∃ a_1, a < a_1 ∧ a_1 < x x : α ax : a < x ⊢ ∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 ≤ x rcases (vu x).2 (by refine' not_imp_comm.1 (fun h => _) h exact ⟨x, fun b => ⟨fun ab => le_of_not_lt fun h' => h ⟨b, ab, h'⟩, lt_of_lt_of_le ax⟩⟩) with ⟨a', h₁, h₂⟩ α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a : α h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b v : Set { a' // a < a' } hv : Encodable ↑v vu : ∀ (x : α), (∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 < x) ↔ ∃ a_1, a < a_1 ∧ a_1 < x x : α ax : a < x ⊢ ∃ a_1, a < a_1 ∧ a_1 < x refine' not_imp_comm.1 (fun h => _) h α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a : α h✝ : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b v : Set { a' // a < a' } hv : Encodable ↑v vu : ∀ (x : α), (∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 < x) ↔ ∃ a_1, a < a_1 ∧ a_1 < x x : α ax : a < x h : ¬∃ a_1, a < a_1 ∧ a_1 < x ⊢ ∃ a', ∀ (b : α), a < b ↔ a' ≤ b exact ⟨x, fun b => ⟨fun ab => le_of_not_lt fun h' => h ⟨b, ab, h'⟩, lt_of_lt_of_le ax⟩⟩ case intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) H : ∀ (a : α), MeasurableSet (Iio a) a : α h : ¬∃ a', ∀ (b : α), a < b ↔ a' ≤ b v : Set { a' // a < a' } hv : Encodable ↑v vu : ∀ (x : α), (∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 < x) ↔ ∃ a_1, a < a_1 ∧ a_1 < x x : α ax : a < x a' : α h₁ : ∃ x, { val := a', property := (_ : a < a') } ∈ v h₂ : a' < x ⊢ ∃ a_1, (∃ x, { val := a_1, property := (_ : a < a_1) } ∈ v) ∧ a_1 ≤ x exact ⟨a', h₁, le_of_lt h₂⟩ case refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α ⊢ ∀ (t : Set α), t ∈ range Iio → MeasurableSet t rw [forall_range_iff] case refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α ⊢ ∀ (i : α), MeasurableSet (Iio i) intro a case refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α a : α ⊢ MeasurableSet (Iio a) exact GenerateMeasurable.basic _ isOpen_Iio ** Qed
borel_eq_generateFrom_Ioi α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α ⊢ SecondCountableTopology α infer_instance ** Qed
borel_eq_generateFrom_Iic α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α ⊢ borel α = MeasurableSpace.generateFrom (range Iic) rw [borel_eq_generateFrom_Ioi] α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α ⊢ MeasurableSpace.generateFrom (range Ioi) = MeasurableSpace.generateFrom (range Iic) refine' le_antisymm _ _ case refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α ⊢ MeasurableSpace.generateFrom (range Ioi) ≤ MeasurableSpace.generateFrom (range Iic) refine' MeasurableSpace.generateFrom_le fun t ht => _ case refine'_1 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t✝ u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α t : Set α ht : t ∈ range Ioi ⊢ MeasurableSet t obtain ⟨u, rfl⟩ := ht case refine'_1.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u✝ : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α u : α ⊢ MeasurableSet (Ioi u) rw [← compl_Iic] case refine'_1.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u✝ : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α u : α ⊢ MeasurableSet (Iic u)ᶜ exact (MeasurableSpace.measurableSet_generateFrom (mem_range.mpr ⟨u, rfl⟩)).compl case refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α ⊢ MeasurableSpace.generateFrom (range Iic) ≤ MeasurableSpace.generateFrom (range Ioi) refine' MeasurableSpace.generateFrom_le fun t ht => _ case refine'_2 α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t✝ u : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α t : Set α ht : t ∈ range Iic ⊢ MeasurableSet t obtain ⟨u, rfl⟩ := ht case refine'_2.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u✝ : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α u : α ⊢ MeasurableSet (Iic u) rw [← compl_Ioi] case refine'_2.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u✝ : Set α inst✝³ : TopologicalSpace α inst✝² : SecondCountableTopology α inst✝¹ : LinearOrder α inst✝ : OrderTopology α u : α ⊢ MeasurableSet (Ioi u)ᶜ exact (MeasurableSpace.measurableSet_generateFrom (mem_range.mpr ⟨u, rfl⟩)).compl ** Qed
IsGδ.measurableSet α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ h : IsGδ s ⊢ MeasurableSet s rcases h with ⟨S, hSo, hSc, rfl⟩ case intro.intro.intro α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ S : Set (Set α) hSo : ∀ (t : Set α), t ∈ S → IsOpen t hSc : Set.Countable S ⊢ MeasurableSet (⋂₀ S) exact MeasurableSet.sInter hSc fun t ht => (hSo t ht).measurableSet ** Qed
measurable_of_isOpen α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ f : δ → γ hf : ∀ (s : Set γ), IsOpen s → MeasurableSet (f ⁻¹' s) ⊢ Measurable f exact measurable_generateFrom hf ** Qed
measurable_of_isClosed α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ f : δ → γ hf : ∀ (s : Set γ), IsClosed s → MeasurableSet (f ⁻¹' s) ⊢ Measurable f apply measurable_of_isOpen case hf α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ f : δ → γ hf : ∀ (s : Set γ), IsClosed s → MeasurableSet (f ⁻¹' s) ⊢ ∀ (s : Set γ), IsOpen s → MeasurableSet (f ⁻¹' s) intro s hs case hf α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ f : δ → γ hf : ∀ (s : Set γ), IsClosed s → MeasurableSet (f ⁻¹' s) s : Set γ hs : IsOpen s ⊢ MeasurableSet (f ⁻¹' s) rw [← MeasurableSet.compl_iff, ← preimage_compl] case hf α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ f : δ → γ hf : ∀ (s : Set γ), IsClosed s → MeasurableSet (f ⁻¹' s) s : Set γ hs : IsOpen s ⊢ MeasurableSet (f ⁻¹' sᶜ) apply hf case hf.a α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ f : δ → γ hf : ∀ (s : Set γ), IsClosed s → MeasurableSet (f ⁻¹' s) s : Set γ hs : IsOpen s ⊢ IsClosed sᶜ rw [isClosed_compl_iff] case hf.a α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ f : δ → γ hf : ∀ (s : Set γ), IsClosed s → MeasurableSet (f ⁻¹' s) s : Set γ hs : IsOpen s ⊢ IsOpen s exact hs ** Qed
measurable_of_is_closed' α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ f : δ → γ hf : ∀ (s : Set γ), IsClosed s → Set.Nonempty s → s ≠ univ → MeasurableSet (f ⁻¹' s) ⊢ Measurable f apply measurable_of_isClosed case hf α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ f : δ → γ hf : ∀ (s : Set γ), IsClosed s → Set.Nonempty s → s ≠ univ → MeasurableSet (f ⁻¹' s) ⊢ ∀ (s : Set γ), IsClosed s → MeasurableSet (f ⁻¹' s) intro s hs case hf α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ f : δ → γ hf : ∀ (s : Set γ), IsClosed s → Set.Nonempty s → s ≠ univ → MeasurableSet (f ⁻¹' s) s : Set γ hs : IsClosed s ⊢ MeasurableSet (f ⁻¹' s) cases' eq_empty_or_nonempty s with h1 h1 case hf.inr α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ f : δ → γ hf : ∀ (s : Set γ), IsClosed s → Set.Nonempty s → s ≠ univ → MeasurableSet (f ⁻¹' s) s : Set γ hs : IsClosed s h1 : Set.Nonempty s ⊢ MeasurableSet (f ⁻¹' s) by_cases h2 : s = univ case neg α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ f : δ → γ hf : ∀ (s : Set γ), IsClosed s → Set.Nonempty s → s ≠ univ → MeasurableSet (f ⁻¹' s) s : Set γ hs : IsClosed s h1 : Set.Nonempty s h2 : ¬s = univ ⊢ MeasurableSet (f ⁻¹' s) exact hf s hs h1 h2 case hf.inl α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ f : δ → γ hf : ∀ (s : Set γ), IsClosed s → Set.Nonempty s → s ≠ univ → MeasurableSet (f ⁻¹' s) s : Set γ hs : IsClosed s h1 : s = ∅ ⊢ MeasurableSet (f ⁻¹' s) simp [h1] case pos α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α inst✝¹² : TopologicalSpace α inst✝¹¹ : MeasurableSpace α inst✝¹⁰ : OpensMeasurableSpace α inst✝⁹ : TopologicalSpace β inst✝⁸ : MeasurableSpace β inst✝⁷ : OpensMeasurableSpace β inst✝⁶ : TopologicalSpace γ inst✝⁵ : MeasurableSpace γ inst✝⁴ : BorelSpace γ inst✝³ : TopologicalSpace γ₂ inst✝² : MeasurableSpace γ₂ inst✝¹ : BorelSpace γ₂ inst✝ : MeasurableSpace δ f : δ → γ hf : ∀ (s : Set γ), IsClosed s → Set.Nonempty s → s ≠ univ → MeasurableSet (f ⁻¹' s) s : Set γ hs : IsClosed s h1 : Set.Nonempty s h2 : s = univ ⊢ MeasurableSet (f ⁻¹' s) simp [h2] ** Qed
Set.OrdConnected.measurableSet α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u : Set α inst✝¹⁶ : TopologicalSpace α inst✝¹⁵ : MeasurableSpace α inst✝¹⁴ : OpensMeasurableSpace α inst✝¹³ : TopologicalSpace β inst✝¹² : MeasurableSpace β inst✝¹¹ : OpensMeasurableSpace β inst✝¹⁰ : TopologicalSpace γ inst✝⁹ : MeasurableSpace γ inst✝⁸ : BorelSpace γ inst✝⁷ : TopologicalSpace γ₂ inst✝⁶ : MeasurableSpace γ₂ inst✝⁵ : BorelSpace γ₂ inst✝⁴ : MeasurableSpace δ α' : Type u_6 inst✝³ : TopologicalSpace α' inst✝² : MeasurableSpace α' inst✝¹ : LinearOrder α inst✝ : OrderClosedTopology α a b x : α h : OrdConnected s ⊢ MeasurableSet s let u := ⋃ (x ∈ s) (y ∈ s), Ioo x y α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u✝ : Set α inst✝¹⁶ : TopologicalSpace α inst✝¹⁵ : MeasurableSpace α inst✝¹⁴ : OpensMeasurableSpace α inst✝¹³ : TopologicalSpace β inst✝¹² : MeasurableSpace β inst✝¹¹ : OpensMeasurableSpace β inst✝¹⁰ : TopologicalSpace γ inst✝⁹ : MeasurableSpace γ inst✝⁸ : BorelSpace γ inst✝⁷ : TopologicalSpace γ₂ inst✝⁶ : MeasurableSpace γ₂ inst✝⁵ : BorelSpace γ₂ inst✝⁴ : MeasurableSpace δ α' : Type u_6 inst✝³ : TopologicalSpace α' inst✝² : MeasurableSpace α' inst✝¹ : LinearOrder α inst✝ : OrderClosedTopology α a b x : α h : OrdConnected s u : Set α := ⋃ x ∈ s, ⋃ y ∈ s, Ioo x y ⊢ MeasurableSet s have huopen : IsOpen u := isOpen_biUnion fun _ _ => isOpen_biUnion fun _ _ => isOpen_Ioo α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u✝ : Set α inst✝¹⁶ : TopologicalSpace α inst✝¹⁵ : MeasurableSpace α inst✝¹⁴ : OpensMeasurableSpace α inst✝¹³ : TopologicalSpace β inst✝¹² : MeasurableSpace β inst✝¹¹ : OpensMeasurableSpace β inst✝¹⁰ : TopologicalSpace γ inst✝⁹ : MeasurableSpace γ inst✝⁸ : BorelSpace γ inst✝⁷ : TopologicalSpace γ₂ inst✝⁶ : MeasurableSpace γ₂ inst✝⁵ : BorelSpace γ₂ inst✝⁴ : MeasurableSpace δ α' : Type u_6 inst✝³ : TopologicalSpace α' inst✝² : MeasurableSpace α' inst✝¹ : LinearOrder α inst✝ : OrderClosedTopology α a b x : α h : OrdConnected s u : Set α := ⋃ x ∈ s, ⋃ y ∈ s, Ioo x y huopen : IsOpen u ⊢ MeasurableSet s have humeas : MeasurableSet u := huopen.measurableSet α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u✝ : Set α inst✝¹⁶ : TopologicalSpace α inst✝¹⁵ : MeasurableSpace α inst✝¹⁴ : OpensMeasurableSpace α inst✝¹³ : TopologicalSpace β inst✝¹² : MeasurableSpace β inst✝¹¹ : OpensMeasurableSpace β inst✝¹⁰ : TopologicalSpace γ inst✝⁹ : MeasurableSpace γ inst✝⁸ : BorelSpace γ inst✝⁷ : TopologicalSpace γ₂ inst✝⁶ : MeasurableSpace γ₂ inst✝⁵ : BorelSpace γ₂ inst✝⁴ : MeasurableSpace δ α' : Type u_6 inst✝³ : TopologicalSpace α' inst✝² : MeasurableSpace α' inst✝¹ : LinearOrder α inst✝ : OrderClosedTopology α a b x : α h : OrdConnected s u : Set α := ⋃ x ∈ s, ⋃ y ∈ s, Ioo x y huopen : IsOpen u humeas : MeasurableSet u ⊢ MeasurableSet s have hfinite : (s \ u).Finite := s.finite_diff_iUnion_Ioo α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u✝ : Set α inst✝¹⁶ : TopologicalSpace α inst✝¹⁵ : MeasurableSpace α inst✝¹⁴ : OpensMeasurableSpace α inst✝¹³ : TopologicalSpace β inst✝¹² : MeasurableSpace β inst✝¹¹ : OpensMeasurableSpace β inst✝¹⁰ : TopologicalSpace γ inst✝⁹ : MeasurableSpace γ inst✝⁸ : BorelSpace γ inst✝⁷ : TopologicalSpace γ₂ inst✝⁶ : MeasurableSpace γ₂ inst✝⁵ : BorelSpace γ₂ inst✝⁴ : MeasurableSpace δ α' : Type u_6 inst✝³ : TopologicalSpace α' inst✝² : MeasurableSpace α' inst✝¹ : LinearOrder α inst✝ : OrderClosedTopology α a b x : α h : OrdConnected s u : Set α := ⋃ x ∈ s, ⋃ y ∈ s, Ioo x y huopen : IsOpen u humeas : MeasurableSet u hfinite : Set.Finite (s \ u) ⊢ MeasurableSet s have : u ⊆ s := iUnion₂_subset fun x hx => iUnion₂_subset fun y hy => Ioo_subset_Icc_self.trans (h.out hx hy) α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u✝ : Set α inst✝¹⁶ : TopologicalSpace α inst✝¹⁵ : MeasurableSpace α inst✝¹⁴ : OpensMeasurableSpace α inst✝¹³ : TopologicalSpace β inst✝¹² : MeasurableSpace β inst✝¹¹ : OpensMeasurableSpace β inst✝¹⁰ : TopologicalSpace γ inst✝⁹ : MeasurableSpace γ inst✝⁸ : BorelSpace γ inst✝⁷ : TopologicalSpace γ₂ inst✝⁶ : MeasurableSpace γ₂ inst✝⁵ : BorelSpace γ₂ inst✝⁴ : MeasurableSpace δ α' : Type u_6 inst✝³ : TopologicalSpace α' inst✝² : MeasurableSpace α' inst✝¹ : LinearOrder α inst✝ : OrderClosedTopology α a b x : α h : OrdConnected s u : Set α := ⋃ x ∈ s, ⋃ y ∈ s, Ioo x y huopen : IsOpen u humeas : MeasurableSet u hfinite : Set.Finite (s \ u) this : u ⊆ s ⊢ MeasurableSet s rw [← union_diff_cancel this] α : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s t u✝ : Set α inst✝¹⁶ : TopologicalSpace α inst✝¹⁵ : MeasurableSpace α inst✝¹⁴ : OpensMeasurableSpace α inst✝¹³ : TopologicalSpace β inst✝¹² : MeasurableSpace β inst✝¹¹ : OpensMeasurableSpace β inst✝¹⁰ : TopologicalSpace γ inst✝⁹ : MeasurableSpace γ inst✝⁸ : BorelSpace γ inst✝⁷ : TopologicalSpace γ₂ inst✝⁶ : MeasurableSpace γ₂ inst✝⁵ : BorelSpace γ₂ inst✝⁴ : MeasurableSpace δ α' : Type u_6 inst✝³ : TopologicalSpace α' inst✝² : MeasurableSpace α' inst✝¹ : LinearOrder α inst✝ : OrderClosedTopology α a b x : α h : OrdConnected s u : Set α := ⋃ x ∈ s, ⋃ y ∈ s, Ioo x y huopen : IsOpen u humeas : MeasurableSet u hfinite : Set.Finite (s \ u) this : u ⊆ s ⊢ MeasurableSet (u ∪ s \ u) exact humeas.union hfinite.measurableSet ** Qed
generateFrom_Ico_mem_le_borel α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t✝ u : Set α✝ inst✝¹⁹ : TopologicalSpace α✝ inst✝¹⁸ : MeasurableSpace α✝ inst✝¹⁷ : OpensMeasurableSpace α✝ inst✝¹⁶ : TopologicalSpace β inst✝¹⁵ : MeasurableSpace β inst✝¹⁴ : OpensMeasurableSpace β inst✝¹³ : TopologicalSpace γ inst✝¹² : MeasurableSpace γ inst✝¹¹ : BorelSpace γ inst✝¹⁰ : TopologicalSpace γ₂ inst✝⁹ : MeasurableSpace γ₂ inst✝⁸ : BorelSpace γ₂ inst✝⁷ : MeasurableSpace δ α' : Type u_6 inst✝⁶ : TopologicalSpace α' inst✝⁵ : MeasurableSpace α' inst✝⁴ : LinearOrder α✝ inst✝³ : OrderClosedTopology α✝ a b x : α✝ α : Type u_7 inst✝² : TopologicalSpace α inst✝¹ : LinearOrder α inst✝ : OrderClosedTopology α s t : Set α ⊢ MeasurableSpace.generateFrom {S \| ∃ l, l ∈ s ∧ ∃ u, u ∈ t ∧ l < u ∧ Ico l u = S} ≤ borel α apply generateFrom_le case h α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t✝ u : Set α✝ inst✝¹⁹ : TopologicalSpace α✝ inst✝¹⁸ : MeasurableSpace α✝ inst✝¹⁷ : OpensMeasurableSpace α✝ inst✝¹⁶ : TopologicalSpace β inst✝¹⁵ : MeasurableSpace β inst✝¹⁴ : OpensMeasurableSpace β inst✝¹³ : TopologicalSpace γ inst✝¹² : MeasurableSpace γ inst✝¹¹ : BorelSpace γ inst✝¹⁰ : TopologicalSpace γ₂ inst✝⁹ : MeasurableSpace γ₂ inst✝⁸ : BorelSpace γ₂ inst✝⁷ : MeasurableSpace δ α' : Type u_6 inst✝⁶ : TopologicalSpace α' inst✝⁵ : MeasurableSpace α' inst✝⁴ : LinearOrder α✝ inst✝³ : OrderClosedTopology α✝ a b x : α✝ α : Type u_7 inst✝² : TopologicalSpace α inst✝¹ : LinearOrder α inst✝ : OrderClosedTopology α s t : Set α ⊢ ∀ (t_1 : Set α), t_1 ∈ {S \| ∃ l, l ∈ s ∧ ∃ u, u ∈ t ∧ l < u ∧ Ico l u = S} → MeasurableSet t_1 borelize α case h α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t✝ u : Set α✝ inst✝¹⁹ : TopologicalSpace α✝ inst✝¹⁸ : MeasurableSpace α✝ inst✝¹⁷ : OpensMeasurableSpace α✝ inst✝¹⁶ : TopologicalSpace β inst✝¹⁵ : MeasurableSpace β inst✝¹⁴ : OpensMeasurableSpace β inst✝¹³ : TopologicalSpace γ inst✝¹² : MeasurableSpace γ inst✝¹¹ : BorelSpace γ inst✝¹⁰ : TopologicalSpace γ₂ inst✝⁹ : MeasurableSpace γ₂ inst✝⁸ : BorelSpace γ₂ inst✝⁷ : MeasurableSpace δ α' : Type u_6 inst✝⁶ : TopologicalSpace α' inst✝⁵ : MeasurableSpace α' inst✝⁴ : LinearOrder α✝ inst✝³ : OrderClosedTopology α✝ a b x : α✝ α : Type u_7 inst✝² : TopologicalSpace α inst✝¹ : LinearOrder α inst✝ : OrderClosedTopology α s t : Set α this✝¹ : MeasurableSpace α := borel α this✝ : BorelSpace α ⊢ ∀ (t_1 : Set α), t_1 ∈ {S \| ∃ l, l ∈ s ∧ ∃ u, u ∈ t ∧ l < u ∧ Ico l u = S} → MeasurableSet t_1 rintro _ ⟨a, -, b, -, -, rfl⟩ case h.intro.intro.intro.intro.intro α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t✝ u : Set α✝ inst✝¹⁹ : TopologicalSpace α✝ inst✝¹⁸ : MeasurableSpace α✝ inst✝¹⁷ : OpensMeasurableSpace α✝ inst✝¹⁶ : TopologicalSpace β inst✝¹⁵ : MeasurableSpace β inst✝¹⁴ : OpensMeasurableSpace β inst✝¹³ : TopologicalSpace γ inst✝¹² : MeasurableSpace γ inst✝¹¹ : BorelSpace γ inst✝¹⁰ : TopologicalSpace γ₂ inst✝⁹ : MeasurableSpace γ₂ inst✝⁸ : BorelSpace γ₂ inst✝⁷ : MeasurableSpace δ α' : Type u_6 inst✝⁶ : TopologicalSpace α' inst✝⁵ : MeasurableSpace α' inst✝⁴ : LinearOrder α✝ inst✝³ : OrderClosedTopology α✝ a✝ b✝ x : α✝ α : Type u_7 inst✝² : TopologicalSpace α inst✝¹ : LinearOrder α inst✝ : OrderClosedTopology α s t : Set α this✝¹ : MeasurableSpace α := borel α this✝ : BorelSpace α a b : α ⊢ MeasurableSet (Ico a b) exact measurableSet_Ico ** Qed
Dense.borel_eq_generateFrom_Ico_mem α✝ : Type u_1 β : Type u_2 γ : Type u_3 γ₂ : Type u_4 δ : Type u_5 ι : Sort y s✝ t u : Set α✝ inst✝²² : TopologicalSpace α✝ inst✝²¹ : MeasurableSpace α✝ inst✝²⁰ : OpensMeasurableSpace α✝ inst✝¹⁹ : TopologicalSpace β inst✝¹⁸ : MeasurableSpace β inst✝¹⁷ : OpensMeasurableSpace β inst✝¹⁶ : TopologicalSpace γ inst✝¹⁵ : MeasurableSpace γ inst✝¹⁴ : BorelSpace γ inst✝¹³ : TopologicalSpace γ₂ inst✝¹² : MeasurableSpace γ₂ inst✝¹¹ : BorelSpace γ₂ inst✝¹⁰ : MeasurableSpace δ α' : Type u_6 inst✝⁹ : TopologicalSpace α' inst✝⁸ : MeasurableSpace α' inst✝⁷ : LinearOrder α✝ inst✝⁶ : OrderClosedTopology α✝ a b x : α✝ α : Type u_7 inst✝⁵ : TopologicalSpace α inst✝⁴ : LinearOrder α inst✝³ : OrderTopology α inst✝² : SecondCountableTopology α inst✝¹ : DenselyOrdered α inst✝ : NoMinOrder α s : Set α hd : Dense s ⊢ ∀ (x : α), IsBot x → x ∈ s simp ** Qed